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2009, VOLUME 4
PROGRESS IN PHYSICS
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The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics
PROGRESS IN PHYSICS
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OCTOBER 2009

VOLUME 4

CONTENTS
P.-M. Robitaille Kirchhoﬀ’s Law of Thermal Emission: 150 Years . . . . . . . . . . . . . . . . . . . . . 3 P.-M. Robitaille Blackbody Radiation and the Loss of Universality: Implications for
Planck’s Formulation and Boltzman’s Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 P.-M. Robitaille COBE: A Radiological Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 G. C. Vezzoli Active Galactic Nuclei: the Shape of Material Around Black Holes and
the Witch of Agnesi Function. Asymmetry of Neutrino Particle Density . . . . . . . . . . . 43 R. T. Cahill Combining NASA/JPL One-Way Optical-Fiber Light-Speed Data with
Spacecraft Earth-Flyby Doppler-Shift Data to Characterise 3-Space Flow. . . . . . . . . . 50 U. E. Bruchholz Geometry of Space-Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 U. E. Bruchholz Derivation of Planck’s Constant from Maxwell’s Electrodynamics . . . . . 67 S. N. Shapovalov, I. A. Rubinstein, O. A. Troshichev, and S. E. Shnoll Changes in the
Shape of Histograms Constructed from the Results of !'Pu Alpha-Activity Measurements Correlate with the Deviations of the Moon from the Keplerian Orbit . . . . . 68 E. N. Chifu Astrophysically Satisfactory Solutions to Einstein’s R-33 Gravitational Field Equations Exterior/Interior to Static Homogeneous Oblate Spheroidal Masses . . . . . 73 I. Suhendro A New Finslerian Uniﬁed Field Theory of Physical Interactions . . . . . . . . . . . 81 E. Comay Physical Consequences of Mathematical Principles . . . . . . . . . . . . . . . . . . . . . . . . . 91

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October, 2009

PROGRESS IN PHYSICS

Volume 4

Kirchhoﬀ’s Law of Thermal Emission: 150 Years

Pierre-Marie Robitaille
Department of Radiology, The Ohio State University, 395 W. 12th Ave, Suite 302, Columbus, Ohio 43210, USA
E-mail: robitaille.1@osu.edu
In this work, Kirchhoﬀ’s law (Kirchhoﬀ G. Monatsberichte der Akademie der Wissenschaften zu Berlin, sessions of Dec. 1859, 1860, 783–787) is being revisited not only to mark its 150th anniversary but, most importantly, to highlight serious overreaching in its formulation. At the onset, Kirchhoﬀ’s law correctly outlines the equivalence between emission and absorption for an opaque object under thermal equilibrium. This same conclusion had been established earlier by Balfour Stewart (Stewart B. Trans. Royal Soc. Edinburgh, 1858, v. 22(1), 1–20). However, Kirchhoﬀ extends the treatment beyond his counterpart, stating that cavity radiation must always be black, or normal: depending only on the temperature and the frequency of observation. This universal aspect of Kirchhoﬀ’s law is without proper basis and constitutes a grave distortion of experimental reality. It is readily apparent that cavities made from arbitrary materials (" < ) are never black. Their approach to such behavior is being driven either by the blackness of the detector, or by black materials placed near the cavity. Ample evidence exists that radiation in arbitrary cavities is sensitive to the relative position of the detectors. In order to fully address these issues, cavity radiation and the generalization of Kirchhoﬀ’s law are discussed. An example is then taken from electromagnetics, at microwave frequencies, to link results in the resonant cavity with those inferred from the consequences of generalization.

1 Introduction
Kirchhoﬀ’s law is one of the simplest and most misunderstood in thermodynamics [1, 2]. It is widely considered to be the ﬁrst of the laws of thermal emission [3–7]. In simple mathematical terms, Kirchhoﬀ’s law can take on several formulations, which stem from the equivalence between the
coeﬃcients of emission, ", and absorption, , at thermal equi-
librium. The most general expression of Kirchhoﬀ’s law for opaque objects is, in fact, a statement of Stewart’s law [6],
namely, " = 1 , where  corresponds to the coeﬃcient of
reﬂection. However, Kirchhoﬀ’s law [1, 2] is much farther reaching than Stewart’s [6], in requiring that radiation within an enclosure, or cavity, must always be black, or normal [5].
Kirchhoﬀ conceives that the ratio of emissive power, e, to absorptive power, a, of all bodies can be described by a universal function, f, common to all radiation within enclosures: e=a = f (T; ). Furthermore, this must be the case in a man-
ner which is independent of the nature and shape of the enclo-
sure, and which depends only on the temperature, T , of the system and the wavelength, , of observation [1, 2, 5, 7].
Kirchhoﬀ’s law constitutes an attempt to summarize the state of knowledge in radiative heat transfer during the mid1800’s. At the time, physicists created blackbodies from graphite plates, by lining the interior of cavities with soot, or by coating objects with black paint containing soot [8]. Contrary to Gustav Kirchhoﬀ [1, 2], Balfour Stewart, in 1858 [6], stated that radiation in thermal equilibrium depends on the constituents involved and his treatment did not lead to a universal function. If Kirchhoﬀ’s law can be expressed as

e=a = f (T; ), then Stewart’s would be e=a = f (T; ; N), where N represents all factors linked to the nature of the emitter itself and f  is not universal. Like Kirchhoﬀ, Stew-
art based his ideas on Pre´vost’s theory of exchanges [9, 10], which was ultimately linked to the study of radiation within enclosures. The distinctions between Stewart’s formulation and Kirchhoﬀ’s are profound [11, 12]. Kirchhoﬀ’s ideas advocate a universal function [5]. Stewart’s do not [6, 11, 12].
Today, 150 years after its formulation [1, 2], the foundation of Kirchhoﬀ’s law still rests on condensed matter physics. Blackbodies continue to be highly specialized objects [13–25] constructed from absorbers which are nearly perfect over the frequency range of interest. Yet, if Kirchhoﬀ was correct about the nature of radiation within cavities, it should be possible to assemble a blackbody from any mate-
rial. Surely, the presence of the universal function, f, dictates
that cavity radiation must always be black, or normal [5]. All that should be theoretically required is thermal equilibrium with the walls of an enclosure. The attributes of the walls, or its contents, should be inconsequential. However, the body of experimental knowledge, relative to the assembly of blackbodies in the laboratory, stands ﬁrmly opposed to this concept [13–25]. True blackbodies [13–25] are extremely diﬃcult to produce and testify against Kirchhoﬀ’s universal formulation [1, 2, 5]. Stewart’s law [6] alone, not Kirchhoﬀ’s [1, 2], is supported by a careful consideration of experimental reality [8, 12–41]. Still, a cursory review of the literature, relative to cavity emission, would suggest that arbitrary cavities can appear black. Furthermore, the trend towards blackness appears to increase as “truer” cavities are produced. This seems to

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be the case, irrespective of the emissivity of the cavity walls. The subject is a fascinating problem in physics.

2 Cavity radiation

While ideal blackbodies do not exist in nature, laboratory examples approach theoretical performances, especially when narrow frequency and temperature ranges are considered [8, 13–25]. Typically, the best laboratory blackbodies are con-
 structed from highly absorbing walls ( ) usually contain-
ing soot, carbon black, or graphite [8, 13–25]. Cavities which operate in the far infrared may also be lined with metals, metal blacks, or metal oxides [35–41]. Blackbody enclosures are often made isothermal using water, oil, or molten metal baths. Alternatively, metal freezing point techniques or electrical heating elements may ensure isothermal operation. The vast body of the laboratory evidence supports the idea that standard blackbodies are always made from highly absorbing materials set to function in an isothermal state.
Nonetheless, in treating cavity radiation from a theoretical standpoint, Planck invokes the perfectly reﬂecting enclosure [7, 8]. This is an interesting approach, since perfectly reﬂecting enclosures are adiabatic by deﬁnition and cannot therefore participate in the exchange of heat, either through emission or absorption. Planck, though, requires that the interior of such cavities contains black radiation [7; §51–52], in conformity with Kirchhoﬀ’s law [1, 2]. In so insisting, Planck makes constant recourse [8] to a minute particle of carbon [7; §51–52]. He inserts the particle into the cavity, in order to ensure that the latter appropriately holds black radiation. Planck invokes carbon, despite the fact that Kirchhoﬀ’s law should have ensured the presence of the radiation sought. In the end, and though carbon particles are perfect absorbers, Planck treats them simply as catalysts, and ignores their importance to the blackbody problem [7, 8].
It remains commonly acknowledged that all cavity radiation must be black. This is the case even though cavities with arbitrary walls of low emissivity are never used as laboratory blackbody standards [13–25]. Clearly, there is more to the understanding of arbitrary cavities than the belief that they are black [1, 2, 5]. In any case, when arbitrary cavities are analyzed with radiometric detectors, they do appear to become black, as seen in classic texts [i.e. 28] and the references they contain [29–34, 42–48]. Ample theoretical work reinforces this position [i.e. 42–48]. Monte Carlo calculations on lambertian spherical arbitrary cavities constructed from walls of low emissivity provide a good example [28]. Such calculations lead to apparent cavity emissivities approaching 1 [28]. These amazing results hint at proof, at least on the surface, that Kirchhoﬀ’s law is fully valid. Unfortunately, it can be shown that such conclusions are erroneous.
Let us return for a moment to Planck’s treatment [7] and the perfectly reﬂecting cavity containing a carbon particle [8]. A schematic representation of this situation is presented in

Fig. 1: Schematic representation of a perfectly reﬂecting cavity A) containing a carbon particle, B) with a carbon particle near the aperture, C) with a carbon particle farther from the aperture, and D) with the carbon particle replaced by a physical detector. The eye represents a point of detection. Note that if perfectly reﬂecting cavities contain any radiation whatsoever, it is solely because they have been ﬁlled with photons either from the carbon particle or the detector.
Figure 1A. Since the cavity wall is perfectly reﬂecting, one can treat it as an adiabatic boundary producing no radiation of its own. All of the radiation which comes to ﬁll the cavity is being produced by the carbon particle [12]. As a result, if one examines the contents of the cavity through a small hole, the radiation it contains will obviously be black. Now, let us displace the carbon particle, such that it is located just outside the aperture leading to the cavity (see Fig. 1B). From this position, the particle will once again be able to ﬁll the cavity with photons, and the observer will ﬁnd that its interior contains black radiation. Finally, let us place the carbon particle well outside the cavity itself, such that its radiation can still penetrate the cavity (see Fig. 1C). In this instance, the observer will record that the cavity is black, but not because it was able to become black on its own. It is black simply because the carbon particle has ﬁlled the cavity with radiation.
Returning to the days of Kirchhoﬀ, it is evident that limited experimental means existed. As a result, cavity radiation was monitored through a combination of prisms, for frequency diﬀerentiation, and thermometers, for energy detection. These thermometers were always blackened with soot, as Langley reminds us in 1888: “I may reply that we have lately found an admirable check on the eﬃciency of our optical devices in the behavior of that familiar substance lampblack, which all physicists use either on the thermometers, thermopiles, or bolometers” [49]. Consequently, by sampling the cavity with a thermometer coated with lampblack, every experimentalist brought about for himself the result which he sought. All cavities appeared black, because all cavities were being ﬁlled unintentionally with black radiation. Adding the carbon particle directly to the interior of the cavity simply helped to bring about the desired experimental scenario.
In Fig. 1D, a cavity is represented along with a radiometric detector. In order to maintain a logical progression, let us assume that the cavity is perfectly reﬂecting in its interior.

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In this case, the cavity itself cannot emit any photons [12]. A small hole is made into the cavity, and the radiation contained within it can be sampled with the radiometer. The cavity will be found to contain black radiation [12]. Yet, if the cavity was a perfect reﬂector, then how could its interior be black? The answer, of course, is similar to what Planck had done with the small carbon particle. A carbon particle, no matter how tiny [8, 12], will instantly ﬁll an experimental cavity with black radiation. Planck, in fact, relies on this reality [7; §51– 52]. Now, consider our radiometric detector. This instrument must have high photon capture rates. That is to say, it must possess an elevated absorptivity. As a result, by Stewart’s law [6], it must also possess a high emissivity. Thus, if the cavity appears black, it is only because it has been ﬁlled with black radiation by the detector. Again, the experimentalist inadvertently produced the expected result.
In order to more fully appreciate the role of the detector in generating black radiation within cavities, let us consider the classic works by De Vos [32, 33] and Ono [28, 34]. Even though he is addressing arbitrary cavities, De Vos emphasizes that: “The radiation emerging from the hole of observation in the blackbody should be an approximation, as well as possible, to the theoretical blackbody radiation” [32]. A cursory examination of these studies would lead one to believe that all arbitrary cavities are indeed black. However, upon closer analysis, these investigators have not distinguished themselves from their predecessors. De Vos elegantly links mathematical and experimental results obtained from cavities [32]. If the cavities appear black under certain viewing conditions, it is simply because black radiation has been injected into them using detectors. De Vos notes that in order to sample black radiation in a spherical cavity of arbitrary construction: “It is necessary to take care that the surface element observed is not perpendicular to the direction of observation” [32]. The reason for this statement is evident. If the surface element was perpendicular, most of the radiation introduced by the detector into the cavity would undergo normal specular reﬂection back out of the cavity and the latter would not appear black. In subsequently describing the tubular blackbody (see Figure 2A), De Vos states that: “The actual value of the quality will be better than calculated in this way but only slightly better since the radiant intensity decreases rapidly towards the ends of the tube” [32]. Of course, the detector is pumping radiation into the hole at the center of the tube. It is, therefore, simple to understand why radiation must fall rapidly towards the ends of the tube. Clearly, the tubular cavity is manifesting the performance of the detector. In fact, De Vos himself unintentionally makes the point: “Owing to the small hole in the tungsten tube a small quantity of energy was available only. Hence it was necessary to use radiation receivers of high sensitivity” [33]. De Vos might have more appropriately written that it was important for the detector to provide an ample supply of photons. For his part, Ono has demonstrated that the apparent emission of the tubular cav-

Fig. 2: A) Schematic representation of a tubular cavity and a detector. B) Illustration of the type of result seen with the detector as a function of angle from the normal. Note how there is less emission measured at  and ! .
ity depends on the position of the detector itself. Ono writes:
= “The apparent emissivity has deep minima around  0 at
which specularly reﬂected radiation escapes through the lateral hole. The shallow minima around 30 are also due to specular reﬂection eﬀects where incident radiation escapes after two successive specular reﬂections” [28, p. 605]. This situation is reproduced schematically in Fig. 2B. Of course, the incident radiation arises from the detector. It alone is ﬁlling the cavity with black radiation. The cavity itself is not producing this radiation for, if it did, the position of the detector would be immaterial. This is certain proof that Kirchhoﬀ’s law does not hold. Much depends on the detector, not on the cavity.
The point is further ampliﬁed by considering the work of Sparrow and Heinisch [30]. The authors demonstrate that the normal emission from a cylindrical cavity is absolutely dependent on the distance of the detector from the cavity. They fail to examine the cavity as a function of detector angle. Still, it is obvious that distance variations should not be occurring. Again, the detector is critically important in ﬂooding the cavity with radiation.
Vollmer’s studies [29] help us to understand that arbitrary cavities are not black, despite the fact that, at least on the surface, they point to the contrary. His work is particularly interesting, as it aims to reconcile theoretical foundations, stemming from Buckley’s classic paper [42], with experimental data. Surprising agreement is obtained between theory and experiment. In the limit, these results appear to re-emphasize that cylindrical cavities of suﬃcient size, made from arbitrary materials, will indeed behave as blackbodies. Everything seems to rest on solid footing, until the experimental setup is carefully examined. In order to reach agreement with theory, the apparatus used not only supplied the typical detector radiation, but also a black bellows, a black water cooled shutter, and a black water cooled cylinder [42]. Given these many possible sources of black radiation in front of the cavity opening, there can be little wonder that the cavity begins to appear black. In reality, the contrary position should have

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been adopted. How surprising that, bombarded with black radiation, some cavities still fail to be able to appear fully black.
R. E. Bedford, though he believes in the validity of Kirchhoﬀ’s law, re-emphasizes the point that arbitrary cavities are simply not black [28; p. 678]: “A blackbody is a lambertian emitter; with the exception of a spherical cavity, none of the blackbody simulators we will discuss will radiate directionally as does a blackbody”. Yet, as seen above for the spherical cavity, “It is necessary to take care that the surface element observed is not perpendicular to the direction of observation” [32]. Consequently, when these two excerpts are taken together, Bedford’s statement constitutes a direct refutation of Kirchhoﬀ’s law. The situation deteriorates further: “At some angle of view away from the normal to the cavity aperture (the angle depending on the particular cavity shape), the cavity radiance will begin to drop sharply from its axial value as that part of the wall becomes visible where =O near the aperture is much lower than =N deep within the cavity. In most cases this deﬁciency in emitted energy will be signiﬁcant only at angles of view larger than are subtended by most pyrometers” [28; p. 678]. In any event, the point is made. None of the cavities modeled can ever truly be considered blackbodies. Arbitrary materials are not lambertian and their emissivity can never be black [5]. Spherical cavities must be monitored with careful attention to the angle of observation. This should not occur if they were truly blackbodies.
If Monte Carlo simulations and other calculations reveal that arbitrary cavities move to blackness independent of wall emissivities, it is strictly because such methods ﬁll the cavities with black radiation [42–48]. Once again, blackbodies are unique in possessing lambertian surfaces. Thus, models which utilize lambertian surfaces of low emissivity represent situations which have no counterparts in nature. In addition, there can be no diﬀerence between placing a carbon particle in a cavity, in order to ensure the presence of black radiation, and simply ﬁlling the cavity with black radiation without physically making recourse to carbon. Monte Carlo simulations introduce black photons into cavities. Hence, they become black. The process is identical to placing a highly emitting carbon particle, or radiometer, at the opening of a cavity. No proof is provided by computational methods that arbitrary cavities contain black radiation.
It can be stated that Monte Carlo simulations obtain similar answers by modeling the repeated emission of photons directly from the cavity walls. In this case, computational analysis relies on internal reﬂection to arrive at a cavity ﬁlled with black radiation. The problem is that this scenario violates the ﬁrst law of thermodynamics and the conservation of energy. It is not mathematically possible to maintain an isothermal cavity while, at the same time, enabling its walls to lose a continual stream of photons. Such approaches build up the photon density in the cavity at the expense of wall cooling. These methods must therefore be forbidden on grounds that they violate the 1st law of thermodynamics.

Fig. 3: Schematic representations typically used to argue that cavity radiation is always black. Figure A is similar to Figure 6.1 in [50]. Figure B is similar to 5.6 in [51]. Note that ﬁgures illustrating immediate reﬂection back out of the cavity (C and D) are never invoked. This is precisely because they represent direct physical proof that arbitrary cavities are not black.
It is commonly argued [50, 51] that a cavity with a suﬃciently small hole contains black radiation. For example, in his classic text on the photosphere D. F. Gray writes: “Let us begin with a container that is completely closed except for a small hole in one wall. Any light entering the hole has a very small probability of ﬁnding its way out again, and eventually will be absorbed by the walls of the container or the gas inside the container. . . We have constructed a perfect absorber” [50; p. 100]. In reality, the maintenance of thermal equilibrium requires that if a photon enters the cavity, another photon must exit. The experimentalist will never be able to discern whether the exiting photon was 1) the same, 2) a photon that was newly emitted without reﬂection, 3) a photon that had previously undergone several reﬂections before exiting the cavity, or 4) a photon that had undergone a nearly inﬁnite number of internal reﬂections before exiting the cavity. Each of these cases corresponds to diﬀerent types of cavities, made either from arbitrary walls, perfectly absorbing walls, or perfectly reﬂecting walls. In any case, a photon must exit to maintain thermal equilibrium and nothing has been learned about the internal nature of the cavity. Clearly, given thermal equilibrium and the ﬁrst law of thermodynamics, we cannot be sure that the radiation inside the cavity was black. Such arguments [50; p. 100–101] are unsound a priori. Notice, for instance, the types of ﬁgures typically associated with such rhetoric: the photon is usually drawn such that normal and immediate specular reﬂection back out of the cavity is discounted (see Figure 3A–B). This is precisely because immediate specular reﬂection of the photon back out of the cavity provides a sound logical defeat of such arguments (see Figure 3C–D).
In summary, the radiation contained inside arbitrary cavities is not black and depends exclusively on 1) the nature of the cavity, and 2) the nature of the radiation which is permitted to enter. If excellent radiometers are used, they will be

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good emitters, and will act to ﬁll the cavities with black radi-
ation. As such, it seems logical, although counterintuitive,
that the sampling of cavity radiation should be performed
with suboptimal radiometers. Radiometers for these stud-
ies should not have high photon capture rates. Such devices
would provide lower photon emission towards the cavity. In
so doing, they would minimally alter the true nature of the ra-
diation they seek to measure. Perhaps, by using cryogenic de-
vices, it might be possible to build detectors which retain ad-
equate sensitivity. By maintaining lower detector emissions,
the true nature of radiation within cavities might be ascer-
tained. The proper result should echo Stewart, as previously
demonstrated mathematically [12].
3 The generalization of Kirchhoﬀ’s law
The proofs of Kirchhoﬀ’s law are usually limited to the realm
of geometrical optics. In his classic paper [2], Kirchhoﬀ states in a footnote: “The eﬀect of the diﬀraction of the rays by the edges of opening 1 is here neglected. This is allowable if openings 1 and 2, though inﬁnitely small in comparison with their distance apart, be considered as very great in comparison with the length of a wave.” Since Planck’s
treatment of Kirchhoﬀ’s law is also based on geometric optics, Planck writes: “Only the phenomena of diﬀraction, so far at least as they take place in space of considerable dimensions, we shall exclude on account of their rather complicated nature. We are therefore obliged to introduce right at the start a certain restriction with respect to the size of the parts of space to be considered. Throughout the following discussion it will be assumed that the linear dimensions of all parts of space considered, as well as the radii of curvature of all surfaces under consideration, are large compared to the wave lengths of the rays considered. With this assumption we may, without appreciable error, entirely neglect the inﬂuence of diﬀraction caused by the bounding surfaces, and everywhere apply the ordinary laws of reﬂection and refraction of light. To sum up: We distinguish once for all between two kinds of lengths of entirely diﬀerent orders of magnitudes — dimensions of bodies and wave lengths. Moreover, even the diﬀerentials of the former, i.e., elements of length, area and volume, will be regarded as large compared with the corresponding powers of wave lengths. The greater, therefore, the wave length of the rays we wish to consider, the larger must be the parts of space considered. But, inasmuch as there is no other restriction on our choice of size of the parts of space to be considered, this assumption will not give rise to any particular diﬃculty” [7; §2]. Kirchhoﬀ and Planck speciﬁcally
excluded diﬀraction. They do so as a matter of mathemati-
cal practicality. The problem of diﬀraction greatly increases
the mathematical challenges involved. As a result, Kirchhoﬀ
and Planck adapt a physical setting where its eﬀects could
be ignored. This is not a question of fundamental physical
limitation.

Nonetheless, the ﬁrst section of Kirchhoﬀ’s law, namely
the equivalence between the absorption and emission of en-
ergy by an opaque material at thermal equilibrium, has been
generalized to include diﬀraction. Correctly speaking, this
constitutes an extension of Stewart’s law, as will be discussed
below.
Much of the eﬀort in generalizing Kirchhoﬀ’s (Stewart’s)
law can be attributed to Sergi M. Rytov, the Russian physi-
cist. Indeed, it appears that eﬀorts to generalize Kirchhoﬀ’s
law were largely centered in Russia [52–55], but did receive
attention in the West [56, 57]. Though Rytov’s classic work
appears initially in Russian [52], later works have been trans-
lated into English [53]. In describing their theoretical results
relative to the generalization of Kirchhoﬀ’s law, Rytov and his associates [53; §3.5] write: “Equations (3.37-39) can be termed Kirchhoﬀ’s form of the FDT (ﬂuctuation-dissipation theorem), as they are a direct generalization of Kirchhoﬀ’s law in the classical theory of thermal radiation. This law is known to relate the intensity of the thermal radiation of a body in any direction to the absorption in that body when exposed to a plane wave propagating in the opposite direction. . . ” The authors continue: “and most important, (3.37– 39) contain no constraints on the relationships between the
wavelength  and characteristic scale l of the problem (the
size of the bodies, the curvature radii of their surfaces, the distances from the body to an observation point, etc.). In other words, unlike the classical theory of thermal radiation, which is bound by the constraints of geometrical optics, we can now calculate the second moments of the ﬂuctuational ﬁeld, that is to say both the wave part (taking into account all the diﬀraction phenomena), and the nonwave (quasistation-
ary) part for any  vs l ratio” [53; §3.5].
A discussion of the ﬂuctuation-dissipation theorem
(FTD), as it applies to thermal radiation, can also be found in
the book by Klyshko [54]. This text provides a detailed pre-
sentation of the generalization of Kirchhoﬀ’s law [54; §4.4
and 4.5]. Apresyan and Kravtsov also address generalization
in their work on radiative heat transfer [55]. They summarize the point as follows: “In this formulation, the Kirchhoﬀ statement — that the radiating and absorbing powers of a body are proportional to each other — as was initially derived in the limit of geometrical optics, is valid also for bodies with dimension below or about the wavelength” [55; p. 406].
It appears that the generalized form of Kirchhoﬀ’s law
has been adapted by the astrophysical community [57]. Like
the Russians before them, Linsky and Mount [56] assume
that the equality between emissivity and absorptivity at ther-
mal equilibrium is a suﬃcient statement of Kirchhoﬀ’s law
[1, 2]. They refer to a Generalized Kirchhoﬀ’s Law (GKL)
as E () = 1  (), where E () is the directional spectral emissivity and  () corresponds to the directional hemi-
spherical reﬂectivity [56]. This statement should properly be
referred to as Stewart’s law [6], since Stewart was the ﬁrst to
argue for the equality between the emissivity and absorptiv-

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ity of an opaque material under conditions of thermal equilibrium. Furthermore, Stewart’s law makes no claim that the radiation within opaque cavities must be black, or normal [5]. Seigel [11] speaks for physics when he outlines the important distinction between Stewart’s law [6] and Kirchhoﬀ’s [1, 2]. He writes: “Stewart’s conclusion was correspondingly restricted and did not embrace the sort of connection between the emissive and absorptive powers of diﬀerent materials, through a universal function of wavelength and temperature which Kirchhoﬀ established” [11; p. 584]. Herein, we ﬁnd the central diﬀerence between Stewart and Kirchhoﬀ. It is also the reason why Kirchhoﬀ’s law must be abandoned. In fact, since universality is not valid, there can be no more room for Kirchhoﬀ’s law in physics.
Returning to Rytov and his colleagues, following their presentation of the generalization of Kirchhoﬀ’s law [53; §3.5], they move rapidly to present a few examples of its use [53; §3.6] and even apply the treatment to the waveguide [53; §3.7]. Interestingly, though the authors fail to discuss the microwave cavity, from their treatment of the waveguide, it is certain that the radiation within the cavity cannot be black. It must depend on the dimensions of the cavity itself. Such a result is a direct conﬁrmation of Stewart’s ﬁndings [6], not Kirchhoﬀ’s [1, 2]. As a consequence, the generalization of Kirchhoﬀ’s law brings us to the conclusion that the radiation within cavities is not black, and the second portion of Kirchhoﬀ’s law is not valid.
These questions now extend to ultra high ﬁeld magnetic resonance imaging [58, 59], and hence the problem of radiation within cavities should be reexamined in the context of the generalization of Kirchhoﬀ’s law [52–55]. Since generalization extends to situations where cavity size is on the order of wavelength, it is appropriate to turn to this setting in magnetic resonance imaging. In fact, this constitutes a ﬁtting end to nearly 10 years of searching to understand why microwave cavities are not black, as required by Kirchhoﬀ’s law.
4 Cavity radiation in magnetic resonance imaging
Prior to treating the resonant microwave cavity, it is important to revisit Kirchhoﬀ’s claims. In his derivation, Kirchhoﬀ initially insists that his treatment is restricted to the study of heat radiation. He reminds the reader that: “All bodies emit rays, the quality and intensity of which depend on the nature and temperature of the body themselves” [2]. Then, he immediately eliminates all other types of radiation from consideration: “In addition to these, however, there may, under certain circumstances, be rays of other kinds, — as, for example, when a body is suﬃciently charged with electricity, or when it is phosphorescent or ﬂuorescent. Such cases are, however, here excluded” [2]. Kirchhoﬀ then proceeds to provide a mathematical proof for his law. Surprisingly, he then reintroduces ﬂuorescence. This is precisely to make the point that, within cavities, all radiation must be of a uni-

versal nature. Moreover, this occurs in a manner which is completely independent of the objects they contain, even if ﬂuorescent, or any other processes. Kirchhoﬀ writes: “The
= equation E=A e cannot generally be true of such a body,
but it is true if the body is enclosed in a black covering of the same temperature as itself, since the same considerations that led to the equation in question on the hypothesis that the body C was not ﬂuorescent, avail in this case even if the body C be supposed to be ﬂuorescent” [2]. Kirchhoﬀ deliberately invokes the all encompassing power of universality and its independence from all processes, provided enclosure is maintained.
Consequently, two important extensions exist. First, given the generalization of Kirchhoﬀ’s law [52–55], it is appropriate to extend these arguments to the microwave cavity. In this experimental setting, the wavelengths and the size of the object are on the same order. Furthermore, assuming thermal equilibrium, it is proper to consider steady state processes beyond thermal radiation. This is provided that a cavity be maintained. In any event, it is established that thermal loses exist within microwave devices. Thus, we can examine the electromagnetic resonant cavity in light of Kirchhoﬀ’s law.
When the use of the blackbody resonator in UHFMRI was advanced [60], it was not possible to reconcile the behavior of such a coil, given the conﬂict between Kirchhoﬀ’s law [1, 2] and the known performance of cavities in electromagnetics [61, 62]. A photograph of a sealed blackbody resonator for UHFMRI [60] is presented in Figure 4. In the simplest sense, this resonant cavity is an enclosure in which radiation can solely enter, or exit through, at a single drive point. The radiation within such cavities should be black, according to Kirchhoﬀ [1, 2]. Nonetheless, measurements of the real cavity show that it does not contain black radiation, as demonstrated experimentally in Figure 5. Resonant cavities are well known devices in electromagnetics [61, 62]. Their radiation is determined purely by the constituent properties of the cavity and its dimensions [61, 62]. This point is aﬃrmed in Figure 5. In its current form, Kirchhoﬀ’s law [1, 2] stands at odds against practical microwave techniques [61, 62]. Since this knowledge should not be discounted, something must be incorrect within Kirchhoﬀ’s law. Everything about the blackbody resonator presented in Figure 4 echoes Planck, yet the radiation it contains is not black [5]. The type of radiation within this cavity is being determined by electromagnetics [61, 62], not by Kirchhoﬀ’s law. Only the attributes of any substance present and that of the enclosed resonant elements, along with the size and shape of the enclosure itself, govern the type of radiation. For example, as seen in Figs. 4 and 5, the simple addition of echosorb acts to signiﬁcantly alter the resonances within such cavities. The associated loses are thermal. Of course, at these frequencies, echosorb is not a perfect absorber and the radiation inside the cavity cannot easily be made black. Still, in partial deference to Kirchhoﬀ, if a perfect absorber could be found, the radiation within cav-

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Fig. 4: A) End-view photograph of a sealed blackbody resonator [60] for use in UHFMRI studies. This device behaves as a resonant cavity [61, 62] and is constructed by sealing both ends of the wellknown TEM resonator [63, 64]. In this particular case, one of the ends of the resonator was made by sealing an acrylic ring with a thin copper sheet which was then re-enforced with copper tape on the inner and outer surfaces. All other assembly details are as previously reported [60]. When a resonator is sealed at both ends to make a cavity [61, 62], radiation can solely enter or leave the device through a single drive port. As such, the blackbody resonator can be regarded as the electromagnetic equivalent of Kirchhoﬀ’s blackbody [1, 2, 5, 7], with the important diﬀerence, of course, that the radiation inside such a device is never black. This constitutes a direct refutation of Kirchhoﬀ’s law of thermal emission as demonstrated experimentally in Fig. 5. B) Photograph of the interior of the blackbody coil illustrating the TEM rods, the interior lined with copper, and the drive point. Note that for these studies, a matching capacitor [60] was not utilized, as the measurement of interest does not depend on matching a given resonance to 50 ohms. It is the resonant nature of the coil itself which is of interest, not the impedance matching of an individual resonant frequency. C) Photograph of the blackbody coil ﬁlled with pieces of Echosorb. D) Photograph of the blackbody coil connected to an Agilent Technologies N5230C 300kHz – 6 GHz PNA-L Network Analyzer using an RG400 cable and SMA connectors. Since the RF coil was assembled with a BNC connector, an SMA/BNC adaptor was utilized to close the RF chain. The calibration of the analyzer was veriﬁed from 200–400 MHz using a matched load of 50 ohms placed directly on the network analyzer port. In this case, the return loss (S11) was less than 40 dB over the frequency range of interest. The matched load was also placed on the end of the test cable used for these studies and in this case the return loss (S11) was less than 25 dB from 200–400 MHz. The network analyzer provides a continuous steady state coherent source of radiation into the cavity. The coherence of this radiation is critical to the proper analysis of the returned radiation by the network analyzer. This does not alter the conclusions reached. Only the ability to properly monitor cavity behavior is aﬀected by the use of incoherent radiation. The cavity, of course, is indiﬀerent to whether or not the radiation incident upon it is coherent.

Fig. 5: Plot of the return loss (S11) for the blackbody coil (solid line) as measured from 200–400 MHz. Note that even though this cavity is completely closed, the radiation within this device is not black. Several sharp resonances are observed whose resonant position depend on the nature of the resonant cavity itself (dimension of the cavity, quality of the inner copper lining, dimensions of the TEM resonant elements, degree of insertion of the struts into the TEM elements, etc.). It is the presence of such resonances within cavities that forms the basis of practical electromagnetics and enables the use of resonant cavities in both EPR and MRI [61, 62]. If Kirchhoﬀ’s law of thermal emission had been correct, such a resonant device would not exist. The problem is easily rectiﬁed if one adopts Stewart’s formulation for the treatment of thermal emission [6]. The dashed line displays the return loss (S11) for the blackbody coil ﬁlled with the carbon-foam Echosorb as measured from 200– 400 MHz. Note that Echosorb is not a perfect absorber of radiation at these frequencies. But since this foam is somewhat absorbing, the resonance lines are broadened substantially. The return losses at several frequencies are lower, as is to be expected from the introduction of an absorbing object within a resonant cavity. If a perfect absorber could be found at these frequencies, the return loses would become extremely low across the entire frequency range of interest. Given these measurements and access to resonant devices, network analyzers and microwave technology, it is likely that Kirchhoﬀ would have reconsidered the formulation of his law of thermal emission.
ities containing such objects would be black. Nonetheless, only Stewart’s law [6] is formulated in such a way as to conform with results from electromagnetics [61, 62].
5 Conclusions
Tragically, if Kirchhoﬀ believed in universality, it was because he did not properly treat both reﬂection and absorption, as previously highlighted [12]. The correct treatment of radiation at thermal equilibrium was ﬁrst performed by Stewart, in 1858 [6]. Stewart properly addresses reﬂection [6, 8, 12], and does not arrive at universality. Unfortunately, Stewart’s formulation lacked mathematical rigor [6, 12] and this did not help in drafting a central law of thermal emission. At the same time, in deriving Kirchhoﬀ’s law in his treatise, Planck fails to fully treat reﬂection [7; §6]. Like Kirchhoﬀ

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his teacher, Planck is thereby lead erroneously to the con-

cept that all enclosures contain black radiation. Planck begins

his derivation of Kirchhoﬀ’s law by considering elements d

within an extended substance. He then analyzes the radiation

emitted by these elements, but ignores the coeﬃcient of re-

ﬂection,  from  to 

. He writes: “total energy in a
+ d emitted in the time dt in

range of frequency the direction of the

conical element d by a volume element d ” [7; §6] is equal

to dtd d d 2". As a result, he is brought to a universal

function, which is independent of the nature of the object,
and aﬃrms the validity of Kirchhoﬀ’s law: "=a = f (T; ).
In this equation, the coeﬃcient of emission, ", the coeﬃcient of absorbance, a, the temperature, T , and the frequency, ,

alone are considered. Had Planck properly addressed the co-

eﬃcient of reﬂection, , and recognized that the total ra-

diation which leaves an element is the sum produced by the

coeﬃcients of emission, ", and reﬂection, , he would have
obtained (" + )=(a + ) = f (T; ; N), where the na-
ture of the object, N , determined the relative magnitudes of

", a, and . By moving to the interior of an object and

neglecting reﬂection, Planck arrives at Kirchhoﬀ’s law, but

the consequence is that his derivation ignores the known truth

that opaque objects possess reﬂection.

Given thermal equilibrium, the equivalence between the

absorptivity, a, and emissivity, ", of an object was ﬁrst rec-

ognized by Stewart [6]. Stewart’s formulation preserves this

central equivalence. Only, it does not advance the univer-

sality invoked by Kirchhoﬀ [1, 2]. At the same time, it re-

mains fortunate for human medicine that Kirchhoﬀ’s law of

thermal emission does not hold. If it did, MRI within cav-

ities [60] would not be possible. Devices containing solely

black radiation would be of no use, either as microwave

components, or as antenna for human imaging. Physics and

medicine should return thereby, by necessity, to Stewart’s for-

mulation [6] and the realization that radiation within cavities

depends not uniquely on frequency and temperature, as stated

by Kirchhoﬀ [1, 2], but also on the attributes of the cavity it-

self and the materials it contains. This contribution was ﬁrst

brought to physics by Balfour Stewart [6]. Stewart’s law, not

Kirchhoﬀ’s, properly describes physical reality as observed

in the laboratory across all subdisciplines of physics and over

the entire span of the electromagnetic spectrum.

Practical blackbodies are always made from specialized

substances which are nearly perfect absorbers over the fre-

quency range of interest [13–25]. Accordingly, the nature

of the enclosure is important, in opposition to Kirchhoﬀ’s

law which claims independence from the properties of the

walls and its contents. Through the formulation of his law

of thermal emission, Balfour Stewart [6], unlike Kirchhoﬀ,

recognized the individualized behavior of materials in ther-

mal equilibrium. In addition, it is well-established that the

radiation within microwave cavities is not necessarily black.

Rather, it depends on the nature, shape, contents, and dimen-

sions of the enclosure itself. This is in accordance with Stew-

art’s law. Alternatively, if Kirchhoﬀ’s law was correct, cavities should strictly contain blackbody radiation and their use in radio and microwave circuitry would be pointless. Network analyzer measurements of return losses for a sealed enclosure, or blackbody resonator [60], from 200–400 MHz, conﬁrm that Kirchhoﬀ’s law of thermal emission does not hold within arbitrary resonant cavities.
At the same time, the physics community is justiﬁed in taking a cautious approach in these matters. After all, it was Planck [5] who provided the functional form contained in Kirchhoﬀ’s law [1, 2]. As a result, there is an understandable concern, that revisiting Kirchhoﬀ’s law will aﬀect the results of Planck himself and the foundation of quantum physics [5]. There is cause for concern. The loss of the universal function brings about substantial changes not only in astrophysics, but also in statistical thermodynamics.
Relative to Planck’s equation itself, the solution remains valid. It does however, become strictly limited to the problem of radiation within cavities which are known to be black (i.e. made of graphite, lined with soot, etc). Universality is lost. As for the mathematical value of Planck’s formulation for the perfectly absorbing cavity, it is preserved. In describing blackbody radiation, Planck consistently invokes the presence of a perfect absorber. In his treatise [7], he repeatedly calls for a minute particle of carbon [8]. Planck views this particle as a simple catalyst, although it can be readily demonstrated that this is not the case: the carbon particle acted as a perfect absorber [12]. As a result, I have stated that Kirchhoﬀ’s law is not universal [8, 12, 26, 27] and is restricted to the study of cavities which are either made from, or contain, perfect absorbers. Arbitrary cavity radiation is not black [12]. There can be no universal function. Planck’s equation presents a functional form which, far from being universal, is highly restricted to the emission of bodies, best represented on Earth by materials such as graphite, soot, and carbon black [8].
In closing, though 150 years have now elapsed since Kirchhoﬀ and Stewart dueled over the proper form of the law of thermal emission [11, 12], little progress has been made in bringing closure to this issue. Experimentalists continue to unknowingly pump black radiation into arbitrary cavities using their detectors. Theorists replicate the approach with Monte Carlo simulations. At the same time, astrophysicists apply with impunity the laws of thermal emission [1–7] to the stars and the universe. Little pause is given relative to the formulation of these laws [1–7] using condensed matter. The fact that all of electromagnetics stands in ﬁrm opposition to the universality, instilled in Kirchhoﬀ’s law, is easily dismissed as science unrelated to thermal emission [61, 62]. Losses in electromagnetics are usually thermal in origin. Nonetheless, electromagnetics is treated almost as an unrelated discipline. This occurs despite the reality that Kirchhoﬀ himself speciﬁcally included other processes, such as ﬂuorescence, provided enclosures were maintained. Though the generalization of Kirchhoﬀ’s law is widely recognized

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as valid [52–55], its application to the microwave cavity has been strangely omitted [52], even though it is used in treating the waveguide. This is the case, even though waveguides and cavities are often treated in the same chapters in texts on electromagnetics. All too frequently, the simple equivalence between apparent spectral absorbance and emission is viewed as a full statement of Kirchhoﬀ’s law [57, 65], adding further confusion to the problem. Kirchhoﬀ’s law must always be regarded as extending much beyond this equivalence. It states that the radiation within all true cavities made from arbitrary walls is black [1, 2]. The law of equivalence [57, 65] is Stewart’s [6].
Most troubling is the realization that the physical cause of blackbody radiation remains as elusive today as in the days of Kirchhoﬀ. Physicists speak of mathematics, of Planck’s equation, but nowhere is the physical mechanism mentioned. Planck’s frustration remains: “Therefore to attempt to draw conclusions concerning the special properties of the particles emitting rays from the elementary vibrations in the rays of the normal spectrum would be a hopeless undertaking” [7; §111]. In 1911, Einstein echoes Planck’s inability to link thermal radiation to a physical cause: “Anyway, the h-disease looks ever more hopeless” [66; p. 228]. Though he would be able to bring a ready derivation of Planck’s theorem using his coeﬃcients [67], Einstein would never be able to extract a proper physical link [68]. In reality, we are no closer to understanding the complexities of blackbody radiation than scientists were 150 years ago.
Acknowledgements
William F. Moulder from the Electrosciences Laboratory is recognized for measuring return losses speciﬁc to this experiment. Brief access to the Agilent Technologies network analyzer was provided by the Electrosciences Laboratory. Luc Robitaille is acknowledged for ﬁgure preparation.
Dedication
This work is dedicated to my eldest sister, Christine.
Submitted on May 27, 2009 / Accepted on May 29, 2008 First published online on June 19, 2009
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55. Apresyan L. A. and Kravtsov Y. A., Radiation transfer: statistical and wave aspects. Gordon and Breach Publishers, Australia, 1996.
56. Richter F., Florian M., Henneberger K. Generalized radiation law for excited media in a nonequilibrium steady state. Phys. Rev. B, 2008, 205114.
57. Linsky J. L. and Mount G. H. On the validity of a generalized Kirchhoﬀ’s law for a nonisothermal scattering and absorptive medium. Icarus, 1972, v. 17, 193–197.
58. Robitaille P. M. L., Abduljalil A.M., Kangarlu A., Zhang X., Yu Y., Burgess R., Bair S., Noa P., Yang L., Zhu H., Palmer B., Jiang Z.,Chakeres D.M., and Spigos D. Human magnetic resonance imaging at eight tesla. NMR Biomed., 1998, v. 11, 263– 265.
59. Robitaille P. M. L. and Berliner L.J. Ultra high ﬁeld magnetic resonance imaging. Springer, New York, 2006.
60. Robitaille P. M. L. Black-body and transverse electromagnetic (TEM) resonators operating at 340 MHz: volume RF coils for UHFMRI. J. Comp. Assist. Tomogr., 1999, v. 23, 879–890.
61. Pozar D. M. Microwave engineering. John Wiley and Sons, New York, 1998.
62. Argence E. Theory of waveguides and cavity resonators. Hart Pub. Co., Oxford, 1968.
63. Roschmann P. K. High-frequency coil system for a magnetic resonance imaging apparatus. US Patent, 1988, no. 4,746,866.
64. Vaughn J. T., Hetherington H., Otu J., Pan J., Pohost G. High frequency volume coils for clinical NMR imaging and spectroscopy. Magn. Reson. Med., 1994, v. 32, 206–218.
65. Kelley F. J. On Kirchhoﬀ’s law and its generalized application to absorption and emission by cavities. J. Res. Nat. Bur. Stand. B — (Math. Math. Phys.), 1965, v. 69B(3), 165–171.
66. Einstein A. The collected papers of Albert Einstein, v. 5. The Swiss years: Correspondence 1902–1914. Princeton University Press, Princeton, N.J., 1995, p. 105.
67. Einstein A. Strahlungs-emission und absorption nach der quantentheorie. Verhandlunger der Deutschen Physikalischen Gesellschaft, 1916, v. 18, 318–323; Einstein A. Phys. Zs., 1917, v. 18, 121 (English translation by ter Haar D.: Einstein A. On the quantum theory of radiation. The old quantum theory, Pergamon Press, 1967, 167–183).
68. Robitaille P. M. L. Comment to the NRC committee on condensed matter and material physics. January 20, 2005. http:// www7.nationalacademies.org/bpa/CMMP2010 Robitaille.pdf

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Blackbody Radiation and the Loss of Universality: Implications for Planck’s Formulation and Boltzman’s Constant

Pierre-Marie Robitaille
Department of Radiology, The Ohio State University, 395 W. 12th Ave, Suite 302, Columbus, Ohio 43210, USA

E-mail: robitaille.1@osu.edu

Through the reevaluation of Kirchhoﬀ’s law (Robitaille P. M. L. IEEE Trans. Plasma Sci., 2003, v. 31(6), 1263–1267), Planck’s blackbody equation (Planck M. Ann. der Physik, 1901, v. 4, 553–356) loses its universal signiﬁcance and becomes restricted to

perfect absorbers. Consequently, the proper application of Planck’s radiation law in-

volves the study of solid opaque objects, typically made from graphite, soot, and carbon

black. The extension of this equation to other materials may yield apparent tempera-

tures, which do not have any physical meaning relative to the usual temperature scales.

Real temperatures are exclusively obtained from objects which are known solids, or

which are enclosed within, or in equilibrium with, a perfect absorber. For this reason,

the currently accepted temperature of the microwave background must be viewed as

an apparent temperature. Rectifying this situation, while respecting real temperatures,

involves a reexamination of Boltzman’s constant. In so doing, the latter is deprived of

its universal nature and, in fact, acts as a temperature dependent variable. In its revised

form, Planck’s equation becomes temperature insensitive near 300 K, when applied to

the microwave background.
With the formulation of his law of thermal emission [1], Planck brought to science a long sought physical order. Though individual materials varied widely in their radiative behaviors, Kirchhoﬀ’s law of thermal emission [2, 3] had enabled him to advance dramatic simpliﬁcations in an otherwise chaotic world [1]. Given thermal equilibrium and enclosure, the blackbody cavity seemed to impart upon nature a universal property, far removed from the confusion prevailing outside its walls [4]. Universality produced conceptual order and brought rapid and dramatic progress in mathematical physics.
In his “Theory of Heat Radiation” [4], Planck outlines the
prize: the existence of the universal constants, h and k. More-

F

F mass ch=f = 5:37 10 # g;

F time fh=c# = 1:33 10 "! s;

1
temperature k

c#h=f = 3:60 10! degree:

Planck continues: “These quantities retain their natural signiﬁcance as long as the law of gravitation and that of the propagation of light in a vacuum and the two principles of thermodynamics remain valid; they therefore must be found always the same, when measured by the most widely diﬀering intelligences according to the most widely diﬀering methods”
[4; §164].

over, he is able to introduce natural units of length, mass,
time, and temperature [4; §164]. He writes: “In contrast with
this it might be of interest to note that, with the aid of the two
constants h and k which appear in the universal law of radia-
tion, we have the means of establishing units of length, mass,

The real triumph of Planck’s equation [1] rested not solely
on solving the blackbody problem, but rather on the univer-
sal nature of h and k. The four fundamental units of scale
for time, length, mass, and temperature profoundly altered
physics. It is in this light, that concern over any fundamen-

time, and temperature, which are independent of special bod- tal change in Kirchhoﬀ’s law [2, 3] and Planck’s equation [1]

ies or substances, which necessarily retain their signiﬁcance must be viewed.

for all time and for all environments, terrestrial and human

The notion that the microwave background [5] is being

or otherwise, and which may, therefore, be described as ‘nat- produced directly by the oceans of the Earth [6–9], brings

ural units’ ” [4; §164]. Planck then presents the values of the with it an immediate realization that universality is lost, and

four fundamental constants [4; §164]:

Kirchhoﬀ’s law is invalid [10–14]. Blackbody radiation is not

Planck’s constant h = 6:415 10 % g cm =sec; Boltzman’s constant k = 1:34 10 $ g cm =sec degree; the speed of light c = 3:10 10 cm=sec; the gravitational constant f = 6:685 10 & cm!=g sec :

a universal process [10–14], as Planck so adamantly advocated [4]. Yet, if the microwave background truly arises from oceanic emissions [5–8], then it is not simple to reconcile a
 temperature at 3 K with a source known to have a physical  temperature of 300 K [10]. Let us examine more closely the
problem at hand, by considering Planck’s formulation (1):

F Finally, he reveals basic units of: length fh=c! = 3:99 10 !! cm;

"


=

2h! c

1
eh=kT

:
1

(1)

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In order to properly ﬁt the microwave background using
this equation, the problem rests in the k T term. It is possible,
for instance, to make that assumption that an apparent temperature exists [10] and to keep the meaning of Boltzman’s constant. In fact, this was the course of action initially proposed [10]. In this way, nothing was lost from the universal
nature of h and k [10]. But, upon further consideration, it
is clear that such an approach removes all physical meaning from temperature itself. The one alternative is to alter Boltzman’s constant directly, and accept the full consequences of the loss of universality. The issue involves a fundamental understanding of how energy is distributed within matter. For the microwave background, this must focus on water [8].
Thus, let us consider a very primitive description of how energy enters, or becomes distributed, within water [8]. Water possesses many degrees of freedom and must be viewed as a complex system. At low temperatures, some of the ﬁrst degrees of freedom to be fully occupied will be associated with the weak intermolecular hydrogen bond (H O HOH) [8]. These involve both stretching and bending processes, resulting in several vibrational-rotational modes. The hydrogen bond (H O HOH) has been advanced as responsible for the microwave background [8], particularly as a result of its predicted bond strength. As energy continues to enter the water system, it will start to populate other degrees of freedom, including those associated with the direct translation and rotation of individual molecules. This is in sharp contrast to graphite, for instance, because the latter never undergoes a solid-liquid phase transition [15]. Eventually, other degrees of freedom, associated with the vibrational and bending modes of the intramolecular hydroxyl bonds (H–OH) themselves, will become increasingly populated. Hydrogen bonds (H O HOH) have bond strengths which are on the order of 100 times lower than hydroxyl bonds (H–OH) [8]. Considering these complexities, it is unreasonable to believe that energy will enter the water system in a manner which ignores the existence of these degrees of freedom, particularly those associated with the liquid state.
Contrary to what Kirchhoﬀ and Planck require for universality [1–3], these complex issues extend throughout nature. Each material is unique relative to the degrees of freedom it has available as a function of temperature [15]. Water possesses two distinct oscillators, the intermolecular hydrogen bond (H O HOH) and the intramolecular hydroxyl bond (H–OH) [7]. These two oscillatory systems have very distinct energies [8] and provide a situation which is quite removed from graphite. Kirchhoﬀ and Planck had no means of anticipating such complexity. In fact, they were relatively unaware of the tremendous atomic variability found at the level of the lattice. As such, it is somewhat understandable that they might seek universal solutions.
In any case, it has been amply demonstrated that Kirchhoﬀ’s law is not valid [10–14]. There can be no universality. In addition, it is extremely likely that the microwave back-

ground is being produced by thermal photons emitted directly

from the oceanic surface and then scattered in the Earth’s at-
 mosphere [6]. This implies that a 300 K source is able to

behave, at least over a region of the electromagnetic spec-
 trum, as a 3 K source. However, since the oceans are not at 3 K, an inconsistency has been revealed in the determina-

tion of temperatures using the laws of thermal emission. The

problem stems from the weakness of the hydrogen bond and

the associated ease with which water enters the liquid state.

Furthermore, it is evident that energy can enter the water sys-

tem and be directed into its translational degrees of freedom,

thereby becoming unavailable for thermal emission. This is a

signiﬁcant problem, which Kirchhoﬀ and Planck did not need

to consider, and of which they were unaware, when treating

graphite boxes [1–4, 10]. Graphite, unlike water, cannot sup-

port convection.
 In any event, the central issue remains that a 3 K tem perature has been obtained from a 300 K source. As men-

tioned above, it is possible to essentially ignore the conse-

quences of this ﬁnding by simply treating the microwave

background as an apparent temperature [10], devoid of physi-

cal meaning. In this way, Planck’s equation and the universal

constants, survive quite nicely [10]. Conversely, if one re-

fuses to abandon the real temperature scale, then a problem

arises. In order to properly ﬁt the microwave background
 with Planck’s equation and a real temperature at 300 K,

then Boltzman’s constant must come a temperature dependent

cvhaarniagbel.e, Inkf(aTc)t,.

it must beThis vari-

able must behave such that when it is multiplied by a range

of temperatures near value independent of

3te0m0pKer,aitturrees(uklts(Tin)

a perfectly constant
T = P , where P is

a constant). Planck’s equation thereby becomes completely

insensitive to temperature ﬂuctuations over the temperature

and frequency ranges of interest, as seen in Eq. (2):

"


=

2h! c

1
eh=P

:
1

(2)

As a result, relative to the microwave background, we

vmaorivaebflero, mk(aTu)n,ivwehriscahl

constant, k, to
acts to render

a temperature sensitive Planck’s equation tem-

perature insensitive. The modern value of the constant, P , for

the microwave background, is approximately 3.762 10 $

ergs. The move away from graphite, into another Planckian

system, has resulted in a profound re-evaluation of the science

of thermodynamics. Boltzman’s constant, therefore, remains

valid only for graphite, soot, or carbon black, and those ma-

terials approaching their performance at a given frequency.

Outside a certain range of temperatures, or frequencies, or

materials, then other constants and/or variables, which are

material speciﬁc, exist. The measure of how much energy

a system can hold at a given temperature, or how temperature

changes as a function of energy, is directly determined by the

makeup of the system itself. The ﬂow of heat within a system

depends on all of the degrees of freedom which eventually

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become available [15]. In this regard, phase transitions bring with them additional degrees of freedom, either translational or rotational, which are simply not available to the solid state [15]. Herein is found the central reason for the loss of universality: phase transitions exist. Nothing is universal, since phase transitions and any available degrees of freedom [15] are strictly dependent on the nature of matter. Hence, each material must be treated on its own accord. This is the primary lesson of the water/microwave background ﬁndings.
Physics cannot maintain a proper understanding of temperature without abandoning the universal attributes of Boltzman’s constant. Otherwise, the temperature scale itself loses meaning. In order to speciﬁcally address the microwave background, Boltzman’s constant, in fact, can become a temperature dependent variable. At the same time, since many materials contain covalent bonds with bond strengths near those found within graphite, it is likely that many material speciﬁc constants will, in fact, approach Boltzman’s. Nonetheless, relative to the microwave background, a temperature dependent variable exists which acts to completely remove all temperature sensitivity from Planck’s equation at earthly temperatures. This explains why Penzias and Wilson [5] ﬁrst reported that the microwave background was devoid of seasonal variations.
As regards to Planck’s constant, and the fundamental units of time, mass, and length, they appear to remain unaltered by the ﬁndings prompted by the microwave background. Perhaps they will be able to retain their universal meaning. However, a careful analysis of individual physical processes is in order, such that the consequences of the loss of universality can be fully understood.
Dedication
This work is dedicated to my wife, Patricia Anne, and to our sons, Jacob, Christophe, and Luc.
Submitted on May 28, 2009 / Accepted on June 02, 2008 First published online on June 20, 2009

translation by ter Haar D.: Planck M. On the theory of the energy distribution law in the normal spectrum. The old quantum theory. Pergamon Press, 1967, 82–90; also Planck’s December 14, 1900 lecture Zur Theorie des Gesetzes der Energieverteilung in Normalspectrum, which stems from this paper, can be found in either German, or English, in: Kangro H. Classic papers in physics: Planck’s original papers in quantum physics. Taylor & Francis, London, 1972, 6–14 or 38–45).
4. Planck M. The theory of heat radiation. P. Blakiston’s Son & Co., Philadelphia, PA, 1914.
5. Penzias A. A. and Wilson R. W. A measurement of excess antenna temperature at 4080 Mc/s. Astrophys. J., 1965, v. 1, 419– 421.
6. Robitaille P. M. L. The Earth microwave background (EMB), atmospheric scattering and the generation of isotropy. Progr. in Phys., 2008, v. 2, L7–L8.
7. Rabounski D. The relativistic eﬀect of the deviation between the CMB temperatures obtained by the COBE satellite. Progr. in Phys., 2007, v. 1, 24–26.
8. Robitaille P. M. L. Water, hydrogen bonding, and the microwave background. Prog. in Phys., 2009, v. 2, L5–L7.
9. Rabounski D. and Borissova L. On the earthly origin of the Penzias-Wilson microwave background. Prog. in Phys., 2009, v. 2, L1–L4.
10. Robitaille P. M. L. Blackbody radiation and the carbon particle. Prog. in Phys., 2008, v. 3, 36–55.
11. Robitaille P. M. L. On the validity of Kirchhoﬀ’s law of thermal emission. IEEE Trans. Plasma Sci., 2003, v. 31(6), 1263–1267.
12. Robitaille P. M. L. An analysis of universality in blackbody radiation. Prog. in Phys., 2006, v. 2, 22–23; arXiv: physics/ 0507007.
13. Robitaille P. M. L. A critical analysis of universality and Kirchhoﬀ’s law: a return to Stewart’s law of thermal emission. Prog. in Phys. 2008, v. 3, 30–35; arXiv: 0805.1625.
14. Robitaille P. M. L. Kirchhoﬀ’s law of thermal emission: 150 years. Prog. in Phys., 2009, v. 4, 3–13.
15. Robitaille P. M. L. The little heat engine: Heat transfer in solids, liquids, and gases. Prog. in Phys., 2007, v. 4, 25–33.

References
1. Kirchhoﬀ G. U¨ ber den Zusammenhang zwischen Emission und Absorption von Licht und Wa¨rme. Monatsberichte der Akademie der Wissenschaften zu Berlin, sessions of Dec. 1859, 1860, 783–787.
2. Kirchhoﬀ G. U¨ ber das Verha¨ltnis zwischen dem Emissionsvero¨gen und dem Absorptionsvermo¨gen. der Ko¨rper fu¨r Wa¨rme und Licht. Poggendorfs Annalen der Physik und Chemie, 1860, v. 109, 275–301. (English translation by F. Guthrie: Kirchhoﬀ G. On the relation between the radiating and the absorbing powers of diﬀerent bodies for light and heat. Phil. Mag., 1860, ser. 4, v. 20, 1–21).
3. Planck M. U¨ ber das Gesetz der Energieverteilung im Normalspektrum. Annalen der Physik, 1901, v. 4, 553–563. (English

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COBE: A Radiological Analysis
Pierre-Marie Robitaille
Department of Radiology, The Ohio State University, 395 W. 12th Ave, Suite 302, Columbus, Ohio 43210, USA
E-mail: robitaille.1@osu.edu
 The COBE Far Infrared Absolute Spectrophotometer (FIRAS) operated from 30 to    3,000 GHz (1–95 cm ) and monitored, from polar orbit ( 900 km), the 3 K mi-
crowave background. Data released from FIRAS has been met with nearly universal admiration. However, a thorough review of the literature reveals signiﬁcant problems with this instrument. FIRAS was designed to function as a diﬀerential radiometer, wherein the sky signal could be nulled by the reference horn, Ical. The null point occurred at an Ical temperature of 2.759 K. This was 34 mK above the reported sky temperature, 2.725 0.001 K, a value where the null should ideally have formed. In addition, an 18 mK error existed between the thermometers in Ical, along with a drift in temper-
 ature of 3 mK. A 5 mK error could be attributed to Xcal; while a 4 mK error was
found in the frequency scale. A direct treatment of all these systematic errors would
 lead to a 64 mK error bar in the microwave background temperature. The FIRAS  team reported 1 mK, despite the presence of such systematic errors. But a 1 mK er-
ror does not properly reﬂect the experimental state of this spectrophotometer. In the end, all errors were essentially transferred into the calibration ﬁles, giving the appearance of better performance than actually obtained. The use of calibration procedures resulted in calculated Ical emissivities exceeding 1.3 at the higher frequencies, whereas an emissivity of 1 constitutes the theoretical limit. While data from 30–60 GHz was once presented, these critical points are later dropped, without appropriate discussion, presumably because they reﬂect too much microwave power. Data obtained while the Earth was directly illuminating the sky antenna, was also discarded. From 300–660 GHz, initial FIRAS data had systematically growing residuals as frequencies increased. This suggested that the signal was falling too quickly in the Wien region of the spectrum. In later data releases, the residual errors no longer displayed such trends, as the systematic variations had now been absorbed in the calibration ﬁles. The FIRAS team also cited insuﬃcient bolometer sensitivity, primarily attributed to detector noise, from 600–3,000 GHz. The FIRAS optical transfer function demonstrates that the instrument was not optimally functional beyond 1,200 GHz. The FIRAS team did not adequately characterize the FIRAS horn. Established practical antenna techniques strongly suggest that such a device cannot operate correctly over the frequency range proposed. Insuﬃcient measurements were conducted on the ground to document antenna gain and ﬁeld patterns as a full function of frequency and thereby determine performance. The effects of signal diﬀraction into FIRAS, while considering the Sun/Earth/RF shield, were neither measured nor appropriately computed. Attempts to establish antenna side lobe performance in space, at 1,500 GHz, are well outside the frequency range of interest
< for the microwave background ( 600 GHz). Neglecting to fully evaluate FIRAS prior
to the mission, the FIRAS team attempts to do so, on the ground, in highly limited fashion, with a duplicate Xcal, nearly 10 years after launch. All of these ﬁndings indicate that the satellite was not suﬃciently tested and could be detecting signals from our planet. Diﬀraction of earthly signals into the FIRAS horn could explain the spectral frequency dependence ﬁrst observed by the FIRAS team: namely, too much signal in the Jeans-Rayleigh region and not enough in the Wien region. Despite popular belief to the contrary, COBE has not proven that the microwave background originates from the universe and represents the remnants of creation.

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Fig. 1: Schematic representation of the COBE FIRAS instrument reproduced from [38]. The spectrometer is based on an interferometer design wherein the signal from the sky horn is being compared with that provided by the reference horn. Each of the input signals is split by grid polarizers, reﬂected by mirrors, and sent down the arms of the interferometer. Two output ports receive the resultant signal. An internal calibrator, Ical, equipped with two germanium resistance thermometers (GRT), provides signal to the reference horn. During calibration, the external calibrator, Xcal, is inserted into the sky horn. Xcal is monitored by three GRTs. The interferometer assembly includes a single mirror transport mechanism (MTM). Speciﬁc details can be found in [38]. No knowledge about the functioning of FIRAS, beyond that contained in this ﬁgure legend, is required to follow this work. The central elements are simply that FIRAS is made up of a sky horn, a reference horn, Ical (2 thermometers), and Xcal (3 thermometers). Reproduced by permission of the AAS.

1 Introduction
Conceding that the microwave background [1] must arise from the cosmos [2], scientists have dismissed the idea that the Earth itself could be responsible for this signal [3–7]. Most realize that the astrophysical claims are based on the laws of thermal emission [8–12]. Yet, few have ever personally delved into the basis of these laws [13–17]. At the same time, it is known that two satellites, namely COBE [18] and WMAP [19], support the cosmological interpretation [2]. As such, it seems impossible that an alternative explanation of the ﬁndings could ever prevail.
In late 2006, I prepared a detailed review of WMAP which uncovered many of the shortcomings of this instrument [20]. A range of issues were reported, including: 1) the inability to properly address the galactic foreground, 2) dynamic range issues, 3) a lack of signal to noise, 4) poor contrast, 5) yearly variability, and 6) unjustiﬁed changes in processing coeﬃcients from year to year. In fact, WMAP brought only sparse information to the scientiﬁc community, related to the dipole and to point sources.
Nonetheless, the COBE satellite, launched in 1989, continues to stand without challenge in providing empirical proof that the microwave background did come from the universe. If COBE appears immune to criticism, it is simply because scientists outside the cosmological community have not taken the necessary steps to carefully analyze its results. Such an analysis of COBE, and speciﬁcally the Far Infrared Absolute Spectrophotometer, FIRAS, is provided in the pages which

follow. Signiﬁcant problems exist with FIRAS. If anything, this instrument provides tangential evidence for an earthly source, but the data was discounted. A brief discussion of the Diﬀerential Microwave Radiometers, DMR, outlines that the anisotropy maps, and the multipoles which describe them, are likely to represent a signal processing artifact.
1.1 The microwave background
When the results of the Cosmic Background Explorer (COBE) were ﬁrst announced, Stephen Hawking stated that this “was the scientiﬁc discovery of the century, if not of all time” [21, book cover], [22, p. 236]. The Diﬀerential Microwave Radiometers (DMR) were said to have detected “wrinkles in time”, the small anisotropies overlaid on the fabric of a nearly isotropic, or uniform, microwave background [21]. As for the COBE Far Infrared Absolute Spectrophotometer, FIRAS (see Figure 1), it had seemingly produced the most perfect blackbody spectrum ever recorded [23–45]. The blackbody curve deviated from ideality by less than 3.4 10 &
 ergs cm s  sr  cm [35] from 60–600 GHz. Eventually,
the FIRAS team would publish that the “rms deviations are less than 50 parts per million of the peak of the cosmic microwave background radiation” [39]. As seen in Figure 2, the signal was so powerful that the error bars in its detection would form but a slight portion of the line used to draw the spectrum [39]. For its part, the Diﬀerential Microwave Radiometers (DMR), beyond the discovery of the anisotropies [21], had also conﬁrmed the motion of the Earth through the

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Fig. 2: Spectrum of the microwave background reproduced from [39]. This ﬁgure is well known for the claim that the error bars it contains are but a small fraction of the line width used to draw the spectrum. While this curve appears to represent a blackbody, it should be recalled that FIRAS is only sensitive to the diﬀerence between the sky and Xcal. This plot therefore reﬂects that the signal from the sky, after extensive calibration, is indistinguishable from that provided by Xcal. Since the latter is presumed to be a perfect blackbody, then such a spectrum is achieved for the sky. Note that the frequency axis is oﬀset and all data below 2 cm  have been excluded. Reproduced by permission of the AAS.
local group, as established by a microwave dipole [46–49]. Over one thousand professional works have now appeared
which directly utilize, or build upon, the COBE results [22, p. 247]. Yet, sparse concern can be found relative to any given aspect of the COBE project. Eventually, George Smoot and John Mather, the principle investigators for the DMR and FIRAS projects, would come to share the 2006 Nobel Prize in physics. Less than 30 years had elapsed since Arno Penzias and Robert Wilson received the same honor, in 1978, for the
 discovery of the 3 K microwave background [1].
Before the background was oﬃcially reported in the literature [1], the origin of the signal had already been advanced by Dicke et al. [2]. The interpretive paper [2] had immediately preceded the publication of the seminal discovery [1]. If the microwave background was thermal in ori-
 gin [8–12], it implied a source at 3 K. Surely, such a sig-
nal could not come from the Earth. For the next 40 years, astrophysics would remain undaunted in the pursuit of the spectrum, thought to have stemmed from the dawn of creation. Smoot writes: “Penzias and Wilson’s discovery of the cosmic microwave background radiation was a fatal blow to the steady state theory” [21, p. 86]. The steady state theory of the universe [50, 51] was almost immediately abandoned and astrophysics adopted Lemaˆıtre’s concept of the primordial atom [52], later known as the Big Bang. Cosmologists advanced that mankind knew the average temperature of the entire universe. Thanks to COBE, cosmology was thought to have become a precision science [53, 54].

Throughout the detection history of the microwave background, it remained puzzling that the Earth itself never provided interference with the measurements. Water, after all, acts as a powerful absorber of microwave radiation. This is well understood, both at sea aboard submarines, and at home, within microwave ovens. As such, it seemed unlikely that the surface of our planet was microwave silent in every CMB experiment which preceded COBE. The only interference appeared to come from the atmosphere [55–57]. The latter was recognized as a powerful emitter of microwave radiation. The presence of water absorption/emission lines and of the water continuum, within the atmosphere, was well documented [55–57]. Nonetheless, emission from the Earth itself was overlooked.
The microwave signal is isotropic [1], while the Earth is anisotropic. The Earth experiences a broad range of real temperatures, which vary according to location and season. Yet, the background is found to be independent of seasonal variation [1]. The signal is deﬁnitely thermal in origin [9–17]. Most importantly, it is completely free from earthly contamination. The background appears to monitor a source temper-
  ature near 3 K. Earthly temperatures average 300 K and  seldom fall below 200 K, even at the poles. It seems im-
possible that the Earth could constitute the source of this signal [3–7]. Everything can be reconsidered, only if the temperature associated with the microwave background signature is not real. Namely, that the source temperature is much higher than the temperature reported by the photons it emits. Insight in this regard can be gained by returning to the laws of thermal emission [8–12], as I have outlined [13–17].
1.2 Kirchhoﬀ’s law
One hundred and ﬁfty years have now passed, since Kirchhoﬀ ﬁrst advanced the law upon which the validity of the microwave background temperature rests [9]. His law of thermal emission stated that radiation, at equilibrium with the walls of an enclosure, was always black, or normal [9, 10]. This was true in a manner independent of the nature of the enclosure. Kirchhoﬀ’s law was so powerful that it would become the foundation of contemporary astrophysics. By applying this formulation, the surface temperatures of all the stars could be evaluated, with the same ease as measuring the temperature of a brick-lined oven. Planck would later derive the functional form of blackbody radiation, the right-hand side of Kirchhoﬀ’s law, and thereby introduce the quantum of action [10]. However, since blackbody radiation only required enclosure and was independent of the nature of the walls, Planck did not link this process to a speciﬁc physical cause [13–17]. For astrophysics, this meant that any object could produce a blackbody spectrum. All that was required was mathematics and the invocation of thermal equilibrium. Even the requirement for enclosure was soon discarded. Processes occurring far out of equilibrium, such as the radiation of a star, and the alleged

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expansion of the universe, were thought to be suitable candidates for the application of the laws of thermal emission [2]. To aggravate the situation, Kirchhoﬀ had erred in his claim of universality [13–17]. In actuality, blackbody radiation was not universal. It was limited to an idealized case which, at the time, was best represented by graphite, soot, or carbon black [13–17]. Nothing on Earth has been able to generate the elusive blackbody over the entire frequency range and for all temperatures. Silver enclosures could never produce blackbody spectra. Kirchhoﬀ’s quest for universality was futile [13–17]. The correct application of the laws of thermal emission [8–12] requires the solid state. Applications of the laws to other states of matter, including liquids, gases, stars, and primordial atoms, constitute unjustiﬁed extensions of experimental realities and theoretical truths [13–17].
Since the source of the microwave background [1] could not possibly satisfy Kirchhoﬀ’s requirement for an enclosure
 [9], its 3 K temperature might only be apparent [13–17].
The temperature of the source could be very diﬀerent than the temperature derived from its spectrum. Planck, indeed, advanced the same idea relative to using the laws of thermal emission to measure the surface temperature of the Sun. He wrote: “Now the apparent temperature of the sun is obviously nothing but the temperature of the solar rays, depending entirely on the nature of the rays, and hence a property of the rays and not a property of the sun itself. Therefore it would be, not only more convenient, but also more correct, to apply this notation directly, instead of speaking of a ﬁctitious temperature of the sun, which can be made to have a meaning only by the introduction of an assumption that does not hold in reality” [58, §101]. Without a known enclosure, spectra appearing Planckian in nature do not necessarily have a direct link to the actual temperature of the source. The Sun operates far out of thermal equilibrium by every measure, as is evident by the powerful convection currents on its surface [59]. Furthermore, because it is not enclosed within a perfect absorber, its true surface temperature cannot be derived from the laws of thermal emission [59]. These facts may resemble the points to which Planck alludes.
1.3 The oceans of the Earth
The COBE team treats the Earth as a blackbody source of
 emission at 280 K [48]. Such a generalization seems plau-
sible at ﬁrst, particularly in the near infrared, as revealed by the remote sensing studies [60, 61]. However, FIRAS is making measurements in the microwave and far-infrared regions of the spectrum. It is precisely in this region that these assumptions fail. Furthermore, the FIRAS team is neglecting the fact that 70% of the planet is covered with water. Water is far from acting as a blackbody, either in the infrared or in the microwave. Using remote sensing, it has been well established that rainfall causes a pronounced drop in terrestrial brightness temperatures in a manner which is proportional to

the rate of precipitation. In the microwave region, large bodies of water, like the oceans, display brightness temperatures
 which vary from a few Kelvin to 300 K, as a function of
angle of observation, frequency, and polarization (see Figure 11.45 in [62]). Since the oceans are not enclosed, their thermal emission proﬁles do not necessarily correspond to their true temperatures. The oceans of the Earth, like the Sun, sustain powerful convection currents. Constantly striving for equilibrium, the oceans also fail to meet the requirements for being treated as a blackbody [13–17].
In order to understand how the oceans emit thermal radiation, it is important to consider the structure of water itself [6]. An individual water molecule is made up of two hydroxyl bonds, linking a lone oxygen atom with two adjacent hydrogens (H O H). These are rather strong bonds, with
 force constants of 8.45 10# dyn/cm [6]. In the gas phase, it
is known that the hydroxyl bonds emit in the infrared region. The O H stretch can thus be found near 3,700 cm , while the bending mode occurs near 1,700 cm  [63]. In the condensed state, liquid water displays corresponding emission bands, near 3,400 cm  and 1,644 cm  [63, p. 220]. The most notable change is that the O H stretching mode is displaced to lower frequencies [63]. This happens because water molecules, in the condensed state (liquid or solid), can interact weakly with one another, forming hydrogen bonds [63]. The force constant for the hydrogen bond (H O HOH) has been determined in the water dimer to be on the order of
0.108 10# dyn/cm [6, 64, 65]. But, in the condensed state,
a study of rearrangement energetics points to an even lower value for the hydrogen bond force constant [66]. In any event, water, through the action of the hydrogen bond, should be emitting in the microwave and far-IR regions [6, 63]. Yet, this emission has never been detected. Perhaps, the oceanic emission from hydrogen bonds has just been mistaken for a cosmic source [2].
1.4 Ever-present water
1.4.1 Ground-based measurements
From the days of Penzias and Wilson [1], ground-based measurements of the microwave background have involved a correction for atmospheric water contributions (see [56] for an in-depth review). By measuring the emission of the sky at several angles (at least two), a correction for atmospheric components was possible. Further conﬁdence in such procedures could be provided through the modeling of theoretical atmospheres [55, 56]. Overall, ground-based measurements were diﬃcult to execute and corrections for atmospheric contributions could overwhelm the measurement of interest, particularly as higher frequencies were examined. The emission from atmospheric water was easy to measure, as Smoot recalls in the “parking lot testing” of a radiometer at Berkeley: “An invisible patch of water vapor drifted overhead; the scanner showed a rise in temperature. Good: this meant the

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instrument was working, because water vapor was a source of stray radiation” [21, p. 132].
The diﬃculty in obtaining quality measurements at high frequencies was directly associated with the presence of the water continuum, whose amplitude displays powerful frequency dependence [55, 56]. As a result, experiments were typically moved to locations where atmospheric water was minimized. Antarctica, with its relatively low atmospheric humidity, became a preferred monitoring location [55]. The same was true for mountain tops, places like Mauna Kea and Kitt Peak [55]. Many ground-based measurements were made from White Mountain in California, at an elevation of 3800 m [55]. But, there was one circumstance which should have given cosmologists cause for concern: measurements located near the oceans or a large body of water. These were amongst the simplest of all to perform. Weiss writes: “Temperature, pressure, and constituent inhomogeneities occur and in fact are the largest source of random noise in ground-based experiments. However, they do not contribute systematic errors unless the particular observing site is anisotropic in a gross manner — because of a large lake or the ocean in the direction of the zenith scan, for example. The atmospheric and CBR contributions are separable in this case without further measurement or modeling” [67, p. 500]. Surely, it might be of some importance that atmospheric contributions are always a signiﬁcant problem which is only minimized when large bodies of condensed water are in the immediate scan direction.
The interesting interplay between atmospheric emissions and liquid surfaces is brought to light, but in a negative fashion, in the book by Mather [22]. In describing British work in the Canary Islands, Mather writes: “Their job was unusually diﬃcult because Atlantic weather creates patterns in the air that can produce signals similar to cosmic ﬂuctuations. It took the English scientists years to eliminate this atmospheric noise. . . ” [22, p. 246–247]. As such, astronomers recognized that the Earth was able to alter their measurements in a substantial manner. Nonetheless, the possibility that condensed water itself was responsible for the microwave background continued to be overlooked.
1.4.2 U2 planes, rockets, and balloons
As previously outlined, the presence of water vapor in the lower atmosphere makes all measurements near the Wien maximum of the microwave background extremely diﬃcult, if not impossible, from the ground. In order to gain more elevation, astrophysicists carried their instruments skywards using U2 airplanes, rockets, and balloons [21, 22]. All too often, these measurements reported elevated microwave background temperatures. The classic example is given by the Berkeley-Nagoya experiments, just before the launch of COBE [68]. Reﬂecting on these experiments, Mather writes: “A greater shock to the COBE science team, especially to me since I was in charge of the FIRAS instrument, was an

announcement made in early 1987 by a Japanese-American team headed by Paul Richards, my old mentor and friend at Berkeley, and Toshio Matsumoto of Nagoya University. The Berkeley-Nagoya group had launched from the Japanese island of Kyushu a small sounding rocket carrying a spectrometer some 200 miles high. During the few minutes it was able to generate data, the instrument measured the cosmic background radiation at six wavelengths between 0.1 millimeter and 1 millimeter. The results were quite disquieting, to say the least: that the spectrum of the cosmic microwave background showed an excess intensity as great as 10 percent at certain wavelengths, creating a noticeable bump in the blackbody curve. The cosmological community buzzed with alarm” [22, p. 206]. The results of the Berkeley-Nagoya
group were soon replaced by those from COBE. The ori-
gin of the strange “bump” on the blackbody curve was never
identiﬁed. However, condensation of water directly into the
Berkeley-Nagoya instrument was likely to have caused the
interference. In contrast, the COBE satellite was able to op-
erate in orbit, where any condensed water could be slowly
degassed into the vacuum of space. COBE did not have to
deal with the complications of direct water condensation and Mather could write in savoring the COBE ﬁndings: “Rich and Ed recognized at once that the Berkeley-Nagoya results had been wrong” [22, p. 216]. Nonetheless, the Berkeley-
Nagoya experiments had provided a vital clue to the astro-
physical community.
Water seemed to be constantly interferring with mi-
crowave experiments. At the very least, it greatly increased
the complexity of studies performed near the Earth. For instance, prior to ﬂying a balloon in Peru, Smoot reports: “It is much more humid in the tropics, and as the plane descended from the cold upper air into Lima, the chilly equipment condensed the humidity into water. As a result, water collected into the small, sensitive wave guides that connect the diﬀerential microwave radiometer’s horns to the receiver. We had to take the receiver apart and dry it. . . Our equipment had dried, so we reassembled it and tested it: it worked” [21, p. 151].
Still, little attention has been shown in dissecting the un-
derlying cause of these complications [6]. Drying scientiﬁc
equipment was considered to be an adequate solution to ad-
dress this issue. Alternatively, scientists simply tried to pro-
tect their antenna from condensation and added small mon-
itoring devices to detect its presence. Woody makes this apparent, relative to his experiments with Mather: “On the ground and during the ascent, the antenna is protected from atmospheric condensation by two removable windows at the top of the horn. . . At the same time, a small glass mirror allows us to check for atmospheric condensation in the antenna by taking photographs looking down the throat of the horn and cone” [69, p. 16]. Indeed, monitoring condensation
has become common place in detecting the microwave back-
ground using balloons. Here is a recent excerpt from the 2006 ﬂight of the ARCADE 2 balloon: “A video camera mounted

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on the spreader bar above the dewar allows direct imaging of the cold optics in ﬂight. Two banks of light-emitting diodes provide the necessary illumination. The camera and lights can be commanded on and oﬀ, and we do not use data for science analysis from times when they are on” [70]. They continue: “The potential problem with a cold open aperture is condensation from the atmosphere. Condensation on the optics will reﬂect microwave radiation adding to the radiometric temperature observed by the instrument in an unknown way. In the course of an ARCADE 2 observing ﬂight, the aperture plate and external calibrator are maintained at cryogenic temperatures and exposed open to the sky for over four hours. Figure 12 shows time averaged video camera images of the dewar aperture taken two hours apart during the 2006 ﬂight. No condensation is visible in the 3 GHz horn aperture despite the absence of any window between the horn and the atmosphere. It is seen that the eﬄux of cold boiloﬀ helium gas from the dewar is suﬃcient to reduce condensation in the horn aperture to below visibly detectable levels” [70].
The fact that condensation is not visible does not imply
that it is not present. Microscopic ﬁlms of condensation could
very well appear in the horn, in a manner undetectable by the
camera. In this regard, claims of strong galactic microwave
bursts, reported by ARCADE 2 [70, 71] and brought to the
attention of the public [72], must be viewed with caution.
This is especially true, since it can be deduced from the pre-
vious discussion, that the camera was not functional during
this short term burst. In any event, it is somewhat improbable
that an object like the galaxy would produce bursts on such a
short time scale. Condensation near the instrument is a much
more likely scenario, given the experimental realities of the
observations.
It remains puzzling that greater attention is not placed
on understanding why water is a source of problems for mi-
crowave measurements. Singal et al. [70], for instance, be-
lieve that condensed water is a good reﬂector of microwave
radiation. In contrast, our naval experiences, with signal
transmission by submarines, document that water is an ex-
tremely powerful absorber of microwave radiation. There-
fore, it must be a good emitter [8–12].
It is interesting to study how the Earth and water were
treated as possible sources of error relative to the microwave
background. As a direct precursor to the COBE FIRAS horn,
it is most appropriate to examine the Woody-Mather instru-
ment [69, 73]. Woody provides a detailed error analysis, as-
sociated with the Mather/Woody interferometer-based spec-
trometer [69]. This includes virtually every possible source
of instrument error. Both Mather and Woody view earthshine
 as originating from a 300 K blackbody source. They ap-
pear to properly model molecular species in the atmosphere
(H O, O , ozone, etc...), but present no discussion of the ex-
pected thermal emission proﬁle of water in the condensed
state on Earth. Woody [69, p. 99] and Mather [73, p. 121]
do attempt to understand the response of their antenna to the

Earth. Woody places an upper limit on earthshine [69, p. 104] by applying a power law continuum to model the problem. In this case, the Earth is modeled as if it could only produce 300 K photons. Such a treatment generates an error correction which grows with increasing frequency. Woody reaches the conclusion that, since the residuals on his ﬁts for the microwave background are relatively small, even when earthshine is not considered, then its eﬀect cannot be very significant [69, p. 105]. It could be argued that continental emission is being modeled. Yet, the function selected to represent earthly eﬀects overtly dismisses that the planet itself could be producing the background. The oceans are never discussed.
Though Mather was aware that the water dimer exists in the atmosphere [73, p. 54], he did not extend this knowledge to the behavior of water in the condensed state. The potential importance of the hydrogen bond to the production of the microwave background was not considered [73]. At the same time, Mather realized that condensation of water into his antenna created problems. He wrote: “The eﬀect of air condensing into the antenna were seen. . . ” [73, p. 140]. He added: “When the second window was opened, the valve which controls the gas ﬂow should have been rotated so that all the gas was forced out through the cone and horn. When this situation was corrected, emissions from the horn were reduced as cold helium has cooled the surfaces on which the air had condensed, and the signal returned to its normal level” [73, p. 140–141]. Mather does try to understand the eﬀect of diﬀraction for this antenna [73, p. 112–121]. However, the treatment did not model any objects beyond the horn itself.
Relative to experiments with balloons, U2 airplanes, and rockets, the literature is replete with complications from water condensation. Despite this fact, water itself continues to be ignored as the underlying source of the microwave background. It is in this light that the COBE project was launched.
1.4.3 The central question
In studying the microwave background, several important conclusions have been reached as previously mentioned. First, the background is almost perfectly isotropic: it has essentially the same intensity, independent of observation angle [1]. Second, the background is not aﬀected by seasonal variations on Earth [1]. Third, the signal is of thermal origin [8–17]. Finally, the background spectrum (see Figure 2) is clean: it is free from earthly interference. Over a frequency
 range spanning nearly 3 orders of magnitude ( 1–660 GHz),
the microwave background can be measured without any contaminating eﬀect from the Earth. The blackbody spectrum is “perfect” [39]. But, as seen above, liquid water is a powerful absorber of microwave radiation. Thus, it remains a complete mystery as to why cosmology overlooked that the surface of the Earth could not produce any interference in these measurements. The only issue of concern for astrophysics is the atmosphere [55, 56] and its well-known absorption in the mi-

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crowave and infrared bands. The contention of this work is that, if the Earth’s oceans cannot interfere with these measurements, it is precisely because they are the primary source of the signal.
2 COBE FIRAS
For this analysis, the discussion will be limited primarily to the FIRAS instrument. Only a brief treatment of the DMR will follow in section 3. The DIRBE instrument, since it is unrelated to the microwave background, will not be addressed.
2.1 General concerns
Beginning in the late 1980’s, it appeared that NASA would utilize COBE as a much needed triumph for space exploration [22, 24]. This was understandable, given the recent Challenger explosion [22, 24]. Visibility and a sense of urgency were cast upon the FIRAS team. COBE, now unable to use a shuttle ﬂight, was faced with a signiﬁcant redesign stage [22, 24]. Mather outlined the magnitude of the task at hand: “Every pound was crucial as the engineers struggled to cut the spacecraft’s weight from 10,594 pounds to at most 5,025 pounds and its launch diameter from 15 feet to 8 feet” [22, p. 195]. This urgency to launch was certain to have affected prelaunch testing. Mather writes: “Getting COBE into orbit was now Goddard’s No. 1 priority and one of NASA’s top priorities in the absence of shuttle ﬂights. In early 1987 NASA administrator Jim Fletcher visited Goddard and looked over the COBE hardware, then issued a press release stating that COBE was the centerpiece of the agency’s recovery” [22, p. 194–195]. Many issues surfaced. These are important to consider and have been highlighted in detail [22, chap. 14].
After the launch, polite open dissent soon arose with a senior group member. The entire premise of the current paper can be summarized in the discussions which ensued: “Dave Wilkinson, the FIRAS team sceptic, argued eﬀectively at numerous meetings that he did not believe that Ned” (Wright) “and Al” (Kogut) “had proven that every systematic error in the data was negligible. Dave’s worry was that emissions from the earth might be shinning over and around the spacecraft’s protective shield” [22, p. 234]. As will be seen below, Wilkinson never suspected that the Earth could be emitting as
a 3 K source. Nonetheless, he realized that the FIRAS horn
had not been adequately modeled or tested. Despite these challenges, the FIRAS team minimized Wilkinson’s unease. Not a single study examines the interaction of the COBE shield with the FIRAS horn. The earthshine issue was never explored and Wilkinson’s concerns remain unanswered by the FIRAS team to this day.
2.2 Preﬂight testing
A review of the COBE FIRAS prelaunch data reveals that the satellite was not adequately tested on the ground. These

concerns were once brought to light by Professor Wilkinson, as mentioned above. He writes: “Another concern was the magnitude of 300 K Earth emission that diﬀracted over, or leaked through, COBE’s ground screen. This had not been measured in preﬂight tests, only estimated from crude (by today’s standards) calculations” [74]. Unfortunately, Professor
Wilkinson does not give any detailed outline of the question
and, while there are signs of problems with the FIRAS data,
the astrophysical community itself has not published a thor-
ough analysis on this subject.
 Professor Wilkinson focused on the Earth as a 300 K
blackbody source, even if the established behavior of the
oceans in the microwave and far-infrared suggested that the
oceans were not radiating in this manner [62]. Wilkinson
never advanced that the Earth could be generating a signal
 with an apparent temperature of 3 K. This means that the
diﬀraction problems could potentially be much more impor-
tant than he ever suspected. Mather did outline Wilkinson’s
concerns in his book as mentioned above [22, p. 234], but did
not elaborate further on these issues.
Beyond the question of diﬀraction, extensive testing of
FIRAS, assembled in the ﬂight dewar, did not occur. Mather
stated that each individual component of FIRAS underwent
rigorous evaluation [22, chap. 14], however testing was cur-
tailed for the fully-assembled instrument. For instance,
Hagopian described optical alignment and cryogenic perfor-
mance studies for FIRAS in the test dewar [29]. These stud-
ies were performed at room and liquid nitrogen temperatures
 and did not achieve the cryogenic values, 1.4 K, associ-
ated with FIRAS [29]. Furthermore, Hagopian explained: “Due to schedule constraints, an abbreviated version of the alignment and test plan developed for the FIRAS test unit was adopted” [29]. Vibration testing was examined in or-
der to simulate, as much as possible, the potential stresses
experienced by FIRAS during launch and ﬂight. The issue centered on optical alignments: “The instrument high frequency response is however, mainly a function of the wire grid beam splitter and polarizer and the dihedrals of the MTM. The instrument is sensitive to misalignments of these components on the order of a few arc seconds” [29]. In
these studies, a blackbody source was used at liquid nitro-
gen temperatures to test FIRAS performance, but not with
its real bolometers in place. Instead, Golay cell IR detec-
tors were fed through light pipes mounted on the dewar output ports. It was noted that: “Generally, the instrument behaved as expected with respect to performance degradation and alignment change. . . These results indicate that the instrument was successfully ﬂight qualiﬁed and should survive cryogenic and launch induced perturbations” [29]. These ex-
periments did not involve FIRAS in its ﬁnal conﬁguration
within the ﬂight dewar and did not achieve operational tem-
peratures.
A description of the preﬂight tests undergone by COBE
was also presented by L. J. Milam [26], Mosier [27], and Co-

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ladonato et al. [28]. These accounts demonstrate how little testing COBE actually underwent prior to launch. Concern rested on thermal performance and ﬂight readiness. There obviously were some RF tests performed on the ground. In Mather [22, p. 216], it was reported that the calibration ﬁle for Xcal had been obtained on Earth. This was the ﬁle utilized to display the ﬁrst spectrum of the microwave background with FIRAS [22, p. 216]. Nonetheless, no RF tests for sensitivity, side lobe performance, or diﬀraction were discussed for the FIRAS instrument. Given that Fixsen et al. [38] cite work by Mather, Toral, and Hemmati [25] for the isolated horn, as a basis for establishing side lobe performance, it is clear that these tests were never conducted for the fully-assembled instrument. Since such studies were diﬃcult to perform in the contaminating microwave environments typically found on the ground, the FIRAS team simply chose to bypass this aspect of preﬂight RF testing.
As a result, the scientiﬁc community believes that COBE was held to the highest of scientiﬁc standards during ground testing when, in fact, a careful analysis suggests that some compromises occurred. However, given the scientiﬁc nature of the project, the absence of available preﬂight RF testing reports implies that little took place. Wilkinson’s previously noted statement echoes this belief [74].
2.2.1 Bolometer performance
The FIRAS bolometers were well designed, as can be gathered from the words of Serlemitsos [31]: “The FIRAS bolometers were optimized to operate in two frequency ranges. The slow bolometers cover the range from 1 to 20 Hz (with a geometric average of 4.5 Hz), and the fast ones cover the range from 20–100 Hz (average 45 Hz).” Serlemitsos contin-
 ues: “The NEP’s for the FIRAS bolometers are 4.5 10 15  W/Hz1/2 at 4.5 Hz for the slow bolometers and 1.2 10 14
W/Hz1/2 at 45 Hz for the fast ones” [31], where NEP stands for “noise equivalent power”. The FIRAS bolometers were made from a silicon wafer “doped with antimony and compensated with boron” [31]. Serlemitsos also outlined the key element of construction: “IR absorption was accomplished by coating the back side of the substrate with metallic ﬁlm. . . ” made “of 20 Å of chromium, 5 Å of chromium-gold mixture, and 30–35 Å of gold” [31]. Such vaporized metal deposits, or metal blacks, were well known to give good blackbody performance in the far IR [75,76]. Thus, if problems existed with FIRAS, it was unlikely that they could be easily attributed to bolometer performance.
2.2.2 Grid polarizer performance
The FIRAS team also fully characterized the wire grid polarizer [30]. While the grids did “not meet the initial speciﬁcation” their spectral performance did “satisfy the overall system requirements” [30].

2.2.3 Emissivity of Xcal and Ical
The FIRAS team essentially makes the assumption that the two calibrators, Xcal and Ical, function as blackbodies over the entire frequency band. Xcal and Ical are represented schematically in Figure 3 [38, 42]. Both were manufactured from Eccosorb CR-110 (Emerson and Cuming Microwave Products, Canton, MA, 1980 [77]), a material that does not possess ideal attenuation characteristics. For instance, CR110 provides an attenuation of only 6 dB per centimeter of material at 18 GHz [78]. In Hemati et al. [79], the thermal properties of Eccosorb CR-110 are examined in detail over the frequency range for FIRAS. The authors conduct transmission and reﬂection measurements. They demonstrate that Eccosorb CR-110 has a highly frequency dependent decrease in the transmission proﬁle, which varies by orders of mag-
 nitude from 30–3,000 GHz [79]. Hemati et al. [79] also
examine normal specular reﬂection, which demonstrate less variation with frequency. Therefore, when absorption coefﬁcients are calculated using the transmission equation [79], they will have frequency dependence. Consequently, Hemati et al. [79] report that the absorption coeﬃcients for Eccosorb CR-110 vary by more than one order of magnitude over the frequency range of FIRAS.
In addition, it is possible that even these computed absorption coeﬃcients are too high. This is because Hemati et al. [79] do not consider diﬀuse reﬂection. They justify the lack of these measurements by stating that: “For all samples the power response was highly specular; i.e., the reﬂected power was very sensitive with respect to sample orientation” [79]. As a result, any absorption coeﬃcient which is derived from the transmission equation [79], is prone to being overestimated. It is unlikely that Eccosorb CR-110 allows no diﬀuse reﬂection of incoming radiation. Thus, Eccosorb CR-110, at these thicknesses, does not possess the absorption characteristics of a blackbody. It is only through the construction of the “trumpet mute” shaped calibrator that blackbody behavior is thought to be achieved [38].
When speaking of the calibrators, Fixsen et al. [39] state: “The other input port receives emission from an internal ref-
 erence calibrator (emissivity 0.98)” and “During calibra-
tion, the sky aperture is completely ﬁlled by the external calibrator with an emissivity greater than 0.99997, calculated and measured” [39]. Practical experience, in the construction of laboratory blackbodies, reveals that it is extremely diﬃcult to obtain such emissivity values over a wide frequency range. Measured emissivity values should be presented in frequency dependent fashion, not as a single value for a broad frequency range [80]. In the infrared, comparable performance is not easily achievable, even with the best materials [15, 80]. The situation is even more diﬃcult in the far infrared and microwave.
The emissivity of the calibrators was measured, at 34 and 94 GHz, using reﬂection methods as described in de-

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Fig. 3: Schematic representation of Xcal and Ical reproduced from [42]. Note that the calibrators are made from Eccosorb CR-110 which is backed with copper foil. Xcal, which contains three GRTs, is attached to the satellite with a movable arm allowing the calibrator to be inserted into, or removed from, the sky horn. The internal calibrator, Ical, is equipped with two GRTs and provides a signal for the reference horn. Reproduced by permission of the AAS.
tail [42]. However, these approaches are not appropriate for devices like the calibrators. In examining Figure 3, it is evident that Xcal is cast from layers of Eccosorb CR-110, backed with copper foil. For reﬂection methods to yield reliable results, they must address purely opaque surfaces. Eccosorb CR-110 is not opaque at these thicknesses [79] and displays signiﬁcant transmission. The problem is worthy of further discussion.
In treating blackbody radiation, it is understood, from the principle of equivalence [8], that the emission of an object must be equal to its absorption at thermal and radiative equilibrium. Emission and absorption can be regarded as quantum mechanical processes. Therefore, it is most appropriate to state that, for a blackbody, or any body in radiative equi-
librium, the probability of absorption, 2 , must be equal to
= the probability of emission, 2, (2 2). But, given the
combination of the transmittance for Eccosorb CR-110, the presence of a copper lining and the calibrator geometry, the
FIRAS team has created a scenario wherein 2 2. This
is an interesting situation, which is permitted to exist because the copper backing on the calibrator provides a conductive path, enabling Xcal to remain at thermal equilibrium through non-radiative processes. Under these test conditions, Xcal is in thermal equilibrium, but not in radiative equilibrium. It receives incoming photons from the test signal, but can dissipate the heat, using conduction, through the copper backing. Xcal does not need to use emission to balance absorption.
If the FIRAS calibrators provide excellent reﬂection measurements [42], it is because of their “trumpet mute” shape

and the presence of a copper back lining. Radiation incident to the device, during reﬂectance measurements, which is not initially absorbed, will continue to travel through the Eccosorb and strike the back of the casing. Here it will undergo normal specular reﬂection by the copper foil present at this location. The radiation can then re-enter the Eccosorb, where it has yet another chance of being absorbed.
As a result, 2 can be eﬀectively doubled as a consequence
of this ﬁrst reﬂection. Because of the shape of the calibrators, along with the presence of normal specular reﬂection on the copper, the radiation is essentially being pushed further into the calibrator where its chances of being ab-
sorbed are repeated. Consequently, 2 continues to increase
with each reﬂection oﬀ the copper wall, or because photons are being geometrically forced to re-enter the adjacent Eccosorb wall. The situation moves in the opposite direc-
tion for 2 and this probability therefore drops under test
conditions. Note that the copper foil has a low emissivity in this fre-
quency range. Therefore, it is reasonable to assume that it cannot contribute much to the generation of photons. These must be generated within the Eccosorb CR-110 layers. Now, given the geometry of the “trumpet mute”, there exists no
means of increasing the probability of emission, 2. In-
deed, some of the photons emitted will actually travel in the direction of the copper foil. This will lengthen their eﬀective path out of the Eccosorb, since they exit and immediately re-enter, and increases the chance that they are absorbed before ever leaving the surface of the calibrator.
Thus, 2 experiences an eﬀective decrease, because of the presence of the copper foil. The net result is that 2 2
and the FIRAS team has not properly measured the emissivity of their calibrators using reﬂective methods [42]. In fact, direct measures of emissivity for these devices would demonstrate that they are not perfectly black across the frequencies of interest. Nonetheless, the devices do appear black in reﬂection measurements. But this is an illusion which does not imply that the calibrators are truly black when it comes to emission. Reﬂection measurements cannot establish the blackness of such a device relative to emission if the surface observed is not opaque. Geometry does matter in treating either emission or absorption under certain conditions. The problem is reminiscent of other logical errors relative to treating Kirchhoﬀ’s ﬁrst proof for universality [16].
The FIRAS group asserts that they have veriﬁed the blackness of their calibrators with computational methods. Yet, these methods essentially “inject photons” into cavities, which otherwise might not be present [17]. Much like the improper use of detectors and reﬂection methods (on nonopaque surfaces), they can ensure that all cavities appear black [17]. The FIRAS calibrators are not perfectly black, but it is not clear what this implies relative to the measurements of the microwave background.

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2.2.4 Leaks around Xcal
The acquisition of a blackbody spectrum from the sky is based on the performance of Xcal. For instance, Fixsen and Mather write: “It is sometimes stated that this is the most perfect blackbody spectrum ever measured, but the measurement is actually the diﬀerence between the sky and the calibrator” [43]. Mathematically, the process is as follows:
(Sky Ical) (Xcal Ical) = (Sky Xcal) :
Thus, Ical and all instrumental factors should ideally be negligible, contrary to what the FIRAS team experiences. Furthermore, if the calibration ﬁle with Xcal perfectly matches the sky, then a null result occurs. Since Xcal is thought to be a perfect blackbody, the derived sky spectrum is also ideal, as seen in Figure 2. It is extremely important that the calibration ﬁle, generated when Xcal is within the horn, does not contain any contamination from the sky. In the limit, should the sky dominate the calibration, a perfect blackbody shape will be recorded. This would occur because the sky is eﬀectively compared against itself, ensuring a null.
The FIRAS team reminds us that: “When the Xcal is in the sky horn it does not quite touch it. There is a 0.6 mm gap between the edge of the Xcal and the horn, so that the Xcal and the sky horn can be at diﬀerent temperatures. Although the gap is near the ﬂare of the horn and not in the direct line of sight of the detectors, it would result in undesirable leakage at long wavelengths because of diﬀraction. To ensure a good optical seal at all wavelengths, two ranks of aluminized Kapton leaves attached to the Xcal make a ﬂexible contact with the horn” [38] (see Figure 3). The claim that the Kapton leaves make a ﬂexible contact with the horn, at operating temperatures, does not seem logical. The horn is
 operating at cryogenic temperatures ( 2.7 K) and, thus, the
Kapton leaves should not be considered ﬂexible, but rather rigid, perhaps brittle. This might cause a poor contact with the horn during critical calibration events in space. The FIRAS team continues: “An upper limit for leakage around the Xcal was determined in ground tests with a warm cryostat dome by comparing signals with the Xcal in and out of the horn. Leakage is less than 1.5 10 4 in the range 5 <  < 20 cm 1 and 6.0 10 5 in the range 25 <  < 50 cm 1” [38]. The issue of leakage around Xcal is critical to the proper functioning of FIRAS. Consequently, Mather et al. revisit the issue at length in 1999 (see section 3.5.1 in [42]). The seal does indeed appear to be good [42], but it is not certain that these particular ground tests are valid in space.
It is not clear if RF leak testing occurred while FIRAS was equipped with its specialized bolometers. As seen in section 2.2, in some preﬂight testing, Golay cell IR detectors had been fed through light pipes mounted on the dewar output ports. Such detectors would be unable to properly detect signals at the lowest frequencies. In fact, the FIRAS bolometers were made from metal blacks [31, 75, 76] in order to speciﬁ-

cally provide sensitivity in the diﬃcult low frequency range.
As a result, any leak testing performed with the Golay cell
IR detectors might be subject to error, since these may not
have been sensitive to signal, in the region most subject to
diﬀraction.
The FIRAS group also makes tests in ﬂight and states: “The Kapton levels sealing the gap between the sky horn and Xcal were tested by gradually withdrawing the Xcal from the horn. No eﬀect could be seen in ﬂight until it had moved 1.2 cm” [38]. This issue is brought up, once again, by Mather et al.: “A test was also done in ﬂight by removing the calibrator 12 steps, or 17 mm, from the horn. Only a few interferograms were taken, but there was no sign of a change of signal level” [42]. It is interesting that Fixsen et al. [38] claim that
no eﬀect could be seen until the horn had moved 1.2 cm. This
implies that eﬀects were seen at 1.2 cm. Conversely, Mather
et al. assert that no eﬀects were seen up to 17 mm [42]. In any
case, identical results could have been obtained, even if the
seal was inadequate. Perhaps this is why Fixsen et al. write: “During calibration, the sky acts as a backdrop to the external calibrator, so residual transmission is still nearly 2.73 K radiation” [39]. Clearly, if the seal was known to be good, there
should not be any concern about “residual transmission” from
the sky.
Fixsen et al. [39] rely on the sky backdrop providing a per-
fect blackbody spectrum behind Xcal. However, if the signal
was originating from the Earth, the sky signal could be dis-
torted as a function of frequency. This would bring error into
the measurements, should the sky signal leak into the horn.
From their comments, a tight seal by the Kapton leaves can-
not be taken for granted. While in-ﬂight tests, slowly remov-
ing Xcal, indicate that the spectrum changes as the calibrator
was lifted out of the horn, they may not exclude that leakage
exists when it is inside the horn.
It is also interesting that Mather describes signiﬁcant problems with Xcal prior to launch, as follows: “Now without gravity to help hold it in place, the calibrator popped out of the horn every time the test engineers inserted it by means of the same electronic commands they would use once COBE was in orbit. Nothing the engineers tried would keep it in place” [22, p. 202]. In the end, the problem was caused by the
ﬂexible cable to the Xcal [22]. The cable was replaced with
three thin ribbons of Kapton [22, p. 202–204]. COBE under-
went one more cryogenic test, with the liquid helium dewar
at 2.8 K, lasting a total of 24 days ending in June 1989 [26].
Milan’s report does not provide the results of any RF test-
ing [26], but everything must have worked. The satellite was
prepared for shipment to the launch site [22, p. 202–204].
In 2002, Mather reminds us of the vibration problems with COBE: “There were annoying vibrations at 57 and
 8 Hz” [43]. On the ground, the Xcal could “pop out” of
the horn if the satellite was turned on its side [22, p. 202].
Only gravity was holding Xcal in place. Still, in orbit, COBE
experiences very little gravity. As such, the eﬀects of the vi-

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brations in knocking Xcal out of the horn, or in breaking the contact between the Kapton leaves and the horn, are not the same in space. A small vibration, in space, could produce a signiﬁcant force against Xcal, pushing it out of the horn. Thus, all leak testing on the ground has little relevance to the situation in orbit, since both gravity and vibrations aﬀect the Xcal position in a manner which cannot be simulated in the laboratory. The FIRAS team simply cannot be assured that Xcal did not allow leakage from the sky into the horn during calibration.
2.2.4.1 Conclusive proof for Xcal performance
When FIRAS ﬁrst begins to transfer data to the Earth, a calibration ﬁle using Xcal had not been collected in space [22, p. 216]. Nonetheless, a calibration ﬁle existed which had been measured on the ground. Mather provides a wonderful account of recording the ﬁrst blackbody spectrum from the microwave background [22, p. 216]. The text is so powerfully convincing that it would be easy to dismiss the search for any problems with FIRAS. Using the ground-based calibration ﬁle, the FIRAS team generates an “absolutely perfect blackbody curve” [22, p. 216]. However, considering all of the errors present in orbit, it is not clear how the calibration ﬁle gathered on Earth diﬀered, if at all, from the one obtained in space. If the FIRAS team had wanted to bring forth the most concrete evidence that the situation in space, relative to Xcal, was identical to that acquired on the ground, then they could have easily displayed the diﬀerence spectrum between these two ﬁles. Ideally, no diﬀerences should be seen. But, if diﬀerences were observed, then either temperature variations, or leakage, must be assumed. In fact, the diﬀerence between the two ﬁles could have provided a clue as to the nature of the leakage into the FIRAS horn. Mather et al. feel compelled to verify the performance of Xcal on the ground 10 years after launch [42]. This suggests that the calibration ﬁles taken prior to launch did not agree with those acquired in ﬂight.
2.2.5 Design of the FIRAS horn
In examining the FIRAS horn (see Figure 1), it is apparent that this component does not conform to accepted practices in the ﬁeld of antenna design [81–83]. This device is unique,
 meant to operate over a phenomenal range from 30 to 3,000
GHz [32–45]. Since broadband horns generally span no more than 1 or 2 decades in frequency [84, 85], it is doubtful that a comparable antenna can be found in the electromagnetics literature. Even the most modern broadband horns tend to cover very limited frequency ranges and, typically, at the expense of variable gains across the band [84, 85]. Unfortunately, insuﬃcient ground tests were conducted, to demonstrate the expected performance from 30–3,000 GHz. It is highly unlikely that FIRAS was ever able to perform as intended. The FIRAS team provides no test measurements to the contrary. These would have included gain and side lobe performances

spanning the frequency spectrum. Moreover, as will be seen below (see section 2.4.3.1), FIRAS is operating less than optimally over all wavelengths. The idea of using an interferometer for these studies was elegant [32–45]. But, broadband horns with demonstrated performances, over such a range of frequencies, simply do not exist [81–85]. It is interesting in this light, that the WMAP [19] and PLANCK [86] missions have both reverted to the use of narrow band devices to sample the microwave background. As for FIRAS, it functions
 primarily from 30–600 GHz. However, even in this region,
the instrument must deal with horn/shield interactions and the eﬀects of diﬀraction. These eﬀects were never appropriately considered by the FIRAS team.
The testing of the COBE FIRAS antenna pattern was inadequate. Proper tests were never performed to document the interaction of the FIRAS horn with the Sun/Earth/RFI shield. Furthermore, the team conducted no computational modeling of the horn-shield interaction as a function of frequency. This type of documentation would have been central in establishing the reliability of the FIRAS ﬁndings. Without it, the FIRAS team did not eliminate the possibility that the Earth itself is producing the microwave background. The RF shield on COBE could accomplish little more than prevent terrestrial/solar photons, in the visible or near-infrared range, from directly illuminating the dewar which contains FIRAS. The central issue for the Sun/Earth shield appears to be the conservation of helium in the dewar, not the elimination of RF interference [87]. The shield is not corrugated [81, p. 657– 659] and has no special edges to prevent diﬀraction in the far infrared. Given that the FIRAS horn is broadband, it is extremely diﬃcult, if not impossible, to build a good RF shield for such a device. The FIRAS team has not established that an adequate shield was constructed to prevent RF interference from the Earth. The Sun/Earth shield simply prevents direct heating of the dewar, by visible or near infrared light [87]. They comment: “a large external conical shield protects the cryostat and instruments from direct radiation from the Sun and the Earth. The Sun never illuminates the instruments or cryostat, but the COBE orbit inclination combined with the inclination of the Earth’s equator to the ecliptic do allow the Earth limb to rise a few degrees above the plane of the instrument and sunshade apertures during about one-sixth of the orbit for one-fourth of the year. During this period, the sky horn could not be cooled to 2.7 K because of the Earth limb heating” [42]. Nowhere, in the COBE literature, is the RF performance of the “sunshade” analyzed.
2.3 FIRAS in ﬂight
2.3.1 Side lobe performance
Fixsen et al. [38] argue that the FIRAS horn “provides a 7 ﬁeld of view with low side lobes”. They base this statement on work by Mather, Toral, and Hemmati [25]. In this paper, Mather et al. present measured and theoretical evaluations of

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Fig. 4: Plot of the side lobe response for the FIRAS horn, without the presence of the COBE ground shield as reproduced from [25]. The sky lobe response, in preﬂight testing, was evaluated at three wave-
lengths, namely 118, 10, and 0.5 m. Note that only the ﬁrst mea-
 surement at 118 m ( 2,540 GHz) is within the frequency range of
the instrument (30–3000 GHz). The latter two occur in the optical band. The side lobe performance is best at the longer wavelength, in opposition to the expected theoretical result. The FIRAS team also measures the FIRAS horn at 31.4 and 90 GHZ [25], with excellent performance (data is not reproduced herein). However, once again, these results were obtained without the interfering eﬀects of the ground shield. Reproduced with permission of the Optical Society of America from: Mather J.C., Toral M., Hemmati H. Heat trap with ﬂare as multimode antenna. Appl. Optics, 1986, v. 25(16), 2826–2830 [25].
side lobe data at 31.4 and 90 GHz [25]. As expected, the side lobes are lower at the higher frequency. The measurements conform to expected performance, at least at these frequencies. But, these tests were conducted without the RF shield and consequently have limited relevance to the actual situation in ﬂight.
A careful examination of Figure 4 [25] is troubling. In this ﬁgure, Mather et al. [25] characterize the antenna pattern of the isolated FIRAS horn, without the COBE RF shield, at
infrared and optical wavelengths (118, 10, and 0.5 m). It
is not evident why the authors present this data, as only the
 ﬁrst wavelength, 118 m ( 2,540 GHz), is within the usable
bandwidth of the instrument. Nonetheless, in Figure 4, the antenna has the strongest side lobes at the highest frequen-
cies. For instance, at a wavelength of 0.5 m, the antenna
shows a relative response that is decreased by only 20 dB at
10 [25], as shown in Figure 4. At 118 m, the antenna re-
sponse is decreased by nearly 50 db. The authors are demonstrating that the FIRAS horn has better side lobe behavior at longer wavelengths rather than at short wavelengths. This is opposed to the expected performance. Mathematical modeling may well be impossible at these elevated frequencies. Once again, the shield was never considered.

Fig. 5: Plot of the side lobe response obtained for the FIRAS shield on the ground, at 3 cm  (solid line), and in orbit, using the Moon as a source of signal, at 50 cm  (dashed line). This ﬁgure is reproduced from [38]. A detailed discussion is provided in section 2.3.1. Reproduced by permission of the AAS.
Neglecting to characterize the horn-shield interaction on the ground, the FIRAS team attempts to do so in ﬂight. In Fixsen et al. [38], they publish Figure 5. They attempt to determine the antenna pattern in space by monitoring the Moon as a function of angle. Using this approach at 50 cm , they conclude that the satellite provides a maximum side lobe response of “less 38 dB beyond 15 from the center of the beam” [38]. Such a performance is reasonable, at least at this frequency. However, the FIRAS team then compares side
 lobe performance at 50 cm  ( 1,500 GHz) with data ob tained on the ground at 3 cm  ( 90 GHz). In referring to
this ﬁgure in their paper, the FIRAS team writes: “Preliminary results are shown in Figure 4, along with preﬂight measurements at 1 and 1.77 cm 1” [38]. Yet the ﬁgure legend itself states the following: “Antenna pattern for the FIRAS horn as measured on the ground before launch at 3 cm 1
 (solid line) and as measured from in ﬂight Moon data at 50
cm 1 (dashed line)” [38]. Beyond the inconsistency between the text and the ﬁgure legend, there are at least ﬁve concerns relative to this ﬁgure.
First, the data on the ground appears to have measured the FIRAS horn exclusively, not the horn with the RF shield. Second, they are comparing data at frequencies which diﬀer by more than one order of magnitude. Third, they display none of the critical in-ﬂight data for the lowest frequencies, namely those frequencies where one would expect the strongest effects from diﬀraction. Fourth, they fail to present ground data at 50 cm . Finally, the data from Fixsen et al. [38] is also puzzling. It reveals much stronger side lobes at 50 cm  than
 one would have predicted at this frequency ( 1,500 GHz).
Note, in Figure 5, that the Moon data displays a plateau at approximately 45 dB in the range from 20–50 . This is higher than would be expected, based on the excellent side lobe response, even at a much lower 90 GHz, reported for the free

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horn on the ground [25]. This plateau may simply be caused by a lack of sensitivity for the Moon at these angles. It is impossible to determine whether the plateau achieved in detection is a result of this eﬀect. The FIRAS Explanatory Supplement suggests that the Moon can contaminate the microwave background at all frequencies [40, p. 61]. The FIRAS team does not adequately confront the issue and does not publish a work focused on side lobe behavior. Comparing ground data
  at 30 GHz, or even 90 GHz, with in-ﬂight data at 1,500
GHz, has no value relative to addressing the side lobe issue. It is also true that a loss of “Moon signal”, as a function of
angle, could account for the appearance of good side lobe performance. The possibility that the Moon could be reﬂecting terrestrial, or even solar, signals back into the FIRAS horn, through normal specular reﬂection, is not discussed. This process would be angle dependent and might create the illusion of reasonable side lobe behavior. The FIRAS team provides no supportive evidence from the literature that the Moon behaves as a lambertian emitter at 50 cm . The Moon does have phases, which result in diﬀerential heating across its surface. Should the Moon not act as a lambertian emitter, the side lobe performance was not properly evaluated. This would be true, unless the satellite was rapidly turned away from the Moon while maintaining a single orbital position. But, this is unlikely to have been the case, since COBE did not have a propulsion system [22, p. 195]. Thus, the satellite was simply permitted to continue in its orbit, and the angle to the Moon thereby increased. Such a protocol might not accurately assay side lobe behavior. This is because it would depend on the absence of specular reﬂection from the Earth and the Sun, while requiring that the Moon is lambertian. In the end, experiments in space cannot replace systematic testing on the ground in establishing side lobe behavior.
Perhaps more troubling is that the frequencies of interest, relative to the microwave background, extend from less
 <  than 1 cm  to 22 cm  ( 30 to 660 GHz). For exam-
ple, the initial Penzias and Wilson measurements were made near 4 GHz [1]. Consequently, the FIRAS team is showing side lobe performance for a region outside the frequencies of interest. In fact, 1,500 GHz is the region wherein galactic dust would be sampled, not the microwave background [23]. The side lobe performance at this frequency is not relevant to the problem at hand. Furthermore, if there are problems with diﬀraction, they are being manifested by a distortion of signal, primarily in the lower frequency ranges. Hence, it would be critical for the FIRAS team to display in-ﬂight data, or ground data including the shield, in order to fully document side lobe performance in this region. The data, unfortunately, is not provided.
Should access be available to the exact dimensions of the FIRAS horn and the COBE shield, it would, in principle, be possible for an independent group to verify the performance of the satellite relative to this instrument. It is true that the problem of modeling the FIRAS horn/shield interaction is ex-

tremely complex, even at 30 GHz. Nonetheless, given current computational methods, using the Geometric Theory of Diﬀraction, it is diﬃcult to reconcile that the true directional sensitivity of the FIRAS horn was not modeled at any frequency. These studies would depend on obtaining the exact conﬁguration, for the FIRAS horn/shield, and then treating the problem using computational methods. The issue cannot be treated analytically. Furthermore, this is a diﬃcult task. It is achievable perhaps, only at the lowest frequencies of operation.
In 2002, Fixsen and Mather give a summary of the FIRAS results [43], wherein they also describe how a new instrument might be constructed. In order to address the lack of side lobe characterization, they advance that: “we would surround the entire optical system with segmented blackbody radiators to measure the side lobe responses and ensure that the source of every photon is understood” [43]. With COBE, the source of every photon was not understood. The side lobes were never measured in the presence of the shield. The idea of surrounding the optical system with blackbody calibrators is less than optimal. It would be best to simply analyze the horn/shield performance with preﬂight testing.
2.3.2 Establishing temperatures
The FIRAS team presents a dozen values for the microwave background temperature, using varying methods, as shown in Table 1. This occurs over a span of 13 years. Each time, there is a striking recalculation of error bars. In the end, the ﬁnal error on the microwave background temperature drops by nearly two orders of magnitude from 60 mK to 0.65 mK. Yet, as will be seen below, in sections 2.3.3 and 2.3.4, FIRAS was unable to yield proper nulls, either with the sky and Ical, or with Xcal and Ical. Despite the subsequent existence of systematic errors, the FIRAS team minimizes error bars.
The problems with correctly establishing temperatures for Xcal and Ical were central to the mission, as these investigators recognized: “There were two important problems. One was that the thermometers on both the Ical and Xcal did not at all agree. In fact, the disagreement among diﬀerent Xcal thermometers was 3 mK at 2.7 K” [38]. They continue: “The disagreement between the Ical thermometers was 18 mK at 2.7 K. The heat sinking of the Ical thermometer leads was inadequate, and some of the applied heat ﬂowed through part of the Ical” [38].
They try to overcome the reality that the temperature monitors on the external calibrator report a systematic error. The temperature errors on Xcal are ﬁtted with an “arbitrary oﬀset in the Xcal thermometer and the result was 7.4 0.2 mK for this oﬀset” [38]. The FIRAS team realizes that this
 was “considerably larger than the 1 mK expected from the
preﬂight calibration of the thermometers” [38]. They attribute the problem either to having improperly calibrated the thermometers before ﬂight, or due to an unknown systematic

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Reference
Mather et al., ApJ, 1990, v. 354, L37–40 [32] Mather et al., ApJ, 1994, v. 420, 439–444 [35] Fixsen et al., ApJ, 1996, v. 473, 576–587 [39] Fixsen et al., ApJ, 1996, v. 473, 576–587 [39] Fixsen et al., ApJ, 1996, v. 473, 576–587 [39] Fixsen et al., ApJ, 1996, v. 473, 576–587 [39] Mather et al., ApJ, 1999, v. 512, 511–520 [42] Mather et al., ApJ, 1999, v. 512, 511–520 [42] Mather et al., ApJ, 1999, v. 512, 511–520 [42] Mather et al., ApJ, 1999, v. 512, 511–520 [42] Fixsen & Mather, ApJ, 2002, v. 581, 817–822 [43] Fixsen & Mather, ApJ, 2002, v. 581, 817–822 [43]

Temperature
2.735x 2.726x 2.730x 2.7255{
2.717¥ 2.728
2.725x 2.7255{
2.722¥ 2.725 2.725 2.725

Error (mK)
60 10
1 0.09 7 4 5 0.085 12 2 0.65 1

Frequency (cm 1) 1–20# 2–20# 2–21y 2–21y 2–21y 2–21y 2–20z 2–21y 2–20z 2–20z 2–20z 2–20z

x

95% conﬁdence intervals. Measurement using FIRAS

microwave

background

lineshape.

Calibration

sensitive

to

the

thermometers

+ o{7f.M6t9he,ea1se1ux.rt5ee3mr,ne1anl5t.c3ua8sli,ibnargnadtFoI1rR,6XA.4cS2amlc.micro1w[3av9e].background frequency. Calibration relies on CO and C lines at

¥ Measurement using a ﬁt of the dipole spectrum to the 1st derivative of a Planckian function describing

the microwave background with 6cmbr set to 2.728 K.

#yz FFFCrrreeoeqqmquuueepennonccscyyiyterrraavannangglgueeeeuuuossseebeddtdaiiissinsennfdooottrfmrffoooamrrlmmlyaaanllsllatyyalytssesttdaais.tteeoddf

three previous entries.
but appears to be 2–21 but appears to be 2–20

cm cm

11..

Table 1: Summary of microwave background temperatures obtained by the COBE FIRAS instrument.

error. They therefore assign a 4 mK oﬀset to Xcal and raise
to 5 mK its 1 error. Though this might seem negligible, the
FIRAS team is suﬃciently concerned about Xcal that they attempt to recalibrate it on the ground, using a duplicate experiment, nearly ten years after launch [42]. For the present discussion, an error of at least 5 mK can be attributed to Xcal.
The FIRAS Explanatory Supplement outlines an enhanced picture relative to Ical performance [40, p. 42]. An optical temperature drift is modeled as follows:
T  = T + A exp (t=Ical) + To set
where T  is the “raw” Ical temperature, A = 4.26 mK,
 To set = 3.054 mK, and Ical = 104.3 days [40, p. 42].
Given that FIRAS was operational for 259 days [40, p. 28], the drift model accounts for a 48 mK error in Ical by the time the instrument is decommissioned. Yet, in 1999, Mather et al. [42] oﬀer a diﬀerent view [40, p. 42]. While treating Ical,
 they write: “An additional drift of 3 mK was noted in the
early part of the mission” [42]. Thus, it is likely that the equation in the supplement is simply missing a negative sign in the
 exponent. As a result, the 3 mK drift, discussed by Mather
et al., can be attributed to Ical [42] along with errors of 18 mK for temperature diﬀerences between thermometers. In addition, as demonstrated in Figure 6, the emissivity modeled for Ical can exceed the theoretical upper limit of 1 over much of

the FIRAS frequency range. This illustrates that the calibration model adopted by the FIRAS team contains signiﬁcant shortcomings.
2.3.3 Achieving a sky null
As represented in Figure 1, FIRAS functions as a diﬀerential spectrometer, wherein the sky or the external reference, Xcal, are being constantly compared to an internal reference blackbody, Ical. When the system is functioning properly and all temperatures are equal, then a perfect null should be measured in the interferogram. This should take place whether 1) the sky is being compared to Ical set at the temperature of the sky, or 2) the external reference calibrator, Xcal, is being compared to Ical set at the same temperature.
Once COBE ﬁnally reaches orbit, the ﬁrst ﬁnding is that FIRAS is unable to achieve a null when the internal reference Ical is set to the sky temperature. This is demonstrated in Figure 7 [32]. Years later, the FIRAS team discuss the situation: “If both the sky and the Ical were blackbodies, and the interferometer were perfectly symmetrical, one could in principle null the signal from the former simply by adjusting the temperature of the latter. The temperature of the CMBR could then be read from the reference body thermometers. Unfortunately, neither of those conditions prevails” [38]. The FIRAS team continues: “Our Ical and instrument asymmetry com-

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Fig. 6: Calculated emissivity for Ical as a result of calibration reproduced from the FIRAS Explanatory Supplement [40]. Note that emissivity exceeds 1, the theoretical maximum, at many frequencies. Reprinted with permission of John Mather.
 bine to produce a net reﬂectance of 4%, and Galactic emis-
sion from gas and dust contributes to the observed signal. To measure these eﬀects, we must calibrate the instrument” [38]. Note that since the sky temperature would end up being assigned as 2.725 0.001 K [43], the upper trace in Figure 7 indicates that the null point appears with Ical at nearly 34 mK
= above the sky temperature (2.759–2.725 K 34 mK). Conse-
quently, COBE is faced with a 34 mK systematic error based on this fact. It is not clear how much of this error can be attributed to Galactic emissions. These should be primarily sensed at frequencies beyond 20 cm  [23], the cutoﬀ of the low frequency channel [38]. As such, it is doubtful that galactic contributions can fully account for the lack of a proper null in these channels. By the end of the mission, Ical is spending
 most of its time near the null, at 2.758 K and toggling to a  temperature 12 mK higher, 2.770 K [40, p. 28]. The FIRAS
team writes: “In addition, the temperature of Ical was toggled between a “sky null” setting to a setting 12 mK hotter, every 3–4 days, to allow instrumental gain errors to be distinguished” [40, p. 19]. The latter is 45 mK above the temperature reported for the microwave background.
Unable to attain the expected null, the FIRAS team begins to target instrumental problems and calibration [38]. They do not envision that a null could not be achieved, because the sky was not acting as anticipated. Consider, for instance, that the Earth is producing the microwave background and that its diﬀracted signal is coming over the shield of the satellite. In this case, one can assume that the Earth was producing a signal with a nearly perfect Planckian [10] shape. But, at lower frequencies, the microwave background will experience more diﬀraction at the shield. Hence, FIRAS will be most sensitive to low frequency signals. As frequencies are increased, progressively less diﬀraction will occur at the shield and the FIRAS horn will become more forward directional. In so doing, it will be less sensitive to signals arising from beneath the shield. Thus, FIRAS may not sense a true Planckian curve,

Fig. 7: Interferograms obtained in ﬂight with the FIRAS instrument, as reproduced from [32]. The upper trace demonstrates the null con-
= dition between the sky (ﬁnal reported temperature 2.725 0.001 K
[43]) and Ical set at 2.759 K. This trace is not plotted with the same vertical scaling factor as the one displayed in the central portion of the ﬁgure. Such a plot creates the illusion that a better result was achieved than actually obtained. The middle trace displays the interferogram recorded when Ical was set at 2.771 K. This indicates the magnitude of signal “oﬀ the null”. The bottom interferogram was measured when comparing the two calibrators set at nearly the same
= = temperature (Xcal 2.759; Ical 2.750). A null should have been
obtained under these conditions, but did not occur. Once again, the vertical scale does not correspond to that used for the central trace. A correction of a factor of 3–5 should be applied to place the upper and lower interferograms on scale with the central one. This was not mentioned in the original text [32], but points to deviations from the theoretically expected results. Reproduced by permission of the AAS.
but a distorted spectrum displaying too much signal at the lower frequencies, and not enough signal at the higher frequencies. There may be less than the expected signal instensity along with constructive/destructive interference eﬀects. The situation is illustrated schematically in an exaggerated fashion in Figure 8. This scenario would make it impossible to reach a null. The issue is not simply a question of temperature, but of lineshape. If two signals, arising from the sky and Ical, do not have the same lineshape, they can never be nulled. A proper null is never displayed. The underlying cause cannot be ascertained, given the nature of preﬂight testing, instrumental drift, and incoming signal.
In re-examining Figure 7 [32], note that the trace determining the null point is not a good null. The top trace in this ﬁgure is not plotted on the same scale as the bottom two traces, as can be deduced by examining the noise power. It needs to be multiplied by a factor of 3–5 to match the noise seen in the central trace. This gives the illusion that a better null is achieved than is actually obtained in practice. The second trace has much more noise. In fact, an analysis of noise

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Fig. 8: Schematic representation of an ideal blackbody at 2.725 K (solid line). The dashed line is an exaggerated representation of the distortions that might occur if an earthly signal was diﬀracting over the FIRAS ground shield. Since diﬀraction might be expected to have the greatest eﬀects at the lowest frequencies, the points in this region would be elevated. Conversely, as frequencies are increased, less diﬀraction should occur oﬀ the ground shield. The FIRAS horn should become more forward directional at elevated frequencies. As a result, a decreased signal might be sensed in this region. It is difﬁcult to deduce the exact appearance of the eﬀects from diﬀraction. For instance, there could actually be signs of constructive and destructive interference on the acquired spectrum. The nature of the spectrum acquired by FIRAS would also depend on the extent that the sky signal was diﬀracting into the FIRAS horn during calibration with Xcal, due to leakage. In the limit of severe leakage, FIRAS would report a perfect blackbody spectrum from the sky, even with diﬀraction occurring at the ground shield. Further details are provided in the text.

Fig. 9: FIRAS interferogram acquired between the sky and Ical, as reproduced from [38]. The signal is being generated just slightly “oﬀ the null”. Apparently, the point at 355 can be perfectly nulled [38], but it is doubtful that such a result can be obtained while maintaining the null condition over all other points. The FIRAS team does not present a perfect null. A spectrum acquired “just oﬀ the null” yields little scientiﬁc information. Reproduced by permission of the AAS.
ment design [38]. This is because they believe that a perfectly Planckian background must be found in the sky in front of FIRAS. The idea that an ideal blackbody spectrum, produced by the Earth, could have been distorted by diﬀraction over the shield, is not entertained. As a result, they cite that the Ical provides a 4% reﬂectance, to partially account for the lack of a proper null [38].
2.3.4 Achieving a null when TIcal = TXcal

power from these traces establishes that the FIRAS team is not maintaining a constant vertical ampliﬁcation. This should not have escaped the eye of the reviewers. Correct scaling factors should have been provided in the ﬁgure legend.
In any case, the null is not clean. The FIRAS team, for instance, shows a second interferogram in Fixsen et al. [38], reproduced herein as Figure 9. In the ﬁgure legend, they state that the peak at 355 can be nulled within detector noise levels. However, they fail to demonstrate the corresponding interferogram. It is certain that the point at 355 can be nulled. But, it is essential that all the points in the spectrum are simultaneously nulled. The FIRAS team has never been able to present such an interferogram. Moreover, if a proper null exists, they should not display data “just oﬀ the null”. These interferograms are not useful as measures of instrument performance. The issue is not simply one of temperature match. For, if two blackbodies are brought to the same temperature, then ideally, the null must be perfect. Lineshape diﬀerences, generated by diﬀraction on the shield, could account for the discrepancies noted.
Unable to reach a perfect null with the sky and dismissing lineshape eﬀects, the FIRAS team is left to implicate instru-

In analyzing the bottom trace in Figure 7, it is evident that a null cannot be achieved, when Xcal is set at nearly the
= = same temperature as Ical (Xcal 2.750 K, Ical 2.759 K).
Unfortunately, the FIRAS team does not publish a suﬃcient number of interferograms to enable the complete dissection of this question. On the surface, failure to locate a null, when
= 6Ical 6Xcal, would support the idea that the problem was in-
strumental. After all, a second failure to establish a solid null is being reported. The FIRAS team might have been able to supply proof of this contention, using a combination of interferograms with Xcal and Ical at diﬀering temperatures. As it is, no proof exists that Ical was the sole problem with FIRAS.
= Again, failure to attain a null, when 6Ical 6 , Xcal could also
be supported by technical issues with leakage around Xcal. It is vital to understand the exact temperatures for Xcal
and Ical, when a null spectrum is achieved by the two calibrators. However, such data is not presented by the FIRAS team. Furthermore, it is not certain that they were ever able to obtain a null. In order to properly address this issue, the critical data is found in the null spectrum between Xcal and Ical on the ground. It is not known if the null imbalance was documented for FIRAS using preﬂight tests. The data have

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not been published, but are critical to understanding the inability to reach a null between the sky and Ical, as discussed in section 2.3.3. Without it, the FIRAS team cannot defend the hypothesis that galactic contributions, for instance, were responsible for this shortcoming. It is obvious that the galaxy may not be invoked for the lack of a null between the two reference blackbodies. Therefore, for a proper evaluation of these questions, ground data, obtained between Xcal and Ical, should be provided.
2.4 Data processing
Initially, the FIRAS team publishes a spectrum from 1–21 cm  [32]. That spectrum was said to deviate from the intensity of a blackbody by less than 1%. Then, in 1994, Mather et al. [35] advance a new set of data, wherein the intensity deviates from a blackbody by less than 0.03%. The error bar in setting the absolute temperature, using Xcal, drops precipitously from 60 mK to 10 mK (see Table 1). Fixsen et al. [39], in 1996, then report that the “rms deviations are less than 50 parts per million of the peak of the cosmic microwave background radiation”. In 1999, Mather et al. apparently again increase the rms deviation and assert that the deviation of the CMB from the theoretical blackbody is less than 0.01% [42]. Finally, in 2002, Fixsen and Mather [43] advance that “the measured deviation from this spectrum are 50 parts per million (PPM, rms) of the peak brightness of the CMBR spectrum, within the uncertainty of the measurement”. Using technology established in the 1970’s, the FIRAS team reported a spectral precision well beyond that commonly achievable today in the best radiometry laboratories of the world.
Figure 2 [39] is famous for the observation that the uncertainties are a small fraction of the line thickness. This ﬁgure is unusually drawn, as the frequency axis is oﬀset. This makes it less apparent that data is not being shown below 2 cm . The ﬁnal result was obtained with the calibration procedures outlined by Fixsen et al. [38]. In the end, the FIRAS team transfers the error from the spectrum of interest into the calibration ﬁle, as will be discussed in detail below. Using this approach, it would be possible, in principle, to attain no deviations whatsoever from the perfect theoretical blackbody. Given enough degrees of freedom and computing power, errors begin to lose physical meaning. The calibration ﬁle became a repository for everything that did not work with FIRAS. The only problem was that it was now impossible to dissect what the FIRAS microwave background spectrum really looked like. Along these lines, the most serious concern was the omission of data, as discussed in section 2.4.3.
2.4.1 FIRAS calibration
In order to provide data for in-ﬂight calibration, the FIRAS team controls the temperature of four key sources of emission, 1) the internal calibrator, 2) the external calibrator,

3) the sky horn, and 4) the reference horn. The emissivity
of each of these devices could be modiﬁed on demand in the
temperature range from 2–25 K [38]. Other parts of the in-
strument are approximated as Planckian functions [10], pre-
sumably because they are isothermal [38]. Cheng describes the calibration process: “Calibration is accomplished by removing all known instrument eﬀects from the raw spectra. This requires a model of the instrument, with all known imperfections, and suﬃcient calibration data to establish the model parameters. The measured instrument state for the sky data can then be used to predict the instrument characteristics based on the model which is then used to calibrate the sky data. . . The emissivity of various internal components in the instrument are determined by varying their temperatures while observing a constant input signal (e.g. from the external calibrator). These components include the sky horn, reference horn, internal reference load, dihedral mirrors, collimator, and the detector itself. The temperature of the ﬁrst three components can be varied by command so that determining their emissivity is straightforward. The emissivity of the other components are determined by temperature variations during several cryostat temperature transients which occurred early on in the mission” [34].
A critical aspect of the calibration procedure is that the ex-
ternal calibrator, Xcal, is treated as providing a perfect black-
body signal to the rest of the instrument. This approximation
may not be justiﬁed, given the discussion in section 2.2.3.
There are also complications, if the seal between the horn
and the calibrator is not perfect, due to vibration, as addressed
in section 2.2.4. The idea of approximating the thermal be-
havior of the dihedral mirrors, collimator, and detectors with
Planck functions, as Fixsen describes [38], does not rest on
solid grounds. Each material should ideally have been mea-
sured in the laboratory, as real materials do not behave as
blackbody sources [80]. For instance, the FIRAS team de-
scribes harmonic responses in the instrument when radiation
passes through the system more than once. This proves that
the interior components of the instrument cannot be modeled
as perfect blackbodies. They do provide reﬂective surfaces.
 It is noted that 20% of the input signal fails to reach the
output [38]. This is a large number, which represents fre-
quency dependent losses. However, no frequency dependence
is mentioned, presumably because the loss for each interfer-
ogram cannot be dissected in these terms. Both second and
third order harmonics were thought to be signiﬁcant at the
0.1% level [38]. They also report that the frequency scale
for FIRAS does not quite agree with that determined using
known spectral lines. In order to correct the situation, they make a 0.5% adjustment with “the remainder being absorbed by a 4 mK adjustment in the absolute temperature scale” [38].
The discussion relative to the bolometers highlights how
modeling can misrepresent the actual behavior of a device. The FIRAS team writes: “The total of nine parameters with their uncertainties and covariance matrix were determined

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from these tests. The agreement with the determination of the parameters from the FIRAS in-orbit calibration is poor, with normalized  ’s of 80 to 800 in various ﬁts for 9 DOF (degrees of freedom). This is probably due to a deﬁcit in the bolometer model” [38]. In the ﬁnal analysis, the in-ﬂight cal-
ibration procedure is viewed as correct, and the disagreement
with pre-ﬂight data appears to be disregarded. This demon-
strates how the COBE calibration procedures have become
essentially detached from any experimental ﬁndings recorded
on the ground before ﬂight.
The calibration process brings many more degrees of free-
dom for setting error bars and temperatures. Mather et al. thus write: “However, the calibration process corrects other eﬀects of the error to the ﬁrst order. . . ” [42]. Calibration involves: “comparison of the sky with an ideal movable external blackbody calibrator (Xcal) that can ﬁll the aperture of the sky horn. The rest of the calibration process is used to measure gains and oﬀsets that apply if the calibrator spectrum does not match the sky spectrum” [43]. As a result,
the FIRAS team can achieve a perfect ﬁt to the sky spec-
trum. They have suﬃcient degrees of freedom to accomplish
the task by invoking the calibration procedure. The inversion matrix required for the calibration ﬁts is “of such large
 rank ( 4,000)” that it “is not generally tractable” [38]. The
FIRAS team was “able to invert this matrix by taking advantage of its special form. . . This made inversion possible, though still not speedy” [38].
Relative to error analysis, very large degrees of freedom (DOF) were invoked. The FIRAS team writes: “The normalized  resulting from this ﬁt is 2.8218 (27873 DOF) for the left low detector, short slow stroke data (2.27 <  < 21.54 cm 1), and 4.53 for (159353 DOF) for the right high detector, short slow stroke data (2.27 <  < 96.28 cm 1)” [38].
Moreover, it can be deduced that the values are rather high for  /DOF, particularly when operating away from the null position. Cheng [34] reports higher than expected  /DOF
values, of 4 to 10, for the low and high frequency channels
when discussing the calibration data. Apparently [34], it is
only when considering calibration ﬁles near the null condition that  /DOF values near 1 are reached [39]. Of course,
it is easier to ﬁt data near the null, for the precise reason that
the spectrum contains little power in this range. It is solely
by examining the performance of the calibration model away
from the null, that any real insight can be harnessed relative
to the reliability of this method. However, such data appears to give even higher  /DOF values than obtained near the
null [34]. This is not a good sign, relative to the validity of this approach. The inability to ﬁnd good  /DOF values oﬀ
the null might be reﬂecting leakage around Xcal, for instance.
This could become more apparent when Xcal and Ical are at
very diﬀerent temperatures. Fixsen et al. [39] do describe excellent  /DOF perfor-
mance in their Figure 1 (not reproduced herein). An analysis of Table 1 in [39] reveals that  /DOF are generally on

Fig. 10: Plot of various error terms for the FIRAS high frequency channel for a typical sky point, as reproduced from [38]. Separate ﬁts are obtained for each point in the sky. This allows for far too many degrees of freedom in the FIRAS calibration stage. Curve D represents the error arising from detector sensitivity. Note the reso-
 nances at 7, 16, and 20 cm . These may correspond to CO lines
in the galaxy. Such resonances should not be found on functions representing detector sensitivity. They are not found in the detector functions at low frequency [38]. The dashed line, which is not labeled in the original work, represents the calculated errors from the galaxy as can be established using Figure 13. Note that there is little error contribution from the galaxy, below 20 cm . As such, the FIRAS team cannot attribute the failure to achieve a proper null to the presence of contaminating galactic signal in this frequency region. The dotted line, PEP, accounts for error associated with various temperatures within the instrument. Once again, a resonance
 line is observed at 7 cm . Such a resonance line should not be
found on this function. It would, however, permit the FIRAS team to vary the error in this region when trying to correct for contributions from galactic CO. PTP accounts for errors in the absolute temperature scale. PUP error depends on the absolute temperature state of the instrument and is most sensitive to Ical. PUP and PTP are given a blackbody appearance without proper justiﬁcation by the FIRAS team (see text for additional details). Reproduced by permission of the AAS.
the order of 2 or more. Nonetheless, it is noticeable that the  /DOF, listed in this work (see Table 1 in [39]), have improved substantially over those found 2 years earlier (see Table 2 in [38]). It is not clear if this represents anything but better insight into how  /DOF values could be minimized. In the end, there is too much ﬂexibility in these approaches. This places at risk all physically meaningful experimental ﬁndings, reﬂecting systematic errors.
A treatment by Fixsen et al. [38] of the error terms for FIRAS reveals that the FIRAS team considered nearly every possible source of instrumental contribution, while discounting the possibility that errors existed in the shape of the blackbody provided by the sky itself. Such a systematic error could exist if diﬀraction eﬀects were important.
Figure 10 is a reproduction of Figure 9b in [38]. For

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the low frequency channel (ﬁgure not displayed), the ma-
jor term is referred to as PTP. It represents the uncertainty
in the absolute temperature scale. The peak brightness of a
2.7 K blackbody is approximately 120 ergs cm s  sr 
cm [38]. As a result, this error term absorbs about 0.5% of
the deviation from the peak of a blackbody. The most impor-
tant error term for the high frequency channel, D, accounts
for detector noise. The PUP error is linked to the temper-
ature state of the instrument and is primarily dependent on
Ical. The PEP error depends on the temperatures of various emitters in the instrument. “These are: Ical 2.76 0.006 K, MTM 2.0 0.4 K, horns 2.75 0.005 K, mirrors 1.56 0.02 K, and bolometers 1.52 0.017 K” [38]. The FIRAS team writes
that the PEP and PUP error terms are well approximated by
Planckian functions. This claim, however, is without foun-
dation. In fact, there are no references provided for assign-
ing a Planckian shape [10] to either PTP or PUP. Assigning
such shapes to these two terms will help determine the ap-
pearance of the other terms. The entire procedure is with-
out scientiﬁc basis [80]. It is particularly concerning that the
FIRAS team generates such error functions for each point in
the sky. Instrument error should not be dependent on the
scan direction. At the same time, it is true that the instrument experiences temperature ﬂuctuations over time: “Further tests of the calibration are obtained by searching the calibrated map of the sky for features relating to changes of the instrument state. The largest such changes occurred during the time from 1990 May to August. In this time period, it was impossible to keep both the Earth and the Sun below the Sun screen, and the Earth illuminated the top of the instrument during part of the orbit. The data taken with the Earth above the instrument were rejected in the maps, but the thermal transient produced by the heat of the Earth was large and long. As a result, we raised the set point of the horn temperature controllers to as high as 6 K to achieve stability” [38]. Direct visualization of the Earth did impact the
COBE results, but the data were rejected. Yet, if the Earth
was truly silent over the frequency of interest, there could
be no reason to reject this data. Heating by the Earth could
simply be accounted for in a manner similar to that used
for other parts of the orbit. The FIRAS team believes that
the heat transient in the instrument, as a consequence of di-
rect infrared heating, was the only eﬀect. However, it would
have been most interesting to examine the resulting sky in-
terferograms. Perhaps these actually contained direct phys-
ical proof that the Earth had emitted the microwave back-
ground.
In any case, note the nature of the error term, D, for the
high frequency channels. Essentially, there are resonance
   lines at 7, 16, and 20 cm . These features seem to
correspond to the presence of the CO lines in the galaxy [39].
Such lines should not be found within detector noise error. In
addition, curve D for the high frequency channels approaches
10 ergs cm s  sr  cm, at 95 cm . This is an extremely

Fig. 11: Calculated residual errors in the microwave background, as reproduced from [35]. These residuals were generated, using a con-
servative approach, by increasing the statistical errors, forcing 
to 32 [35]. Nonetheless, note the systematic increase in the residuals beyond 15 cm . There is a slight trend towards signal loss in this region as well. In addition, the points below 5 cm , slowly begin to rise away from the reported temperature, and represent signs of excessive signal in this region of the spectrum. The residuals are presented once again in 2001 [44]. At this time, systematic variations have been absorbed by the calibration ﬁles and the residuals are now random and of insigniﬁcant importance. Reproduced by permission of the AAS.
powerful contribution from this term, given that the maximal power of the microwave background itself is on the order of
120 ergs cm s  sr  cm.
2.4.2 Analysis of residual errors
When Mather et al. [35] publish the 1994 FIRAS data release, several unexpected ﬁndings are revealed. Figure 1 of this work [35], a presentation of the CMBR residuals, is reproduced as Figure 11. There are two interesting aspects of this ﬁgure. First, there is a pronounced increase in the error bars associated with the residuals, as the frequencies are raised beyond 15 cm . This increase in variability is systematic, and consequently may represent a real ﬁnding. In fact, there is a slight trend towards decreased temperatures as a function of frequency beyond 15 cm . Second, at the lower frequencies, the data points begin to rise. The FIRAS team comments as follows: “pending further detailed study of possible instrument faults at these low frequencies, we cannot speculate on their nature. We emphasize that the size of the apparent deviations is greatest at those frequencies where diﬀractive eﬀects, interferogram baseline curvature, and very low spectral resolving power and wide spectral sidebands cause the greatest diﬃculties in calibration” [35]. The authors therefore “conservatively increase the statistical errors
by a factor, forcing  to exactly 32, the number of degrees of
freedom in the ﬁt” [35]. Nonetheless, they eventually publish

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new residuals [44], which have now lost the systematic variations displayed in Figure 11. This shows the power of the ﬁtting methods applied.
The FIRAS team believes that they fully understand all systematic errors and that their ﬁts are justiﬁed. However, this is not the case. The fact that an excellent ﬁt can be found, given suﬃcient degrees of freedom, is well recognized in science. The question remains how well justiﬁed were the bases for the ﬁts. Adequate justiﬁcation is based on a complete understanding of the instrument on the ground with calibrated test procedures. This approach was not utilized. Instead, ﬁts are obtained by adjusting gains, oﬀsets, and functions, which have a weak foundation, other than their ability to result in minimal residual errors for the sky. Furthermore, the FIRAS team has not shown that it can minimize residuals, using their ﬁnal calibrations across all ranges of temperatures for Xcal, Ical, the sky horn, and the reference horn. Without explicit demonstration that the ﬁnal calibrations apply to all possible interferograms, the analysis of residuals for the sky alone have little value. It is a complement of all residuals, for all conditions, which is important to visualize, for this alone might help establish the reliability of the approach in the absence of suﬃcient pre-ﬂight testing.
2.4.3 Data omission
The FIRAS data set from 1994 contains a more serious concern: all of the observations at frequencies below 2 cm  are now excluded [35]. Moreover, there is a rise in the residuals below 4 cm  which cannot be accounted for by their error bars. This region is usually the easiest to monitor due to the low frequency range. Never again is the data below 2 cm  re-included in the FIRAS data set. It is only through reading the accompanying calibration work by Fixsen et al. [38], that one might postulate on the causes behind the loss of this data. A single sentence is presented when discussing the reference horn: “However, the measured emission is higher than predicted, particularly at the lowest frequencies” [38].
Though FIRAS was designed to cover the region from 1–2 cm , the FIRAS team omits the data below 2 cm  and ignores the excessive signal. They do not discuss the cause of this anomaly, unless Wilkinson’s concerns about earthshine were a reaction to this problem [74]. At the same time, given the use of calibration ﬁles to correct FIRAS, it may have been that the FIRAS team could not envision a means to account for the spectral behavior below 2 cm . On the surface, ignoring this data might not appear so serious. After all, the entire spectrum beyond 2 cm  was reported.
Given that diﬀraction of a terrestrial signal would produce distortions in the measurement of the microwave background, which include excessive signal at low frequencies and decreased signal as frequencies increase, the dismissal of this data cannot be taken lightly. The FIRAS team also forsakes all data acquired when the Earth was directly illuminating

FIRAS [38], as previously discussed in section 2.4.1. While infrared heating of the instrument did occur at this time, it is not evident that such heating could not be modeled. This is the type of evidence that may have pointed to an earthly source for the microwave background.
2.4.4 Error bars
Despite the presence of systematic errors, the FIRAS team is able to essentially sidestep the recordings of their thermometers and overcome their inaccuracy. E. S. Cheng summarizes the overall approach of the group: “Since the FIRAS is a far more sensitive thermometer that the GRT’s (germanium resistance thermometers), especially at temperatures above 3 K, the thermometer readings can be adjusted, using the calibration data, to provide maximal internal consistency and a reﬁned temperature calibration” [34]. As such, the readings of the physical thermometers could be given less weight.
Initially, it is not evident if they are aware that errors in the thermometers limit the ultimate temperature that can be reported for the microwave background. In 1996, Fixsen et al. arrive at a microwave background temperature of 2.730 0.001 K (see Table 1), which relies on Xcal (see page 581, section 4.1, in [39]). Then, three years later, in 1999, the FIRAS team writes: “A 5 mK error in the temperature determination of Xcal leads directly to a 5 mK error in the temperature determination of the CMBR” [42]. The team apparently realized that it was impossible for Fixsen et al. [39] to claim a 1 mK error bar for this measurement in 1996. But, they continue to discount the 18 mK error between the Ical thermometers [38].
In order to fully restrict the error bars on the determination of the microwave background, the COBE group therefore moves to adopt two additional methods which, at least on the surface, are independent of Xcal. In the ﬁrst instance, they determine the temperature by calibrating the frequencies of the background, using lines from CO and C+ [39]. Few details are provided relative to this approach; however, it may rely on accurately deﬁning a Wien maximum and extracting the temperature from Wien’s law [11]. The method is solid, on the surface at least. Nonetheless, it will depend on correctly setting the peak in the microwave background data, which may in turn depend on Ical and/or Xcal. The ability to detect a proper Wien maximum [11] would also be sensitive to interference eﬀects caused by diﬀraction on the COBE shield, should the signal originate from the Earth. As a result, it is not clear that the frequency method holds any less systematic error than that directly relying on Xcal.
Alternatively, the group also uses the existence of a dipole to extract a monopole temperature [39]. In this way, they can build on the ﬁndings of the DMR relative to the dipole value [46–49]. Once again, the method may appear more accurate, but is also subject to many of the same problems as that based on Xcal. If the use of frequency calibration, or of

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the dipole, seems less prone to systematic error, it may simply be because these have escaped detection by the FIRAS team. It is well established, not only in physics, but across the sciences, that systematic errors can be extremely diﬃcult, even impossible, to detect [88]. Consequently, one must not dismiss those systematic errors which are evident.
Using a combination of these three methods, the FIRAS team ﬁnally arrives at a microwave background temperature of 2.725 .00065 K [43]. Beyond undetected systematic errors, this number circumvents much of the planning built into Xcal and Ical. It also neglects the excessive signal detected below 2 cm . Relative to error bars, the result obtained, using an average of many methods, was analogous to ignoring the existence of known temperature error in the reference calibrators Xcal and Ical. The existence of imperfect nulls was also dismissed, as were all interferograms obtained while the Earth was directly illuminating FIRAS.
In the absence of proper pre-ﬂight testing, it is impossible to account, with certainty, for all possible source of systematic errors associated with inability to ﬁnd a null. Data processing methods do not address the fundamental issue. The FIRAS team believes that it has fully understood all systematic errors and that they can be removed from the ﬁnal error report. But, systematic errors are best treated through the proper design and testing of scientiﬁc instruments on the ground. This was not achieved. The calibration procedure creates the illusion that all systematic error can be taken into account, after completion of data acquisition. This is not a prudent approach to systematic error, especially since they can be nearly impossible to identify [88, p. 93–95]. It is best to report all known systematic errors within the ﬁnal error bar.
In failing to achieve a clear null, FIRAS is pointing to something on the order of a 34 mK error. The overall error in
 Xcal was 5 mK. The error diﬀerence between the Ical ther-
mometers is 18 mK and the drift for Ical is 3 mK. A frequency
 correction of 4 mK exists. Some of these errors may be
related and could be added quadratically [88, p. 93–95]. Di-
 rect addition provides a worse case scenario of 64 mK [88,  p. 93–95]. As such, using direct addition, 64 mK appears to
be a good lower limit on the accuracy of the FIRAS data set, from 2–20 cm . This treatment would discount attempts to lower the error bar to 1 mK in the ﬁnal FIRAS report [43]. In
 fact, 64 mK is not far from the 60 mK error initially used
by the FIRAS team [32]. At the same time, the group asserts that their data is “indistinguishable from a blackbody” [37]. A cursory examination would suggest that this was the case (see Figure 2). An understanding of calibration process has provided the explanation.
2.4.5 The optical transfer function
The FIRAS team ﬁrst presents the optical transfer function in the Explanatory Supplement, in 1997 [40]. This function is critical in processing FIRAS data ﬁles [40, p. 50] and it is

Fig. 12: Illustration of the Optical Transfer Function for FIRAS, as reproduced from the Explanatory Supplement [40]. The features near 20 cm  are due to the position of the ﬁlter cutoﬀ. Nonetheless, this does act to provide a substantial correction for signal beyond the Wien maximum and between 15 and 20 cm . Note the oscillation present below this frequency range. It is not clear why such features should be present on this optical transfer function. These might represent the eﬀect of constructive and destructive interference. It is impossible to truly ascertain their cause with the data provided. Most importantly, the optical transfer function is decreasing exponentially. This is not characteristic of a properly functioning spectrophotometer. This ﬁgure reveals that the FIRAS instrument is
 suboptimal, beyond 30 cm . Reprinted with permission of John
Mather.
reproduced herein as Figure 12. For an ideal spectrometer, the optical transfer function would be unity over the entire frequency range. That is, for every photon which enters the system, one photon is recorded by the detector. This situation does not occur in practice, and transfer functions will deviate from ideality. But, the transfer function for FIRAS is much
< less than ideal. At the lowest frequencies ( 20 cm ), the
transfer function contains a very strange and unexplained oscillation. The FIRAS team does not comment on the cause of this feature. Nonetheless, since the reciprocal of the transfer function is used to process data, this oscillation is signiﬁcant. Although diﬃcult to ascertain, this feature might be a sign of signal diﬀraction into the horn. In any event, the discontinuity near 20 cm  is due to the ﬁlter cutoﬀ between the low and high frequency channels.
The most noteworthy feature of the optical transfer function for FIRAS is that only 1 photon in 10 is being detected, at best. In addition, the plot is on a logarithmic scale. Such behavior is highly unusual and demonstrates that the FIRAS instrument is not linear. It is also not sensitive at the higher frequencies. As a result, when the optical transfer function is applied to process data beyond 30 cm , it results in a pronounced ampliﬁcation of spectral noise. This is revealed in Figure 13 [41], where noise in the ﬁts is ampliﬁed beyond 40 cm . This constitutes a solid illustration that the FIRAS instrument, for practical purposes, is subfunctional in this frequency range.

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the calibrator and that was not at the CMB temperature is the small aperture leading to the spectrometer” [42]. Such a statement cannot be justiﬁed. It is not clear that the sky is at the temperature of the CMB. Should the signal originate from the Earth, it would undergo diﬀerential diﬀraction as a function of frequency, as it travels over the RF shield and into the horn. This would lead to a spectrum which is not blackbody, and the measured sky spectrum would not be at the exact temperature of the microwave background. It would be distorted. Fixsen and Mather cannot assume that the sky is a blackbody at the temperature of the CMB. That is what they are trying to determine.
Work by Battistelli et al. [90] is centered on a computational analysis of Xcal, in order to further reﬁne cosmological parameters. The text does not constitute a criticism of FIRAS. The emissivity values obtained for Xcal, are nearly ideal. Salvaterra and Burigana [91] examine a range of issues in detail, but the text does not raise any real concerns relative to FIRAS.

3 The Diﬀerential Microwave Radiometers (DMR)

Fig. 13: Fit spectra calculated across the high frequency region using the FIRAS instrument, as reproduced from [41]. Note the tremendous increase in random errors beyond 30 cm . This indicates that the spectrometer is suboptimal, in this frequency range. Reproduced by permission of the AAS.
2.4.6 Comments made by other authors
Several Italian authors [89–91] have been interested in the calibration of the FIRAS instrument as Fixsen and Mather highlight [42]. Giorgi, for instance, suggests that there could be an asymmetry of as much as 5% in the two input arms of FIRAS [89]. Fixsen and Mather point out that the measured asymmetry is only 1–3% [42]. In defending FIRAS data, Fixsen and Mather write: “However, one must also consider the source of any reﬂection. The Xcal is part of a closed cavity composed of the calibrator, the sky horn, a small gap between the calibrator and the sky, and a small aperture leading to the spectrometer horn. Consequently, the radiation reﬂected by the calibrator must have originated either from itself, the sky horn, the sky through the gap, or the small aperture to the spectrometer. Three of these sources are effectively at the temperature of the CMB. As the most emissive of the four, the source of most of the reﬂected radiation is the calibrator itself. . . Moreover, since both the horn and the Xcal temperatures were set to match the CMB temperature, the only source of radiation that could be reﬂected by

The COBE satellite is also equipped with Diﬀerential Mi-
crowave Radiometers, the DMR. These constitute three pairs
of narrow band antennae operating at 31.5, 53, and 90 GHz
[46]. The DMR are mounted directly on the sides of the he-
lium dewar containing the FIRAS and DIRBE instruments
[45]. A detailed treatment of the DMR will not be presented,
as many of the issues relative to the DMR have already been
addressed relative to the WMAP satellite [20]. It is clear
that the DMR has measured a dipole. This result is highly
signiﬁcant.
Of all the concerns which the DMR shares with WMAP,
the central issue remains the processing of data and the extraction of the multipoles [20]. These are the “wrinkles on the fabric of time” [21]. Before the multipoles can be an-
alyzed, the signal from both the dipole and galactic fore-
ground must be removed. Importantly, as Smoot discusses
in his popular book [21], these investigators also remove the
quadrupole signal from the underlying maps. It is only at this stage that the multipoles become visible. Smoot writes: “We were conﬁdent that the quadrupole was a real cosmic signal. . . By late January and early February, the results were beginning to gel, but they still did not quite make sense. I tried all kinds of diﬀerent approaches, plotting data in every format I could think of, including upside down and backwards, just to try a new perspective and hoping for a breakthrough. Then I thought, why not throw out the quadrupole — the thing I’d been searching for all those years — and see if nature had put anything else there!” [21, 276–277]. After removing
the quadrupole, the multipoles ﬁnally appeared. Smoot then comments [21, 279]: “Why, I puzzled, did I have to remove the quadrupole to see the wrinkles?”
The answer to this question is one of data processing.

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The raw maps do not contain any systematic signal variations on their own [21, 276–279]. The signals were random in nature. However, when Smoot and his colleagues imposed a systematic removal of signal, they produced a systematic remnant. In essence, the act of removing the quadrupole created the multipoles and the associated systematic anisotropy. Once the quadrupole was removed, the multipoles appeared as extremely consistent variations on the maps. As previously mentioned, these ﬁndings have no relevance to cosmology and are purely an artifact of signal processing. Citing from previous work [20]: “Apparent anisotropy must not be generated by processing”. The sky does have anisotropy. But this anisotropy is likely to remain random, as Smoot initially observed in his data set, before removal of the quadrupole.
4 Conclusion
Through this analysis, unexpected problems with FIRAS and the DMR data have been brought to light. With regard to FIRAS, many issues exist. They include: 1) lack of gain and side lobe characterization for the FIRAS horn, 2) absence of diﬀraction modeling involving the interaction between FIRAS and the shield, 3) rudimentary pre-ﬂight testing, 4) failure to document side lobe performance, in space, at frequencies relevant to the microwave background, 5) inappropriate evaluation of Xcal emissivities, 6) inability to ensure that leakage did not occur around Xcal in ﬂight, given the vibrations present, the lack of gravity, and the nature of the Kapton leaves, 7) existence of a suboptimal transfer function for the instrument, 8) the presence of systematic errors, for the Xcal and Ical thermometers, 9) inability to achieve a proper null between the sky and Ical, 10) inability to reach a proper null between Xcal and Ical, 11) excessive degrees of freedom during the calibration process, 12) lack of justiﬁcation for the error functions PTP and PUP, 13) inappropriate minimization of error bars, 14) omission of data below 2 cm  from all ﬁnal data releases, and 15) omission of data when the Earth was directly illuminating FIRAS.
Given the systematic errors on Xcal, Ical, the frequency drift, and the null temperature, it is reasonable to ascertain that the FIRAS microwave background temperature has a signiﬁcant error bar. As such, an error on the order of 64 mK represents a best case scenario, especially in light of the dismissal/lack of data at low frequency. The report of a microwave temperature of 2.725 0.001 K [43] does not accurately reﬂect the extent of the problems with the FIRAS instrument. Furthermore, the absolute temperature of the microwave background will end up being higher than 2.725 K, when measured without the eﬀect of diﬀraction, and when data below 2 cm  is included. Contrary to popular belief, the FIRAS instrument did not record the most perfect blackbody spectrum in the history of science.
Relative to the DMR, the problems mirror, to a large extent, those I voiced earlier with WMAP [20]. The most

pressing questions are centered on the ability to remove the quadrupole from the maps of the sky. In so doing, it is clear that a systematic residual will be created, which can easily be confounded for true multipoles. In the end, the methods to process the anisotropy maps are likely to be “creating anisotropy” where none previously existed.
It also remains fascinating that the astrophysical community has not expressed greater anxiety relative to the diﬃculties produced by water, in the lower atmosphere. This is perhaps the most serious area of concern. It is certainly true that the Earth is bathed in a ﬁeld with an apparent temperature near 3 K. The existence of the dipole is also ﬁrmly established. Cosmology holds that the monopole signal [1] represents a remnant of creation. Conversely, I maintain, along with my colleagues [5, 7], that it is being produced by the oceans of the Earth. Through this work, it is my hope that others will begin to see that there are legitimate issues with the FIRAS and DMR results on COBE. The thermal emission of water, in the microwave and far infrared, remains incompletely characterized. Our planet has never been eliminated as the source of the microwave background. In the end, the PLANCK satellite [86] should reveal that the Penzias and Wilson monopole [1] was never present in the depth of the Cosmos. The signal belongs to the Earth.
Acknowledgement
John Mather is recognized for granting permission to reproduce ﬁgures on behalf of the COBE team.
Dedication
This work is dedicated to Professor A. J. Christoforidis for the faith he demonstrated relative to my work these many years and for conferring upon me the privilege of becoming a professor of radiology.
Submitted on June 24, 2009 / Accepted on July 03, 2009 First published online on July 16, 2009
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89. Giorgi P.G. Inﬂuence of the angular response on Fourier absolute spectrometry the case of COBE-FIRAS. Infrared Phys. Tech., 1995, v. 36, 749–753.
90. Battistelli E.S., Fulcoli V., and Macculi C. The CMBR spec-
trum: new upper limits for the distortion parameters O and .
New Astronomy, 2000, v. 5, 77–90.
91. Salvatera R. and Burigana C. A joint study of early and late spectral distortions of the cosmic microwave background and of the millimetric foreground. Mon. Not. Royal Astron. Soc., 2002, v. 336(2), 592–610.

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Active Galactic Nuclei: the Shape of Material Around Black Holes and the Witch of Agnesi Function. Asymmetry of Neutrino Particle Density
Gary C. Vezzoli
Department of Science and Mathematics, Lebanon College, Hanover Street, Lebanon, NH 03766, USA Senior Research Consultant Physicist, Institute for the Basic Sciences, West Windsor, VT 05089, USA
E-mail: vezzoli2005@yahoo.com
A mathematical representation is given and physically described for the shape of the very hot material that immediately surrounds a black hole and the warm material located at a greater distance from the black hole, as related to active galactic nuclei. The shape of the material surrounding the black hole is interpreted in terms of asymmetry of the neutrino ﬂux. Detailed experimental measurements on radioactive decay inﬂuenced by astrophysical events are given to support this interpretation.

1 Introduction

Recent work [1] that examined over 200 active galactic nuclei has shown that all have a common shape of the material surrounding the black hole core, and that this shape seems to be independent of the size of the black hole. The Active galactic nuclei (AGN) are cores of galaxies that are energized by disks of hot material that act as ingress/feeder to supermassive black holes. the shape of the hot material that surrounds the black hole was inferred from the observation of x-rays that emanate from very hot material that is close to the black hole, and from infra-red radiation that derives from warm material much further from the core of the black hole.
Through comparing the ratio of x-rays to infrared radiation, the contour shape of the black hole is indirectly mapped [1]. The results are shown in Fig. 1. Inspection of the inferred topology of the surrounding material indicates that although approximate symmetry is shown across the vertical axis, the horizontal axis shows no indication whatsoever of mirror plane symmetry, and thus the upper and lower regions of the 2-d projection must derive from very diﬀerent functional representations. Stars, planets, and moons do not show a signiﬁcant asymmetry, other than equatorial bulge. The non-symmetry of the material surrounding the black hole appears thus at ﬁrst surprising, however, when considered in terms of a collision-induced gravity model [2], the asymmetry could be hypothesized to be a consequence of observing the black hole from a location closer to the centre of the universe where the neutrino ﬂux density is far greater than at position coordinates that are associated with the expansion of the periphery region of the universe, even though that locus of positions is considered unbounded. Asymmetries, such as shown in Fig. 1 are generally thought to be associated with tidal eﬀects — and in the case at hand, this would mean gravitational interactions, such as a form of lensing. Although there is a consideration of the red shift associated with the receding of the galaxies, the cores of which are powered by disks of very hot material “feeding” the supper massive black

Fig. 1: Material shape near black hole. Courtesy of Anna Morton, moderator of 4D WorldX Yahoo Science Groups. See [1].
hole, I do not think that the asymmetry shown in Fig. 1 arises purely from considerations of Relativity, but instead arises at least to some signiﬁcant level from collision criteria [2].
2 Analysis and interpretation
The event horizon associated with a black hole refers to the surface that surrounds the black hole, having the property that any visible light cannot escape from the super dense mass because of the strength of the gravitational ﬁeld [3]. In terms of collision-induced gravity, the term “ﬁeld” is not employed because gravity is considered to be particle-based and the escape-inability of photons at energies less than x-rays is due the increase in collision cross-section between neutrinos and photons that accompanies the super dense packing of mass in a black hole that has developed from a neutron star. The accretion disc of a black hole refers to how accretion onto a neutron star takes place from from a matter input from the Roche lobe of a primary star in the binary system. This passing of matter when occurring from the primary to the secondary star through a Lagrangian point [4] establishes a nonsymmetry, but of a diﬀerent form than that of the black hole shown in Fig. 1, yet these asymmetries may be ultimately related through the physical processes associated with the involvement of the black hole. The vertical asymmetry of the material that surrounds the black hole may also arise from the phenomena that are associated with the periodic ejecta of

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Fig. 2: The Witch of Agnesi function.

material from the black hole, which may be inﬂuenced by the
magnetic properties of the super dense collapsed star. The
comparatively slight asymmetry in the horizontal direction
must relate to inhomogeneous temperatures and non-isotropic
mass distributions of the black hole because of speciﬁc local
conditions at the xyzt spatial-temporal location of the highly
dense aggregating of matter.
The 2-d geometry shown in Fig. 1 above the horizontal
axis that passes through the extrema of the inferred contoured
distribution of mass, shows the appearance of the mathe-
matical function known as the Witch of Agnesi. (The term
“witch” is an involvement of a misnomer, caused by an in-
correct translation of the work of Maria Agnesi who devel-
oped the function geometrically in 1768). The Agnesi func-
tion (Fig. 2) is generally given by, y = [(8a!)=(x + 4a )], where a is the radius of the circle that is utilized to geometri-
cally form the functional curve. In polar coordinates the Ag-
nesi function is given by x = 2a cot , and y = 2 sin . The
function can be generated geometrically by rotating the radius
of the circle whereby the y-coordinate of the function is the y-value of the radial vector as it sweeps the associated circle, and the x-coordinate is the x-value of the ordered pair that
represents the intersection of the extrapolation of the radial
vector with the line, y = a. Although many world class math-
ematicians explored the geometric development of this func-
tion, including Fermat, no application in astrophysics to the
author’s knowledge was established for what became known
as the Witch of Agnesi function, until now — general appli-
cations of the function being conﬁned to probability theory.
Some properties of the Agnesi function are associated
with gravitational criteria, such as the x-squared term appear-
ing in the denominator, and suggestive of an inverse square
relationship, which in Newtonian gravity derives from New-
ton’s postulate of a central force, which he interpreted from
Kepler’s First Law of Planetary Action-namely that the orbits
of the planet must be elliptical from consideration of years of
visual data of Tycho Brahe. The inverse square relationship in the collision-induced gravity model/theory derives from the
properties of a ﬂux, as in the photon inverse-square light intensity fall-oﬀ, or the equivalent for the distance dependency of the amplitude/intensity of magnetic or electrostatic prop-
erties. The relationship of the sweeping rotating radius of the
function-forming circle, and its extension to intersect the line

Fig. 3: Representation of the shape of material near a black hole using Agnesi function contours and quadratic function.
y = a can be arguably topologically associated with the no-
tion of accretion and event horizon, and continuous processes.
The asymptotes of the function (the positive and negative x-
axes) relate to the convergence of the shape of the constituent material as temperature decreases because of distance from the “donut” core of the black hole.
The region of Fig. 1 below the horizontal axis can in 2-d projection be well represented by a wide parabola that opens upward. Thus the combined representation of the 2-d geometry shown in Fig. 1 requires the use of a two-function coalescence, and implies the involvement of two diﬀerent physical phenomena, whereby the quadratic is typically associated with gravitational interactions but the Agnesi function is not.
Using the Agnesi function, and varying the value of the
radius, a, combined with the parabola, y = ax k, where a
 is a very small positive constant 1, the contoured represen-
tation shown in Fig. 3 is readily developed. The knee shaped curve given also in Fig. 3 represents the calculation of volume of integration of the region surrounding the black hole as
a function of the position coordinate, x, showing a threshold
eﬀect above which the volume increases rapidly with high slope. The volume function involves an arc tangent term which which is consistent with involvement of an event horizon.
It has been proposed [5] that when emission from an inner accretion disk around a black hole is occulted by a companion star, the observed light curve becomes asymmetric at ingress and egress on a time scale of 0.1–1 sec. The lightcurve analysis is claimed [5] to provide a means of verifying the relativistic properties of the accretion ﬂow which is based on both the Special Relativity and General Relativity that is associated with black holes. It is reported [5] that the “skewness” for the eclipsing light curve is approximately zero for what are called slim disks because the innermost part is selfocculted by the outer rim of the disk. This self occulting is a very important property of the black hole, yet these criteria do not uniquely and exclusively seem capable of explaining the major asymmetry shown in the geometry inferred from the x-ray and infra-red data [1] given in Fig. 1.

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Fig. 4: Radioactive decay data for Po-210 during August-September 2001 measured at Harvard University using the Rad-7 solid-state detector.
On the other hand, it has been reported [6] that propagation of fermions in curved space-time generates gravitational interaction due to the coupling of its spin with space time curvature connection, and causes a CPT violating term in the Lagrangian, generating an asymmetry between the left-handed and the right-handed partners under the CPT transformation. (CPT refers to charge conjugation, space reversal, and time interval, and thus deals with parity). It is interpreted [6] that in the case of neutrinos this property can generate neutrino asymmetry in the Universe, causing the dispersion energy relation for the neutrino and its anti-neutrino to be diﬀerent giving rise to diﬀerences in their number density, and associated with the left-hand helicity of the neutrino. These eﬀects may have an inﬂuence in contributing to the asymmetry shown in Fig. 1. It has also been shown [7] that particle interactions in the black-hole accretion disks cause an excess production of positrons as compared to electrons, however, this disparity alone, without emission directionality considerations, does not constitute a non-conservation of parity.
Although the behaviour of each type of galaxy or AGN is dependent upon the angle of observation relative to the accretion plane of the black hole core, the asymmetry shown in Fig. 1 is common to all 200 AGN’s that were studied in [1], yet the angles of observation relative to the accretion zones had to be diﬀerent, and the azimuths from the observation coordinates also had to be diﬀerent.
Our own work [8] has suggested that near the periphery of the current universe, gravitational interactions must have a net repulsive, rather than attractive, dependence — this owing to the far lower neutrino ﬂux in the far distant regions of the

 universe ( # km). Thus, though arguably at very small  length scales ( : mm), gravitational interactions may be
described by an inverse fourth dependence [9], and at typical solar system and galactic length scales by inverse square dependence, yet at length scales of n km (where n > "), the dependence is likely not to be attractive at all, and instead repulsive near the outer zones of the universe. Thus, relative to the line of centres (a curved Riemannian arc) of the earth born measurement laboratory and the very distant black holes, the neutrino ﬂux that is emanating from the outer regions of the universe, and opposing the escape of both xrays and infra-red radiation toward the observer, has a higher particle density, than the neutrino ﬂux that is opposing (due to collisions and associated net exchange of total momenta) the escape of electromagnetic radiation in the direction of the periphery of the universe. This higher level of particles per square centimetre per second escaping toward the periphery of the universe diﬀuses in curved directions because of the collision basis of gravity, and the net result contributes to the asymmetry detected by the observer, as in Fig. 1, and shown functionally in Figs. 2 and 3.
3 Supporting evidence for the signiﬁcance of the neutrino ﬂux
In a work previously published in this journal [10] I presented the explanation of the physical cause of the decades of radioactive decay data histograms determined by Shnoll et al. [11–13] which reported characteristic histograms for the decay of Pu-239 which were periodic over a 24 h interval
 (the solar day, thus the spin of the Earth), a 28 day in-
terval (the lunar month, thus the period of the Moon), and the sidereal year, and also reported characteristic histograms of radioactive decay rate associated with a New Moon and a total solar eclipse. My explanation [10] was based on the Moon and/or the Earth periodically interrupting through scattering and capture some of the neutrinos that emanated from the Sun, and which would have otherwise transferred their momentum to the radioactive source, the decay rate of which was being studied in the experiments (taking place in Moscow, and aboard two research ships that travelled all over the world, including the polar regions). Also, the Sun and Moon intercept neutrinos emanating from deep space.
The Shnoll work [11–13] prompted me to lease a Rad-7 solid-state detector through Dr. Derek Lane-Smith at Durridge Corporation (Bedford, MA) for the purpose of exploring further the Shnoll conclusions. The Rad-7 detector is utilized worldwide as the principle detector of alpha particles decaying from radon gas, and as such is ideally suited also to study the daughter isotopes of Radon. Amongst these, Po210 has the ideal half life compatible with the purposes of my work. The detector was set up for a 4 week period at the Farlow Herbarium at Harvard University, where I was a research aﬃliate at the time, conducting work at the Arnold Ar-

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Fig. 5: Data showing three decreases in radioactive decay data of Po-210, 24 hours apart, corresponding to period of 6–10 AM Sept. 7–10 during time interval of Jupiter eclipsing quasar JO842+1835. Reproduced from [16].

boretum, studying the negative geotropism of a heavy vine, Aristolochia macrophylla. Although, conclusions could not be deﬁnitely established regarding a diurnal variation of the radioactive decay, a clear peak was observed in the 12:00– 4:00 PM time interval on 26 August 2001, far exceeding twosigma in alpha particles per 4 hour interval. These data are given in Fig. 4, and are digitally reproduced from [14]. It was not until over a year later that I learned that on 26–27 August 2001, radiation from the explosion of supernova SN 2001 dz (in UGC 47) reached the Earth [15]. A supernova explosion is associated with a very signiﬁcant release of neutrinos, and I interpret that the radioactive alpha particle decay rate peak, shown in Fig. 4, is a consequence of the impingement of the neutrinos, associated with the supernova explosion burst, upon the radioactive isotope source which then pertubed and further de-stabilized a nucleus that was already unstable due to the ratio of neutrons to protons.
During the period September 7–11, 2002, the planet Jupiter eclipsed the deep space quasar JO842+1835, and measurements of alpha particle decay rate were conducted by Dr. Lane-Smith in the Boston area at my request. The averaged data are given in Fig. 5 (digitally reproduced from [16]), showing a decrease in decay rate from approximately 6:00AM to 10:00 AM every 24 hours during the 3-day time interval of the eclipsing event. This variation is attributed to the rotation of the Earth such that once per day Jupiter, interrupted the particle-path from the deep-space quasar to the earth laboratory where the radioactive source was located for the experiment. This interruption of neutrinos, due to the nucleons of Jupiter scattering and inelastically capturing some small, but non-trivial, proportion of particles and/or radiation causes a decrease in radioactive decay rate because of the consequent decrease in the particle ﬂux transferring momen-

tum to the nuclei of Po-210. The x-axis scale is the number of four-hour periods in to the experiment starting at 1:00 AM EDT Sept 7, 2002, and showing decreases at abscissa values of 9, 15, and 21 — these being six 4-hr intervals (24 h) apart.
On 4 Dec 2002, a total solar eclipse occurred, during which the radioactive decay rate of Co-60 was measured at Pittsburg State University in southeastern Kansas [17], and the radioactive decay rate of Po-210 [18] was measured in the Boston area, both at/near the time of totality in southern Australia. The decay data [14] are plotted in Fig. 6, and show dips in decay rate at the time when the umbra of the eclipse was closest to the location of the source isotopes (on the opposite side of the Earth from totality). The inset shows very recent data [19] on the decay of Cs-137 during the annular solar eclipse of 26 January 2009, also in southeast Kansas, at the time when the eclipse was at peak darkness in Australia, also showing a dip in decay rate when the umbra passed closest
= to the source isotope (time 4.06 days into the experiment).
The 2009 data plot (inset) shows also the envelope of the negative percent changes. The circled data points are analogous to the leading-edge signal and the trailing-edge signal that corresponded to dips in gravity upon ﬁrst contact and upon last contact associated with the total solar eclipse in China in March 1997 (see [20, 20]). These consistent decreases in decay rate (using three diﬀerent isotopes) during two diﬀerent solar eclipses can only be explained by the mass of the Moon and the mass of the Earth interupting the ﬂux of neutrinos coming from the Sun, and thus some of the neutrinos associated with the ﬂux, never reaching the source isotope. Hence these scattered and captured neutrinos do not cause any further de-stabilization of the weak cohesive interaction of mesons and of gluons that hold the nucleus intact/together, normally ascribed to the weak force — an internal interaction

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Fig. 6: Decrease in radioactive decay rate for Co-60 and for Po-210 during total eclipse of 4 Dec. 2002, and decrease in radioactive decay rate for Cs-137 during annular solar eclipse of 26 Jan. 2009.

— but now shown to depend upon momenta transfer from externally impinging particles including primarily the muon neutrino and the electron neutrino.
Additional supporting data regarding the signiﬁcance of the neutrino ﬂux on radioactivity, and highly supportive of my own work and interpretations given herein, are as follows:
1. A major multi-year study by Purdue University researchers at Brookhaven National Laboratory clearly show that the radioactive decay rates of many isotopes correlate very well with the distance of the source isotope from the Sun, as well as changes in radioactive decay rate correlated with major solar ﬂares [22].
2. Positron annihilation measurements [23] that show periodic variation with the phases of the Moon, yielding peaks associated with the New Moon (which approximates a solar eclipse), and troughs correlated with the presence of the Full Moon. The source of positrons in this study was Na-22, and the dependent variable of the experiment measured the yield of molecular iodine (I ).
Thus the peaks in I % correlated with the presence of the New Moon, and hence the interference by the Moon of a ﬂow of particles from the Sun and from space. The data also showed a general trend increase in I production over the course of the months of the experiment (November through February), that the authors tentatively attribute to seasonal changes of the distance between the Earth and the Sun. The exact phenomena causing the peaks is not yet established

since in this case an interruption of neutrino ﬂow by the Moon enhances positronium production. It is possible that the peaks are due to more molecular iodine being produced associated with a diﬀerent collision crosssection caused by change-in-ﬂavor of the neutrino due to collision with nucleons of the Moon.
3. Periodic oscillations have been reported [24] in Pm142 which show an oscillating sinusoidal decay for electron capture (as contrasted to a conventionally established exponential decay) which the authors attribute to modulations caused by the oscillation of neutrinos between two diﬀerent mass states (ﬂavors), that of the electron neutrino emitted in the original decay, and that of the muon neutrino which is observed in decays of the muon (a particle 200 times more massive than the electron).
4. The standard deviation of decay rate of radioactive isotopes is periodic with respect to the phases of the Moon, being maximum at Full Moon (whereby external particle impingement from the Sun is unobstructed) and minimum at New Moon (whereby external particle impingement is obstructed by the Moon [25]), akin an eclipse condition.
4 Conclusion
Thus based on all of the above considerations, in the current work, the asymmetry in neutrino ﬂux is identiﬁed as the prin-

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cipal cause of the non-symmetry shown in Fig. 1, owing to neutrino-photon collisions in the AGN or black hole regimes where the collision cross-sections of neutrinos and photons is many orders of magnitude higher than in the solar system regime. This conclusion is supported by our previous experimental work using both very-close-proximity gravitational pendula, and a magnetic pendulum system, interrogated by laser scattering, showing asymmetry in gravitational parti-
X Y cle/wave impinging ﬂux in the - plane as compared to Z the zed ( ) direction [16].
Note added in proof
Recent work by G. C. Vezzoli and R. Morgan has shown that the 1444 minute annually periodic histogram reported by Shnol and Rubenstein in this journal for the period 24 July 2005 into August correlates with the NASA report of the Sun beginning the occulting of Saturn on that date; and thus also correlates with the work of Vezzoli reporting a dip in gravity on 18 May 2001 when earth, Sun, and Saturn were in syzygy [15]. The Morgan-Vezzoli work will be reported in a Letter-to-the-Editor of this journal authored by Morgan.
Acknowledgements
The author wishes to gratefully acknowledge the assistance, collegial cooperation, fruitful discussions of Prof. C. Blatchley, Dr. Derek Lane-Smith, Dr. William Stanley, Sandra Smalling, and the faculty, administration, staﬀ, and students at Lebanon College.
Submitted on February 24, 2009 / Accepted on June 03, 2009
References
1. McKernan B. American Museum of Natural History and Manhattan Community College, City University of New York, 17 Dec 2008 See also Kaku M. and Thompson J. Beyond Einstein. Anchor, Random House, New York, 1995.
2. Stanley W. and Vezzoli G.C. Induced gravity model based on external impinging neutrinos: calculation of G in terms of collision phenomena and inferences to inertial mass and atomicquantization. arXiV: astro-ph/0102109
3. The new physics. Ed. by P. Davies, Cambridge University Press, 1989.
4. Longair M. Ibid., p.141 5. Watarai K-Y. Ecipsing light-curve asymmetry for black hole ac-
cretion ﬂows. Publ. Astron. Soc. Jpn., 2005, v. 57(5), 827–833. 6. Banibrata M. Neutrino asymmetry around black holes. Modern
Physics Letters A, 2005, v. 20(28), 2145–2155. 7. Akujar C.E. Contribution of black hole accretion disks to
symmetric lepton production. Astrophysics and Space Science, 1984, v. 100, 1–2, 413–415.

8. Vezzoli G.C. Note on indication of nano local short range order in liquid iron from analysis of recovered products from molten state: geophysical inferences from the materials science, Mat. Res. Innov., 2002, v. 5, 222–225.
9. Hoyle C.D. Phys. Rev. Lett., 2001, v. 86(8), 1418.
10. Vezzoli G.C. On the explaination of the physical cause of the Shnoll characteristic histograms and observed ﬂuctuations. Progress in Physics, 2008, v. 2, 158–161.
11. Shnoll S., Kolombet V., Pozharski E., Zenchenko T., Zvereva I., and Konradov A. Realization of discrete states during ﬂuctuations in macroscopic processes. Physics-Uspekhi, 1998, v. 162(10), 1139–1140.
12. Shnoll S., Zenchenko T., Zenchenko K., Pozharski E., Kolombet V., and Konradov A. Regular variation of the ﬁne structure of statistical distributions as a consequence of cosmophysical agents. Physics-Uspekhi, 2000, v. 43(2), 205–209.
13. Shnoll S., Zenchenko K., Shapovalov S., Gorshkov S., Makarevich., and Troshichev O.A. The speciﬁc form of histograms presenting the distribution of data of alpha decay measurements appears simultaneously in the moment of New Moon in diﬀerent points from Artic to Antartic. arXiv: physics/0412152.
14. Vezzoli G.C. Radioactive decay of Po-210 and Co-60 at two U.S. Observation Stations in the Path of the Umbra/Penumbra of the total eclipse of the Sun of December 4, 2002 in Southern Australia. Inﬁnite Energy, 2005, v. 11(61), 48–53.
15. Photograph of this supernova was taken by T. Boles and G. M. Hurst in Basingstoke, England, August 27–28 2001. The supernova SN 2001 is described at website http://www. theastronomer.org/supernovae.html. Code symbols are SN 2001dw-20010825-tb.jpg or SN 2001dz.
16. Vezzoli G.C. Gravitational data during the syzygy of May 18, 2001 and related studies. Inﬁnite Energy, 2004, v. 9(53), 18–27.
17. Co-60 was conducted by Professor C. Blatchley at Pittsburg State University during time interval of total solar eclipse of 4 Dec 2002 showing a decrease in decay rate at the time of totality in Southern Australia. Dr. Blatchley asked me to point out in this citation that he is not certain whether his data are supportive of my conclusions because of inherent noise eﬀects and barometric eﬀects that relate to these type of radioactive decay measurements.
18. Po-210 study was conducted by Dr. Derek Lane-Smith of Durridge Corporation in Bedford, MA, during time interval of total solar eclipse of 4 Dec 2002 showing a decrease in radioactive decay rate at the time of totality in southern Australia.
19. Cs-137 decay study was conducted by Professor C. Blatchley at Pittsburg State University during the time interval of the annular solar eclipse of 26 January 2009, indicating a small decrease in decay rate at the time of totality in Australia, corresponding to 4.06 days since the beginning of the experimental measurements. Dr. Blatchley does not share the same opinion as the author regarding the signiﬁcance of the Cs-137 radioactive decay data as related to the total physics of a solar eclipse, partly because the detected signal does not show a variation exceding two-sigma. However, it must be pointed out that the variation in the measuring of a property such as radioactive decay, or gravity, during totality of a solar eclipse must indeed necessarily

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be extremely small. Regarding gravity, this variation was measured to be about 10 microgal (see Refs. 21 and 22).
20. Wang Q., Yang X., Wu C., Guo H., Liu H., and Hua C. Precise measurement of gravity variation during a total solar eclipse. Phys. Rev. D., 2000, v. 62, 041101R.
21. Yang X. and Wang Q. Gravity anomaly during the Mohe total solar eclipse and new constraint on gravitational shielding parameter. Astrophysics and Space Science, 2002, v. 285, 245– 253; Yang X. Private communications, 2005–2007.
22. Jenskin J., Fischbach E., Buncher J., Gruenwald J., Krause D., and Mattes J. Evidence for correlations between nuclear decay rates and earth-Sun distance. arXiv: 00808.3283.
23. Vikin B.P. Gravitational pertubations as a possible cause for instability in the measurements of positron annihilation. Progress in Physics, 2008, v. 2, 76–77.
24. Litvinov Y.A. et al. Phys. Rev. Lett., 2008, v. 664, 162–168.
25. Goleminov N. Private communications, 2006.

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Combining NASA/JPL One-Way Optical-Fiber Light-Speed Data with Spacecraft Earth-Flyby Doppler-Shift Data to Characterise 3-Space Flow

Reginald T. Cahill
School of Chemistry, Physics and Earth Sciences, Flinders University, Adelaide 5001, Australia E-mail: Reg.Cahill@ﬂinders.edu.au
We combine data from two high precision NASA/JPL experiments: (i) the one-way speed of light experiment using optical ﬁbers: Krisher T.P., Maleki L., Lutes G.F., Primas L.E., Logan R.T., Anderson J.D. and Will C.M. Phys. Rev. D, 1990, v. 42, 731–734, and (ii) the spacecraft earth-ﬂyby Doppler shift data: Anderson J.D., Campbell J.K., Ekelund J.E., Ellis J. and Jordan J.F. Phys. Rev. Lett., 2008, v. 100, 091102, to give the
= solar-system galactic 3-space average speed of 486 km/s in the direction RA 4.29D, = Dec 75.0 . Turbulence eﬀects (gravitational waves) are also evident. Data also
reveals the 30 km/s orbital speed of the Earth and the Sun inﬂow component at 1AU of 42 km/s and also 615 km/s near the Sun, and for the ﬁrst time, experimental measurement of the 3-space 11.2 km/s inﬂow of the Earth. The NASA/JPL data is in remarkable agreement with that determined in other light speed anisotropy experiments, such as Michelson-Morley (1887), Miller (1933), Torr and Kolen (1981), DeWitte (1991), Cahill (2006), Munera (2007), Cahill and Stokes (2008) and Cahill (2009).

1 Introduction
In recent years it has become clear, from numerous experiments and observations, that a dynamical 3-space exists [1, 2]. This dynamical system gives a deeper explanation for various observed eﬀects that, until now, have been successfully described, but not explained, by the Special Relativity (SR) and General Relativity (GR) formalisms. However it also oﬀers an explanation for other observed eﬀects not described by SR or GR, such as observed light speed anisotropy, bore hole gravity anomalies, black hole mass spectrum and spiral galaxy rotation curves and an expanding universe without dark matter or dark energy. Herein yet more experimental data is used to further characterise the dynamical 3-space, resulting in the ﬁrst direct determination of the inﬂow eﬀect of the Earth on the ﬂowing 3-space. The 3-space ﬂow is in the main determined by the Milky Way and local galactic cluster. There are also components related to the orbital motion of the Earth and to the eﬀect of the Sun, which have already been extracted from experimental data [1].
The postulate of the invariance of the free-space speed of light in all inertial frames has been foundational to the physics of the 20th Century, and so to the prevailing physicist’s paradigm. Not only did it provide computational means essential for the standard model of particle physics, but also provided the spacetime ontology, which physicists claim to be one of the greatest of all discoveries, particularly when extended to the current standard model of cosmology, which assumes a curved spacetime account of not only gravity but also of the universe, but necessitating the invention of dark matter and dark energy.
The nomenclature 3-space is used to distinguish this dynamical 3-dimensional space from other uses of the word space.

It s usually assumed that the many successes of the resulting Special Theory of Relativity mean that there could be very little reason to doubt the validity of the invariance postulate. However the spacetime formalism is just that, a formalism, and one must always be careful in accepting an ontology on the basis of the postulates, as in the case of the speed of light, because the postulate never stipulated how the speed of light was to be measured, in particular how clock retardation and length contraction eﬀects were to be corrected. In contrast to the spacetime formalism Lorentz gave a diﬀerent neo-Galilean formalism in which space and time were not mixed, but where the special relativity eﬀects were the consequence of absolute motion with respect to a real 3-space. Recently [3] the discovery of an exact linear mapping between the Minkowski-Einstein spacetime class of coordinates and the neo-Galilean class of time and space coordinates was reported. In the Minkowski-Einstein class the speed of light is invariant by construction, while in the Galilean class the speed is not invariant. Hence statements about the speed of light are formalism dependent, and the claim that the successes of SR implies that the speed of light is invariant is bad logic. So questions about of the speed of light need to be answered by experiments.
There have been many experiments to search for light speed anisotropy, and they fall generally into two classes — those that successfully detected anisotropy and those that did not. The reasons for this apparent disparity are now understood, for it is important to appreciate that because the speed of light is invariant in SR — as an essential part of that formalism, then SR cannot be used to design or analyse data from light speed anisotropy experiments. The class of experiments that failed to detect anisotropy, such as those using vacuum Michelson interferometers, say in the form of reso-

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nant vacuum cavities [4], suﬀer a design ﬂaw that was only discovered in 2002 [5, 6]. Essentially there is a subtle cancellation eﬀect in the original Michelson interferometer, in that two unrelated eﬀects exactly cancel unless the light passes through a dielectric. In the original Michelson interferometer experiments the dielectric happened, fortuitously, to be a gas, as in [7–11,15], and then the sensitivity is reduced by the fac-
tor k = n 1, where n is the refractive index of the gas, compared to the sensitivity factor k = 1 used by Michel-
son in his calculation of the instrument’s calibration constant,
using Newtonian physics. For air, with n = :1 00029, this factor has value k = :0 00058 which explained why the original
Michelson-Morley fringe shifts were much smaller than expected. The physics that Michelson was unaware of was the reality of the Lorentz-Fitzgerald contraction eﬀect. Indeed the null results from the resonant vacuum cavities [4] experiments, in comparison with their gas-mode versions, gives explicit proof of the reality of the contraction eﬀect . A more sensitive and very cheap detector is to use optical ﬁbers as the light carrying medium, as then the cancellation eﬀect is overcome [16]. Another technique to detect light speed anisotropy has been to make one-way speed measurements; Torr and Kolen [12], Krisher et al. [18], DeWitte [13] and Cahill [14]. Another recently discovered technique is to use the Doppler shift data from spacecraft earth-ﬂybys [19]. Using the spacetime formalism results in an unexplained earth-ﬂyby Doppler shift anomaly, Anderson et al. [20], simply because the spacetime formalism is one that explicitly speciﬁes that the speed of the EM waves is invariant, but only wrt a peculiar choice of space and time coordinates.
Here we combine data from two high precision NASA/JPL experiments: (i) the one-way speed of light experiment using optical ﬁbers: Krisher et al. [18], and (ii) the spacecraft earth-ﬂyby Doppler shift data: Anderson et al. [20], to give the solar-system galactic 3-space average speed of 486 km/s in the direction RA = 4.29D, Dec = 75 . Turbulence eﬀects (gravitational waves) are also evident. Various data reveal the 30 km/s orbital speed of the Earth and the Sun inﬂow component of 615 km/s near the Sun, and 42 km/s at 1AU, and for the ﬁrst time, experimental evidence of the 3-space inﬂow of the Earth, which is predicted to be 11.2 km/s at the Earth’s surface. The optical-ﬁber and restricted ﬂyby data give, at this stage, only an average of
12:4 5 km/s for the Earth inﬂow — averaged over the space-
craft orbits, and so involving averaging wrt distance from earth and RF propagation angles wrt the inﬂowy. The optical ﬁber — ﬂyby data is in remarkable agreement with the
As well the null results from the LIGO-like and related vacuum-mode Michelson interferometers are an even more dramatic conﬁrmation. Note that in contrast the LISA space-based vacuum interferometer does not suﬀer from the Lorentz contraction eﬀect, and as a consequence would be excessively sensitive.
yA spacecraft in an eccentric orbit about the Earth would permit, using the high-precision Doppler shift technology, a detailed mapping of the 3-space inﬂow.

spatial ﬂow characteristics as determined in other light speed anisotropy experiments, such as Michelson-Morley (1887), Miller (1933), DeWitte (1991), Torr and Kolen (1981), Cahill (2006), Munera (2007), Cahill and Stokes (2008) and Cahill (2009). The NASA data enables an independent calibration of detectors for use in light speed anisotropy experiments and related gravitational wave detectors. These are turbulence effects in the ﬂowing 3-space. These ﬂuctuations are in essence gravitational waves, and which were apparent even in the Michelson-Morley 1887 data [1, 2, 21].

2 Flowing 3-space and emergent quantum gravity

We give a brief review of the concept and mathematical for-

malism of a dynamical ﬂowing 3-space, as this is often con-

fused with the older dualistic space and aether ideas, wherein

some particulate aether is located and moving through an un-

changing Euclidean space — here both the space and the

aether were viewed as being ontologically real. The dy-

namical 3-space is diﬀerent: here we have only a dynamical

3-space, which at a small scale is a quantum foam system

without dimensions and described by fractal or nested homo-

topic mappings [1]. This quantum foam is not embedded in

any space — the quantum foam is all there is and any met-

ric properties are intrinsic properties solely of that quantum

foam. At a macroscopic level the quantum foam is described
by a velocity ﬁeld v(r; t), where r is merely a 3-coordinate

within an embedding space. This space has no ontological

existence — it is merely used to (i) record that the quan-

tum foam has, macroscopically, an eﬀective dimension of 3,

and (ii) to relate other phenomena also described by ﬁelds, at

the same point in the quantum foam. The dynamics for this

3-space is easily determined by the requirement that observ-

ables be independent of the embedding choice, giving, for zero-vorticity dynamics and for a ﬂat embedding spacez

H

@v @t + (v


H)v +

D D + (tr ) tr( ) =

9
4G ; >>>>>>>=

8
H v = 0;

Dij

=

1 2

 @vi @xj

+

@vj @xi


;

>>>>>>>;

(1)

where (r; t) is the matter and EM energy densities expressed as an eﬀective matter density. Borehole g mea-
surements and astrophysical blackhole data has shown that
 1=137 is the ﬁne structure constant to within observa-
tional errors [1, 2, 24, 25]. For a quantum system with mass

zIt is easy to re-write (1) for the case of a non-ﬂat embedding space, such as an 5!, by introducing an embedding 3-space-metric Cij r, in place of the Euclidean metric ij . A generalisation of (1) has also been suggested in [1] when the vorticity is not zero. This vorticity treatment predicted an
additional gyroscope precession eﬀect for the GPB experiment, R. T. Cahill, Progress in Physics, 2007, v. 3, 13–17.

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m the Schro¨dinger equation is uniquely generalised [24] with
the new terms required to maintain that the motion is intrinsically wrt to the 3-space, and not wrt to the embedding space, and that the time evolution is unitary

i

@

(r;
@t

t)

=

2m H

(r; t)

i

 v

H

+

1 2

H

 v

(r; t) :

(2)

The space and time coordinates Bt; x; y; zC in (1) and (2)
ensure that the separation of a deeper and uniﬁed process into diﬀerent classes of phenomena — here a dynamical 3-space (quantum foam) and a quantum matter system, is properly tracked and connected. As well the same coordinates may be used by an observer to also track the diﬀerent phenomena. However it is important to realise that these coordinates have no ontological signiﬁcance — they are not real. The velocities v have no ontological or absolute meaning relative to this coordinate system — that is in fact how one arrives at the form in (2), and so the “ﬂow” is always relative to the internal dynamics of the 3-space. A quantum wave packet propagation analysis of (2) gives the acceleration induced by wave refraction to be [24]

g

=

@v @t

+

(v

H)v + (H

v)

v4 ;

(3)

v4(r (t); t) = v (t) v(r (t); t) ;

(4)

where v4 is the velocity of the wave packet relative to the 3-space, and where v and r are the velocity and position
relative to the observer, and the last term in (3) generates the
Lense-Thirring eﬀect as a vorticity driven eﬀect. Together
(2) and (3) amount to the derivation of gravity as a quantum
eﬀect, explaining both the equivalence principle (g in (3) is
independent of m) and the Lense-Thirring eﬀect. Overall we
see, on ignoring vorticity eﬀects, that

H g = 4G 8 (trD) tr(D ) ; (5)

which is Newtonian gravity but with the extra dynamical term
whose strength is given by . This new dynamical eﬀect
explains the spiral galaxy ﬂat rotation curves (and so doing
away with the need for “dark matter”), the bore hole g anoma-
lies, the black hole “mass spectrum”. Eqn. (1), even when
 = 0, has an expanding universe Hubble solution that ﬁts
the recent supernovae data in a parameter-free manner with-
out requiring “dark matter” nor “dark energy”, and without
the accelerating expansion artifact [25, 26]. However (5) can-
not be entirely expressed in terms of g because the fundamen-
tal dynamical variable is v. The role of (5) is to reveal that
if we analyse gravitational phenomena we will usually ﬁnd
that the matter density  is insuﬃcient to account for the ob-
served g. Until recently this failure of Newtonian gravity has

been explained away as being caused by some unknown and

undetected “dark matter” density. Eqn. (5) shows that to the

contrary it is a dynamical property of 3-space itself. Here

we determine various properties of this dynamical 3-space

from the NASA optical-ﬁber and spacecraft ﬂyby Doppler

anomaly data.

Signiﬁcantly the quantum matter 3-space-induced “grav-

itational” acceleration in (3) also follows from maximising
the elapsed proper time wrt the wave-packet trajectory r (t),

see [1],

Zr
 = dt 1

v4(r (t); t)
c

(6)

and then taking the limit v4=c  0. This shows that (i)

the matter ‘gravitational’ geodesic is a quantum wave refrac-

tion eﬀect, with the trajectory determined by a Fermat least

proper-time principle, and (ii) that quantum systems undergo

a local time dilation eﬀect — which is used later herein in

connection with the Pound-Rebka experiment. A full deriva-

tion of (6) requires the generalised Dirac equation.

3 3-space ﬂow characteristics and the velocity superposition approximation

This paper reports the most detailed analysis so far of data

from various experiments that have directly detected the
3-space velocity ﬁeld v(r; t). The dynamics in (1) is nec-

essarily time-dependent and having various contributing ef-

fects, and in order of magnitude: (i) galactic ﬂows associated

with the motion of the solar system within the Milky Way, as

well as ﬂows caused by the supermassive black hole at the

galactic center and ﬂows associated with the local galactic

cluster, (ii) ﬂows caused by the orbital motion of the Earth

and of the inﬂow caused by the Sun, and (iii) the inﬂow asso-

ciated with the Earth. An even smaller ﬂow associated with

the Moon is not included in the analysis. It is necessary to

have some expectations of the characteristics of the ﬂow ex-

pected for an earth based observer. First consider an isolated
spherical mass density (r), with total mass M, then (1) has
a stationary ﬂow solution, for r > R, i.e outside of the mass,

v(r) =

^r

r

2GM

(1

+
r

+:::)

(7)

which gives the matter acceleration from (3) to be

g(r) =

^r

GM

(1

+
r

+:::)

(8)

H corresponding to a gravitational potential, via g = Φ,

(r) =

GM(1 +
r

+:::) :

(9)

This special case is Newton’s law of gravity, but with
some 0.4% of the eﬀective mass being caused by the - dy-
namics term. The inﬂow (7) would be applicable to an iso-
lated and stationary sun or earth. At the surface of the Sun

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615 42 this predicts an inﬂow speed of km/s, and km/s at the 11 2 Earth distance of 1AU. For the Earth itself the inﬂow speed at
the Earth’s surface is predicted to be : km/s. When both
occur and when both are moving wrt the asymptotic 3-space, then numerical solutions of (1) are required. However an approximation that appears to work is to assume that the net ﬂow in this case may be approximated by a vector superposition [27]
= + + + v vgalactic vsun vorbital vearth : : : (10)
which are, in order, translational motion of the Sun, inﬂow into the Sun, orbital motion of the Earth (the orbital motion produces an apparent ﬂow in the opposite direction — hence the -ve sign; see Fig. 4), inﬂow into the Earth, etc. The ﬁrst three have been previously determined from experimental data, and here we more accurately and using new data determine all of these components. However this superposition cannot be completely valid as (1) is non-linear. So the superposition may be at best approximately valid as a time average only. The experimental data has always shown that the detected ﬂow is time dependent, as one would expect, as with multi-centred mass distributions no stationary ﬂows are known. This time-dependence is a turbulence eﬀect — it is in fact easily observed and is seen in the Michelson-Morley 1887 data [2]. This turbulence is caused by the presence of any signiﬁcant mass, such as the galaxy, sun, earth. The NASA/JPL data discussed herein again displays very apparent turbulence. These wave eﬀects are essentially gravitational waves, though they have characteristics diﬀerent from
( ) those predicted from GR, and have a diﬀerent interpretation.
Nevertheless for a given ﬂow v r; t , one can determine the corresponding induced spacetime metric g which generates
the same matter geodesics as from (5), with the proviso that this metric is not determined by the Hilbert-Einstein equations of GR. Signiﬁcantly vacuum-mode Michelson interferometers cannot detect this phenomenon, which is why LIGO and related detectors have not seen these very large wave effects.

4 Gas-mode Michelson interferometer

( ) The Michelson interferometer is a brilliant instrument for
measuring v r; t , but only when operated in dielectric mode.
This is because two diﬀerent and independent eﬀects exactly cancel in vacuum mode; see [1, 2, 5]. Taking account of the geometrical path diﬀerences, the Fitzgerald-Lorentz armlength contraction and the Fresnel drag eﬀect leads to the travel time diﬀerence between the two arms, and which is detected by interference eﬀects , is given by

t=k

LvP c!

cos

2(

);

(11)

The dielectric of course does not cause the observed eﬀect, it is merely
a necessary part of the instrument design physics, just as mercury in a thermometer does not cause temperature.

C

❄

L

gas

(a)

✻

✲

✲✛

A ❄D L B

gas (b) ✲

A✄✄✄✗✲✄✄✄✄❈❈❈C❈A❈❲❈❈❈✟✲❈❈vP✟D✟✛✟✯B

Fig. 1: Schematic diagrams of the gas-mode Michelson Interferometer, with beamsplitter/mirror at A and mirrors at B and C mounted on arms from A, with the arms of equal length L when at rest. D is the detector screen. In (a) the interferometer is at rest in space. In (b) the 3-space is moving through the gas and the interferome-
= ter with speed v2 in the plane of the interferometer and direction  relative to AB arm. Interference fringes are observed at D when mirrors C and D are not exactly perpendicular. As the interferometer is rotated in the plane shifts of the fringes are seen in the case of absolute motion, but only if the apparatus operates in a gas. By measuring fringe shifts the speed v2 may be determined. In
= general the v2 direction has angle  wrt the local meridian, and the
arm AB has angle relative to the local meridian, so that  is angle between v2 and one-arm. The diﬀerence in travel times t is given in (11), but with temperature changes and non-orthogonal mirrors by (12). In vacuum the fringes do not shift during rotation .

( ) where speciﬁes the direction of v r; t projected onto
the plane of the interferometer, giving projected value vP ,
= ( 2) relative to the local meridian, and where k n ( 1) n =n. Neglect of the relativistic Fitzgerald-Lorentz 1   contraction eﬀect gives k n! for gases, which is es-

sentially the Newtonian theory that Michelson used.

However the above analysis does not correspond to how

the interferometer is actually operated. That analysis does not actually predict fringe shifts, for the ﬁeld of view would be uniformly illuminated, and the observed eﬀect would be a changing level of luminosity rather than fringe shifts.

As Michelson and Miller knew, the mirrors must be made

slightly non-orthogonal with the degree of non-orthogonality

determining how many fringe shifts were visible in the ﬁeld

of view. Miller experimented with this eﬀect to determine a comfortable number of fringes: not too few and not too many. Hicks [22] developed a theory for this eﬀect – however it is not necessary to be aware of the details of this analysis in

using the interferometer: the non-orthogonality reduces the

symmetry of the device, and instead of having period of 180

cos( ) the symmetry now has a period of 360 , so that to (11) we

must add the extra term a 

in

t=k

L(1

+ e)vP
c!

cos

2(

)+

(12)

+ a(1 + e) cos( ) + f :

1 + The term e models the temperature eﬀects, namely
that as the arms are uniformly rotated, one rotation taking
several minutes, there will be a temperature induced change
in the length of the arms. If the temperature eﬀects are linear

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in time, as they would be for short time intervals, then they are
linear in . In the Hick’s term the parameter a is proportional
to the length of the arms, and so also has the temperature fac-
tor. The term f simply models any oﬀset eﬀect. Michelson
and Morley and Miller took these two eﬀects into account when analysing his data. The Hick’s eﬀect is particularly apparent in the Miller and Michelson-Morley data.
The interferometers are operated with the arms horizon-
tal. Then in (12)  is the azimuth of one arm relative to the
local meridian, while is the azimuth of the absolute motion velocity projected onto the plane of the interferometer,
with projected component vP . Here the Fitzgerald-Lorentz
contraction is a real dynamical eﬀect of absolute motion, unlike the Einstein spacetime view that it is merely a spacetime perspective artifact, and whose magnitude depends on the choice of observer. The instrument is operated by rotating at a rate of one rotation over several minutes, and observing the shift in the fringe pattern through a telescope during the rotation. Then fringe shifts from six (Michelson and Morley) or twenty (Miller) successive rotations are averaged to improve the signal to noise ratio, and the average sidereal time noted. Some examples are shown in Fig. 2, and illustrate the incredibly clear signal. The ongoing claim that the MichelsonMorely experiment was a null experiment is disproved. And as well, as discussed in [1, 2, 21], they detected gravitational waves, viz 3-space turbulence in 1887. The new data analysed herein is from one-way optical ﬁber and Doppler shift spacecraft experiments. The agreement between these and the gas-mode interferometer techniques demonstrate that the Fitzgerald-Lorentz contraction eﬀect is a real dynamical effect. The null results from the vacuum-mode interferome-
ters [4] and LIGO follow simply from having n = 1 giving k2 = 0 in (11).
5 Sun 3-space inﬂow from Miller interferometer data
Miller was led to the conclusion that for reasons unknown the existing theory of the Michelson interferometer did not reveal
true values of vP , and for this reason he introduced the param-
eter k, with k herein indicating his numerical values. Miller
had reasoned that he could determine both vgalactic and k by observing the interferometer-determined vP and over
a year because the known orbital speed of the Earth about the Sun of 30 km/s would modulate both of these observables, giving what he termed an aberration eﬀect as shown in Fig. 11, and by a scaling argument he could determine the absolute velocity of the solar system. In this manner he ﬁ-
nally determined that jvgalacticj = 208 km/s in the direction
( = 4h54m, = 70 33). However now that the theory of the Michelson interferometer has been revealed an anomaly
becomes apparent. Table 2 shows v = vM =kair, the speed
determined using (11), for each of the four epochs. How-
ever Table 3 also shows that k and the speeds v = vM =k
determined by the scaling argument are considerably diﬀer-

Fig. 2: (a) A typical Miller averaged-data from September 16, 1925, "D" Local Sidereal Time (LST) — an average of data from 20 turns of the gas-mode Michelson interferometer. Plot and data after ﬁtting using (12), and then subtracting both the temperature drift and Hicks eﬀects from both, leaving the expected sinusoidal form. The error bars are determined as the rms error in this ﬁtting procedure, and show how exceptionally small were the errors, and which agree with Miller’s claim for the errors. (b) Best result from the Michelson-Morley 1887 data — an average of 6 turns, at %D LST on July 11, 1887. Again the rms error is remarkably small. In both cases the indicated speed is L2 — the 3-space speed projected onto the plane of the interferometer. The angle is the azimuth of the 3-space speed projection at the particular LST. The speed ﬂuctuations from day to day signiﬁcantly exceed these errors, and reveal the existence of 3-space ﬂow turbulence — i.e gravitational waves.

ent. We denote by vM the notional speeds determined from
(11) using the Michelson Newtonian-physics value of k = 1.
The vM values arise after taking account of the projection eﬀect. That k is considerably larger than the value of kair in-

dicates that another velocity component has been overlooked.

Miller of course only knew of the tangential orbital speed of

the Earth, whereas the new physics predicts that as-well there
is a 3-space radial inﬂow vsun = 42 km/s at 1AU. We can

approximately re-analyse Miller’s data to extract a ﬁrst ap-
F proximation to the speed of this inﬂow component. Clearly
it is vR = vs2un + vo2rbital that sets the scale, see Fig. 4 and not vorbital, and because k = vM=vorbit and kair = vM=vR

I are the scaling relations, then

I

vsun = vorbital

vR2 vo2rbital

1 = vorbital

k2
ka2ir

1 : (13)

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Epoch 1925/26 vM

k

M air M p = v v =k

= v v =k v = 3v

vsun

February 8 April 1 August 1 September 15

9.3 km/s 10.1 11.2 9.6

0.048 0.051 0.053 0.046

385.9 km/s 419.1 464.7 398.3

193.8 km/s 198.0 211.3 208.7

335.7 km/s 342.9 366.0 361.5

51.7 km/s 56.0 58.8 48.8

 air M Table 1: The k anomaly: k k = 0:0241, as the 3-space inﬂow eﬀect. Here v and k come from ﬁtting the interferometer data using Newtonian physics (with vorbital = 30 km/s used to determine k), while v and v are computed speeds using the indicated scaling. The average of the Sun inﬂow speeds, at 1AU, is vsun = 54 6 km/s, compared to the predicted inﬂow speed of 42 km/s from (7). From
column 4 we obtain the average galactic ﬂow of v = 417 50 km/s, compared with the NASA-data determined ﬂow of 486 km/s.

vorbital

✛

❆

❆

vsun

❆ ❆

vR

❄ ❆❯

sun

Fig. 4: Orbit of earth about the Sun deﬁning the plane of the ecliptic
with tangential orbital velocity vorbital and the Sun inﬂow velocity vsun. Then vR = vsun vorbital is the velocity of the 3-space relative to the Earth, but not showing the vgalactic contribution.

Fig. 3: Speeds vP , of the 3-space velocity v projected onto the hor-
izontal plane of the Miller gas-mode Michelson interferometer located atop Mt.Wilson, plotted against local sidereal time in hours, for a composite day, with data collected over a number of days in September 1925, [8]. The data shows considerable ﬂuctuations, from hour to hour, and also day to day, as this is a composite day. The dashed curve shows the non-ﬂuctuating best-ﬁt variation over one day, as the Earth rotates, causing the projection onto the plane of the interferometer of the velocity of the average direction of the space ﬂow to change. The maximum projected speed of the curve is 417 km/s (using the STP air refractive index of n = 1:00029 in (11) (atop Mt. Wilson the better value of n = 1:00026 is suggested by the NASA data), and the min/max occur at approximately 5hrs and 17hrs local sidereal time (Right Ascension). Note from Fig. 11 and Table 2 that the Cassini ﬂyby in August gives a RA= 5:15D, close to the RA apparent in the above plot. The error bars are determined by the method discussed in Fig. 2. The green data points, with error bars, at 7D and 13D are from the Michelson-Morley 1887 data, from averaging (excluding only the July 8 data for 7D because it has poor S/N), and with same rms error analysis. The ﬁducial time lines at 5D and 17D are the same as those shown in Figs. 6 and 11. The speed ﬂuctuations are seen to be much larger than the statistically determined errors, conﬁrming the presence of turbulence in the 3-space ﬂow, i.e gravitational waves, as ﬁrst seen in the Michelson-Morley experiment.

Using the k values in Table 1 and the value of kair we

54 6 obtain the vsun speeds shown in Table 1, which give an aver-

aged speed of

km/s, compared to the predicted inﬂow

42 speed of km/s. Of course this simple re-scaling of the

Miller results is not completely valid because the direction of

vR is of course diﬀerent to that of vorbital, nevertheless the

= 54 5 Sun inﬂow speed of vsun

km/s at 1AU from this

42 analysis is reasonably close to the predicted value of km/s.

6 Generalised Maxwell equations and the Sun 3-space inﬂow light bending

One of the putative key tests of the GR formalism was the

gravitational bending of light by the Sun during the 1915

solar eclipse. However this eﬀect also immediately follows

from the new 3-space dynamics once we also generalise the

Maxwell equations so that the electric and magnetic ﬁelds are

excitations of the dynamical space. The dynamics of the elec-

tric and magnetic ﬁelds must then have the form, in empty

space,
H
H

E= H=


 @H + v HH ;

 @t


@E + v HE ;

H E = 0 ; 9>>>= H H = 0 ; >>>; (14)

@t

We have not modiﬁed this value to take account of the altitude eﬀect or

temperatures atop Mt.Wilson. This weather information was not recorded by

= 1 0+0 00029 PP TT Miller. The temperature and pressure eﬀect is that n : :

 ,

0 where T is the temperature in K and P is the pressure in atmospheres.

= 273 = T

K and P 1atm. The NASA data implies that atop Mt. Wilson

= 1 00026 the air refractive index was probably close to n :

.

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✲ ✲❅❅❘✬ ❅❅❄ ❳✩ ✠❳✛❳❳ ③❳❳❳

❅

✫❅✪

✒

❅■

✻❅

where v is the inﬂow speed at distance d and d is the impact
parameter. This agrees with the GR result except for the
correction. Hence the observed deﬂection of 8:4 10 $ ra-
dians is actually a measure of the inﬂow speed at the Sun’s
surface, and that gives v = 615 km/s, in agreement with
(7). These generalised Maxwell equations also predict gravi-
tational lensing produced by the large inﬂows associated with
the new “black holes” in galaxies.

Fig. 5: Shows bending of light through angle by the inhomogeneous spatial inﬂow, according to the minimisation of the travel time in (18). This eﬀect permits the inﬂow speed at the surface of the Sun to be determined to be $# km/s. The inﬂow speed into the Sun at the distance of the Earth from the Sun has been extracted from the Miller data, giving #" $ km/s.
which was ﬁrst suggested by Hertz in 1890 [23], but with v being a constant vector ﬁeld. Suppose we have a uniform ﬂow of space with velocity v wrt the embedding space or wrt an observer’s frame of reference. Then we can ﬁnd plane wave solutions for (14):
E(r; t) = Eeik r !t H(r; t) = Heik r !t (15)
with
! (k; v) = cj~kj + v k where c = 1= p : (16)

7 Torr and Kolen RF one-way coaxial cable experiment
A one-way coaxial cable experiment was performed at the Utah University in 1981 by Torr and Kolen [12]. This involved two rubidium vapor clocks placed approximately 500 m apart with a 5 MHz sinewave RF signal propagating between the clocks via a nitrogen ﬁlled coaxial cable buried
 in the ground and maintained at a constant pressure of 2
psi. Torr and Kolen observed variations in the one-way travel time, as shown in Fig. 7 by the data points. The theoretical predictions for the Torr-Kolen experiment for a cosmic speed
of 480 km/s in the direction ( ; = 5D = 70 ), and includ-
ing orbital and in-ﬂow velocities, are shown in Fig. 7. The maximum/minimum eﬀects occurred, typically, at the predicted times. Torr and Kolen reported ﬂuctuations in both the magnitude, from 1–3 ns, and time of the maximum variations in travel time, just as observed in all later experiments — namely wave eﬀects.

Then the EM group velocity is

vEM = r~ k ! (k; v) = ck^ + v :

(17)

So the velocity of EM radiation vEM has magnitude c
only with respect to the space, and in general not with re-
spect to the observer if the observer is moving through space.
These experiments show that the speed of light is in general
anisotropic, as predicted by (17). The time-dependent and
inhomogeneous velocity ﬁeld causes the refraction of EM ra-
diation. This can be computed by using the Fermat least-time
approximation. Then the EM ray paths r(t) are determined
by minimising the elapsed travel time:

j j Z sB
=
sE

ds

dr ds

c v^R(s) + v(r(s); t(s)

;

(18)

 dr vR = dt


v(r(t); t) ;

(19)

by varying both r(s) and t(s), ﬁnally giving r(t). Here s is
a path parameter, and vR is the 3-space vector tangential to the path. For light bending by the Sun inﬂow (7) the angle of
deﬂection is

v =2c

4GM (1 +

=

c d

+:::) +:::

(20)

8 Krisher et al. one-way optical-ﬁber experiment
The Krisher et al. one-way experiment [18] used two hydrogen maser oscillators with light sent in each direction through optical ﬁber of length approximately 29 km. The optical ﬁber was part of the NASA DSN Deep Space Communications Complex in the Mojave desert at Goldstone, California. Each maser provided a stable 100-MHz output frequency. This signal was split, with one signal being fed directly into one channel of a Hewlett-Packard Network Analyzer. The other signal was used to modulate a laser carrier signal propagated along a 29 km long ultrastable ﬁber optics link that is buried ﬁve feet underground. This signal was fed into the second channel of the other Network Analyzer at the distant site. Each analyzer is used to measure the relative phases of the
masers,  and  . The data collection began on Novem-
ber 12 1988 at 20:00:00 (UTC), with phase measurements made every ten seconds until November 17 1988 at 17:30:40 (UTC). Figs. 6(a) and (f) shows plots of the phase diﬀerence
  and phase sum  +  , in degrees, after remov-
ing a bias and a linear trend, as well as being ﬁltered using a Fast Fourier Transform. The data is plotted against local sidereal time. In analysing the phase data the propagation path
ww35ahso2st5aek3leo3nn :tg3oi7tbu)edaeanlaodnnd(g2la4a3stittr0ua6dige4h0at rlei:n3ge7i;bvee3nt5wbe1ye4n(52t1h4e3:8tw21)2o. 2m1Faisg:e6.r5s6;,

56

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Fig. 6: Data from ﬁve diﬀerent EM speed anisotropy experiments showing earth rotation wrt local preferred frame, as shown by sidereal
time phasing, together with wave eﬀects. In all cases a zero bias was removed and low-pass ﬁltering was applied. (a): Krisher [18] optical
ﬁber phase diﬀerence data   , in degrees. (b): DeWitte [13] RF coaxial cable phase data, in ns. The DeWitte cable ran NS. (c):
Cahill [14] hybrid optical-ﬁber/RF coaxial-cable data, in ps, from August 2006. Cable ran NS. (d): Cahill [16, 17] optical-ﬁber Michelson
interferometer, in photodiode mV, from September 18, 2007. (e): Cahill RF coaxial-cable data, in ps, from May 2009. Cable ran NS. (f):
+ Krisher [18] optical ﬁber phase sum data   , in degrees. In each case the (red) sinusoidal curves shows the phase expected for a RA of 5D, but with arbitrary magnitudes. The vertical lines are at local sidereal times of 5D and 17D, on successive days, corresponding to the RAs 4 96 6 09 shown in red in Fig. 11. The Krisher data gives a local sidereal time of : D, corresponding to a RA of : D for November — caused by
the 42 azimuth angle of the optical ﬁber to the local meridian. This RA was used in combination with the spacecraft earth-ﬂyby Doppler
shift data. Note the amplitude and phase ﬂuctuations in all the data — these are gravitational wave eﬀects.

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❅■v

✓✏ c ✛vi✲ ✒✑ c + vi Earth

❅E❅ ✛VE

❅ ❅

SC

❅

Fig. 7: Data from the 1981 Torr-Kolen experiment at Logan, Utah [12]. The data shows variations in travel times (ns), for local sidereal times, of an RF signal travelling through 500 m of coaxial cable orientated in an EW direction. Actual days are not indicated but the experiment was done during February-June 1981. Results are for a typical day. For the 1st of February the local time of 12:00 corresponds to 13:00 sidereal time. The predictions are for February, for
a cosmic speed of 480 km/s in the direction ( = 5:0D; = 70 ),
and including orbital and in-ﬂow velocities but without theoretical turbulence. The vertical lines are at local sidereal times of 5D and 17D, corresponding to the RAs shown in red in Figs. 6 and 11.
shows as well the corresponding phase diﬀerences from other experiments. Krisher only compared the phase variations with that of the Cosmic Microwave Background (CMB), and noted that the phase relative to the local sidereal time differed from CMB direction by 6 hrs, but failed to notice that it agreed with the direction discovered by Miller in 1925/26 and published in 1933 [8]. The phases from the various experiments show that, despite very diﬀerent longitudes of the experiments and diﬀerent days in the year, they are in phase when plotted against local sidereal times. This demonstrates that the phase cycles are caused by the rotation of the Earth relative to the stars — that we are observing a galactic phenomenon, being that the 3-space ﬂow direction is reasonably steady wrt the galaxy . Nevertheless we note that all the phase data show ﬂuctuations in both the local sidereal time for maxima/minima and also ﬂuctuations in magnitude. These wave eﬀects ﬁrst appeared in experimental data of Michelson and Morley in 1887.
From the November Krisher data in Figs. 6(a) and (f) the Right Ascension of the 3-space ﬂow direction was obtained from the local sidereal times of the maxima and minima, giving a RA of 6:09D, after correcting the apparent RA of 4:96D for the 42 inclination of the optical ﬁber to the local meridian. This RA was used in combination with the spacecraft earth-ﬂyby Doppler shift data, and is shown in Fig. 11.
The magnitudes of the Krisher phases are not used in de-

Fig. 8: Asymptotic ﬂyby conﬁguration in earth frame-of-reference, with spacecraft (SC) approaching Earth with velocity VE. The departing asymptotic velocity will have a diﬀerent direction but the same speed, as no force other than conventional Newtonian gravity is assumed to be acting upon the SC. The dynamical 3-space
velocity is v(r; t), though taken to be time independent during the
Doppler shift measurement, which causes the outward EM beam to
have speed c vE(r), and inward speed c + vE(r), where vE(r) =
 v(r)  cos( E), with E the angle between v and V. A similar descrip-
tion applies to the departing SC, labeled i f .
✬❅■❅ ✩ v☎✉✥S(DC✥)✥✾V✥✥E ☎D
Earth ☎❡TS ✠ ✫✪ VB
Fig. 9: Spacecraft (SC) earth ﬂyby trajectory, with initial and ﬁnal asymptotic velocity V, diﬀering only by direction. The Doppler shift is determined from Fig. 8 and (32). The 3-space ﬂow velocity at the location of the SC is v. The line joining Tracking Station (TS) to
SC is the path of the RF signals, with length D. As SC approaches earth v(D) changes direction and magnitude, and hence magnitude of projection vE(D) also changes, due to earth component of 3-space
ﬂow and also because of RF direction to/from Tracking Station. The SC trajectory averaged magnitude of this earth in-ﬂow is determined from the ﬂyby data and compared with theoretical prediction.
termining the RA for November, and so are not directly used in this report. Nevertheless these magnitudes provide a check on the physics of how the speed of light in optical ﬁbers is affected by the 3-space ﬂow. The phase diﬀerences   in Fig. 6a, which correspond to a 1st order in v=c experiment in which the Fresnel drag eﬀect must be taken into account, are shown to be consistent with the determined speed for November, noting that the use of phase comparators does not allow the determination of multiple 360 contributions to the phase diﬀerences. The analysis of the Krisher phase sum  +  in Fig. 6f, which correspond to a 2nd order in v=c experiment, requires the Lorentz contraction of the optical ﬁbers. as well as the Fresnel drag eﬀect, to be taken into account. The physics of optical ﬁbers in relation to this and other 3space physics will be discussed more fully elsewhere.
9 3-space ﬂow from Earth-ﬂyby Doppler shifts

The same eﬀect is observed in Ring Lasers [29] — which detect a side- The motion of spacecraft relative to the Earth are measured by real period of rotation of the Earth, and not the solar period. Ring Lasers observing the direction and Doppler shift of the transponded

cannot detect the 3-space direction, only a rate of rotation.

RF transmissions. This gives another technique to determine

58

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the speed and direction of the dynamical 3-space as mani-

fested by the light speed anisotropy [19]. The repeated de-

tection of the anisotropy of the speed of light has been, until

recently, ignored in analysing the Doppler shift data, causing

the long-standing anomalies in the analysis [20]. The use of

the Minkowski-Einstein choice of time and space coordinates

does not permit the analysis of these Doppler anomalies, as

they mandate that the speed of the EM waves be invariant.

Because we shall be extracting the Earth inﬂow eﬀect we

need to take account of a spatially varying, but not time-

varying, 3-space velocity. In the Earth frame of reference,

see Fig. 8, and using clock times from earth-based clocks, let

the transmitted signal from earth have frequency f . The time

for one RF maximum to travel distance D to SC from earth

is, see Fig. 9,

Z,

dr

= ( ) t

E  c

: vr

(21)

= 1 The next RF maximum leaves time T =f later and

arrives at SC at time, taking account of SC motion,

= + ( ) Z , 8 6

dr

t

T


E:
c vr

(22)

The period at the SC of the arriving RF is then

=

T

t

= + ( ) Z , 8 6

t T

,

dr
E  c v r

( ) E c v D

V

 ( ) E c v D

T:

(23)

Essentially this RF is reﬂected by the SC. Then the 1st

RF maximum takes time to reach the Earth

= + ( ) 

Z

dr

t

, 8 6 c Ev r

(24)

( ) Ignoring the projected 3-space velocity vE D , that is, as-
suming that the speed of light is invariant as per the usual lit-
eral interpretation of the Einstein 1905 light speed postulate,
we obtain instead

= ( ) 

c

f

f:

cV

(28)

The use of (28) instead of (27) is the origin of the putative anomalies. Expanding (28) we obtain

= = 2 

f

f

f

V

:

(29)

f

f

c

However expanding (27) we obtain, for the same Doppler

shift,

= = 1 + ( ) 2 + 


f

f

f

vD

V

: : : (30)

f

f

c

c

2 It is the prefactor to V =c missing from (29) that ex-
plains the spacecraft Doppler anomalies, and also permits yet
( ) another determination of the 3-space velocity v D , viz at
the location of the SC. The published data does not give the
( ) Doppler shifts as a function of SC location, so the best we can
do at present is to use a SC trajectory-averaged v D , namely
vE and vB , for the incoming and outgoing trajectories, as fur-
ther discussed below.
From the observed Doppler shift data acquired during a
ﬂyby, and then best ﬁtting the trajectory, the asymptotic hy-
perbolic speeds VE and VB are inferred from (29), but in-
correctly so, as in [20]. These inferred asymptotic speeds may
be related to an inferred asymptotic Doppler shift

= = 2 + fE f f

VE :::

(31)

f

f

c

and the 2nd RF maximum takes time

= + ( )   Z 

dr

t

T

, 86 86 c

E:
vr

(25)

Then the period of the returning RF at the Earth is


T

= =


t T


+


Zt

=
,

,

86 86

86
+ ( )  dr
E c v r

+ ( ) c v D

V

 E T :

+ ( ) E c v D

(26)

Then overall we obtain the return frequency to beO

1 + ( ) ( )  = = + ( ) ( ) f

T 

E c v D

E c v D

V

E c v D

E c v D

f: V

(27)

In practice a more complex protocol is used. OThis corrects the corresponding expression in [19], but without aﬀecting the ﬁnal results.

which from (30) gives


 2 = 1 + + VE

fE c

f

v E

V

:::

c

(32)

where V is the actual asymptotic speed. Similarly after the

ﬂyby we obtain

!

 2 = 1 + + VB

fB c
f

v B

V

:::

c

(33)

and we see that the “asymptotic” speeds VE and VB must
diﬀer, as indeed reported in [20]. We then obtain the expres-
sion for the so-called ﬂyby anomaly

= = V

VB VE

v B

v E V

(34)

c

 where here V V to suﬃcient accuracy, where V is the
average of VE and VB, The existing data on v permits ab

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initio predictions for V. As well a separate least-squares-

ﬁt to the individual ﬂybys permits the determination of the

average speed and direction of the 3-space velocity, relative

to the Earth, during each ﬂyby. These results are all remark-

ably consistent with the data from the various laboratory ex-
periments that studied v. We now indicate how vi and vf

+ were parametrised during the best-ﬁt to the ﬂyby data. In
(10) vgalactic vsun vorbital was taken as constant dur-

ing each individual ﬂyby, with vsun inward towards the Sun,
42 with value km/s, and vorbital as tangential to earth orbit 30 with value km/s — consequentially the directions of these

two vectors changed with day of each ﬂyby. The earth inﬂow

vearth in (10) was taken as radial and of an unknown ﬁxed

trajectory-averaged value. So the averaged direction but not

the averaged speed varied from ﬂyby to ﬂyby, with the in-
coming and ﬁnal direction being approximated by the ( i; i)
( ) and ( f ; f ) asymptotic directions shown in Table 2. The pre-
dicted theoretical variation of vearth R is shown in Fig. 10.

To best constrain the ﬁts to the data the ﬂyby data was used in

conjunction with the RA from the Krisher optical ﬁber data.

This results in the aberration plot in Fig. 11, the various ﬂyby

data in Table 2, and the Earth in-ﬂow speed determination in

Fig. 12. The results are in remarkable agreement with the re-

sults from Miller, showing the extraordinary skill displayed

by Miller in carrying out his massive interferometer exper-

iment and data analysis in 1925/26. The only eﬀect miss-

ing from the Miller analysis is the spatial in-ﬂow eﬀect into

the Sun, which aﬀected his data analysis, but which has been

partially corrected for in Sect. 5. Miller obtained a galactic

= 4 52 = 70 5 ﬂow direction of

: hrs,

: , compared to

= 4 29 that obtained herein from the NASA data of

: hrs,

= 75 0 5 :  , which diﬀer by only .

10 Earth 3-space inﬂow: Pound and Rebka experiment
The numerous EM anisotropy experiments discussed herein demonstrate that a dynamical 3-space exists, and that the
speed of the earth wrt this space exceeds 1 part in 1000 of c,
namely a large eﬀect. Not surprisingly this has indeed been
500 detected many times over the last 120 years. The speed of
nearly km/s means that earth based clocks experience a real, so-called, time dilation eﬀect from (6) of approximately 0.12 s per day compared to cosmic time. However clocks may be corrected for this clock dilation eﬀect because their speed
v though space, which causes their slowing, is measurable
by various experimental methods. This means that the absolute or cosmic time of the universe is measurable. This very much changes our understanding of time. However because of the inhomogeneity of the Earth 3-space in-ﬂow component the clock slowing eﬀect causes a diﬀerential eﬀect for clocks at diﬀerent heights above the Earth’s surface. It was this effect that Pound and Rebka reported in 1960 using the Harvard
tower [28]. Consider two clocks at heights h and h , with

Fig. 10: Earth 3-space inﬂow speed vs distance from earth in earth radii, as given in (7), plotted only for R > :. Combining the NASA/JPL optical ﬁber RA determination and the ﬂyby Doppler shift data has permitted the determination of the angle- and distanceaveraged inﬂow speed, to be  :" #km/s.

h=h

h, then the frequency diﬀerential follows from (6),

f f

=

H
1

v (h )
c

H
1

v

(h)
c


 v (h)2c v (h ) + =

=

1 2c

dv (r)
dr

h

+

=

=

g(r)
c

h

+

=

= c +:::

(35)

= = 
 
H using (3) with v v H v H for zero vorticity v 0,

and ignoring any time dependence of the ﬂow, and where ﬁ-

nally, is the change in the gravitational potential. The

actual process here is that, say, photons are emitted at the top

of the tower with frequency f and reach the bottom detec-

tor with the same frequency f — there is no change in the

= 0 frequency. This follows from (23) but with now V

giv-

= ing T T . However the bottom clock is running slower

because the speed of space there is faster, and so this clock

determines that the falling photon has a higher frequency, ie.

appears blue shifted. The opposite eﬀect is seen for upward

travelling photons, namely an apparent red shift as observed

by the top clock. In practice the Pound-Rebka experiment

used motion induced Doppler shifts to make these measure-

ments using the Mo¨ssbauer eﬀect. The overall conclusion is
that Pound and Rebka measured the derivative of v wrt to

height, whereas herein we have measured that actual speed,

but averaged wrt the SC trajectory measurement protocol. It

is important to note that the so-called “time dilation” eﬀect is

really a “clock slowing” eﬀect — clocks are simply slowed

by their movement through 3-space. The Gravity Probe A

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= 4 52 = 70 5 Fig. 11: South celestial sphere with RA and Dec shown. The red dotted circle shows the Miller aberration path discovered in 1925/26,

from [8]. The red point at

: hrs,

: shows the galactic ﬂow direction determined by Miller, after removing earth-

orbit aberration eﬀect. The dark blue circle shows the aberration path from best-ﬁtting the Earth-ﬂyby Doppler shift data and using the

12 4 5 optical-ﬁber RA data point for November from Krisher [18], see Fig. 12. This corresponds to a best ﬁt averaged earth inﬂow speed of
: km/s. The blue aberration paths show the best-ﬁt if (a) upper circle: earth inﬂow speed = 0 km/s, (b) = 4.0 km/s, (c) = 8.0 km/s

and (d) = 12.4 km/s (thick blue circle). The actual 3-space ﬂow directions are shown by light-blue background to labels for the ﬂybys in

= 4 29 = 75 0 Aug, Dec, Jan and Mar, and given in Table 2. The red point at

: hrs,

: shows the optical-ﬁber/earth-ﬂyby determined

5 galactic ﬂow direction, also after removal of earth-orbit aberration eﬀect, and is only from the above mentioned Miller direction. The

miss-ﬁt angle  between the best-ﬁt RA and Dec for each ﬂyby is given in Table 2, and are only a few degrees on average, indicating

the high precision of the ﬁt. This plot shows the remarkable concordance between the NASA/JPL determined 3-space ﬂow characteristics

and those determined by Miller in 1925/26. It must be emphasised that the optical-ﬁber/ﬂyby aberration plot and galactic 3-space ﬂow
6 09 direction is obtained completely independently of the Miller data. The blue line at : D is the orientation corrected Krisher RA, and has 1 an uncertainty of D, caused by wave/turbulence eﬀects. The ﬁducial RA of 5D and 17D, shown in red, are the ﬁducial local sidereal times

= 1 00026 shown in Figs. 3, 6 and 7. The point EP is the pole of the ecliptic. The speed and declination diﬀerences between the Miller and NASA data
arise from Miller being unaware of the Sun 3-space inﬂow eﬀect — correcting for this and using an air refractive index of n :

atop Mt. Wilson increases the Miller data determined speed and moves the declination slightly southward, giving an even better agreement

with the NASA data. Here we have merely reproduced the Miller aberration plot from [8].

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Parameter

GLL-I

GLL-II

NEAR

Cassini

Rosetta

M’GER

Date
V km/s
E deg E deg B deg B deg
L hrs L deg
v km/s (O) V mm/s (P) V mm/s (P)  deg

Dec 8, 1990 8.949 266.76 -12.52 219.97 34.15
5.23 80.3 490.6
3.92 0.3 4.07 1

Dec 8, 1992 8.877 219.35 34.26 174.35 4.87
5.23 80.3 490.6
4.6 1.0 5.26 1

Jan 23, 1998 6.851 261.17 20.76 183.49 71.96
3.44 80.3 497.3
13.46 0.01 13.45 2

Aug 18, 1999 16.010 334.31 12.92 352.54 4.99
5.18 70.3 478.3
21 0.76 4

Mar 4, 2005 3.863 346.12 2.81 246.51 34.29
2.75 76.6 499.2
1.80 0.03 0.86 5

Aug 2, 2005 4.056 292.61 31.44 227.17 31.92
4.89 69.5 479.2
0.02 0.01 4.56 —

Table 2: Earth ﬂyby parameters from [20] for spacecraft Galileo (GLL: ﬂybys I and II), NEAR, Cassini, Rosetta and MESSENGER

(M’GER). V is the average osculating hyperbolic asymptotic speed, and are the right ascension and declination of the incoming (i)

and outgoing (f) osculating asymptotic velocity vectors, and (O) V is the putative “excess speed” anomaly deduced by assuming that

the speed of light is isotropic in modeling the Doppler shifts, as in (31). The observed (O) V values are from [20], and after correcting

for atmospheric drag in the case of GLL-II, and thruster burn in the case of Cassini. (P) V is the predicted “excess speed”, using (34),

after least-squares best-ﬁtting that data using (34): L and L and v are the right ascension, declination and the 3-space ﬂow speed for each

= 4 29 ﬂyby date, which take account of the Earth-orbit aberration and earth inﬂow eﬀects, and correspond to a galactic ﬂow with

: hrs,

= 75 0 = 486 : and v

km/s in the solar system frame of reference.  is the error, in the best ﬁt, for the aberration determined ﬂow

direction, from the nearest ﬂyby ﬂow direction. In the ﬁtting the MESSENGER data is not used, as the data appears to be anomalous.

experiment [33] also studied the clock slowing eﬀect, though again interpreted diﬀerently therein, and again complicated by additional Doppler eﬀects.

11 CMB direction

Fig. 12: The weighted angle- and distance-averaged earth 3-space
inﬂow speed vA=HJD, see Fig. 10, as determined from NASA data, up-

= 4 96 = 6 09 per green plot. Uses the averaged Right Ascension from the Krisher

et al. data for November,

: D, but corrected to

:D

for orientation eﬀect of the optical ﬁber, shown by the thick blue

line, with uncertainty range from wave eﬀects shown by two thin

blue lines, compared with the predicted RA from ﬁtting the ﬂyby

data, as shown in Fig. 11. The red plot shows that prediction for

various averaged inﬂow speeds, with +ve speeds being an inﬂow,

while -ve speeds are an outﬂow. The earth ﬂyby aberration ﬁts for
= 0 +4 0 +8 0 +12 4 vA=HJD ; : , : and : km/s are shown in Fig. 11. The+11 2 ory gives that the inﬂow speed is : km/s at the Earth’s surface —

shown by lower green plot. So the detected averaged inﬂow speed

seems to be in good agreement with an expected averaged value.

This is the ﬁrst detection of the Earth’s spatial inﬂow, and the accel-

eration of this ﬂow is responsible for the Earth’s gravity. Note that

the ﬂyby data clearly mandates an inﬂow (+ve values in this ﬁgure

and not an out-ﬂow — having -ve values).

The Cosmic Microwave Background (CMB) velocity is often confused with the Absolute Motion (AM) velocity or lightspeed anisotropy velocity as determined in the experiments discussed herein. However these are unrelated and in fact point in very diﬀerent directions, being almost at 90 to each other, with the CMB velocity being 369 km/s in direction
( = 11:2D, = 7:22 ). The CMB velocity vector was
ﬁrst determined in 1977 by Smoot et al. [30]. The CMB velocity is obtained by deﬁning a frame of
reference in which the thermalised CMB 3 K radiation is isotropic, that is by removing the dipole component, and the CMB velocity is the velocity of the Earth in that frame. The CMB velocity is a measure of the motion of the solar system relative to the last scattering surface (a spherical shell) of the universe some 13.4Gyrs in the past. The concept here is that at the time of decoupling of this radiation from matter that matter was on the whole, apart from small observable ﬂuctuations, on average at rest with respect to the 3-space. So the CMB velocity is not motion with respect to the local 3-space now; that is the AM velocity. Contributions to the AM velocity would arise from the orbital motion of the solar system within the Milky Way galaxy, which has a speed of some 250 km/s, and contributions from the motion of the Milky

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Way within the local cluster, and so on to perhaps super clusters, as well as ﬂows of space associated with gravity in the Milky Way and local galactic cluster etc. The diﬀerence between the CMB velocity and the AM velocity is explained by the spatial ﬂows that are responsible for gravity at the galactic scales.
12 Conclusions
We have shown that the NASA/JPL optical ﬁber and spacecraft earth ﬂyby data give another independent determination of the velocity of the solar system through a dynamical 3space. The resulting direction is in remarkable agreement with the direction determined by Miller in 1925/26 using a gas-mode Michelson interferometer. The Miller speed requires a better knowledge of the refractive index of the air atop Mt. Wilson, where Miller performed his experiments, but even using the STP value we obtain reasonable agreement with the NASA/JPL determined speed. Using an air refractive index of 1.00026 in place of the STP value of 1.00029 would bring the Miller speed into agreement with the NASA data determined speed. As well the NASA/JPL data has permitted the ﬁrst direct measurement of the ﬂow of 3-space into the Earth, albeit averaged over spacecraft trajectory during their ﬂybys. This is possible because the inﬂow component is radially inward and so changes direction relative to the other ﬂow components during a ﬂyby, making the ﬂyby Doppler shifts sensitive to the inﬂow speed.
It must be emphasised that the long-standing and repeated determinations of the anisotropy of vacuum EM radiation is not in itself in contradiction with the Special Relativity formalism — rather SR uses a diﬀerent choice of space and time variables from those used herein, a choice which by construction mandates that the speed of EM radiation in vacuum be invariant wrt to that choice of coordinates [3]. However that means that the SR formalism cannot be used to analyse EM radiation anisotropy data, and in particular the ﬂyby Doppler shift data.
The discovery of absolute motion wrt a dynamical 3-space has profound implications for fundamental physics, particularly for our understanding of gravity and cosmology. It shows that clocks, and all oscillators, whether they be classical or quantum, exhibit a slowing phenomenon, determined by their absolute speed though the dynamical 3-space. This “clock slowing” has been known as the “time dilation” eﬀect — but now receives greater clarity. It shows that there is an absolute or cosmic time, and which can be measured by using any clock in conjunction with an absolute speed detector — many of which have been mentioned herein, and which permits the “clock slowing” eﬀect to be compensated. This in turn implies that the universe is a far more coherent and nonlocally connected process than previously realised, although a model for this has been proposed [1]. It also shows that the now standard discussion of the limitations of simultaneity

were really misleading — being based on the special space and time coordinates invoked in the SR formalism, and that simultaneity is a fact of the universe, albeit an astounding one.
As well successful absolute motion experiments have always shown wave or turbulence phenomena, and at a signiﬁcant scale. This is a new phenomena that is predicted by the dynamical theory of 3-space. Ongoing development of new experimental techniques to detect and characterise these wave phenomena will be reported elsewhere.
Submitted on June 18, 2009 / Accepted on July 10, 2009
References
1. Cahill R.T. Process physics: from information theory to quantum space and matter. Nova Science Pub., New York, 2005.
2. Cahill R.T. Dynamical 3-space: a review. In: Ether Space-time and Cosmology: New Insights into a Key Physical Medium, Duﬀy M. and Le´vy J., eds., Apeiron, 2009, 135–200.
3. Cahill R.T. Unravelling Lorentz covariance and the spacetime formalism. Progress in Physics, 2008, v. 4, 19–24.
4. Braxmaier C. et al. Phys. Rev. Lett., 2002, v. 88, 010401; Mu¨ller H. et al. Phys. Rev. D, 2003, v. 68, 116006-1-17; Mu¨ller H. et al. Phys. Rev. D, 2003, v. 67, 056006; Wolf P. et al. Phys. Rev. D, 2004, v. 70, 051902-1-4; Wolf P. et al. Phys. Rev. Lett., 2003, v. 90, 060402; Lipa J.A. et al. Phys. Rev. Lett., 2003, v. 90, 060403.
5. Cahill R.T. and Kitto K. Michelson-Morley experiments revisited. Apeiron, 2003, v. 10(2), 104–117.
6. Cahill R.T. The Michelson and Morley 1887 experiment and the discovery of absolute motion. Progress in Physics, 2005, v. 3, 25–29.
7. Michelson A.A. and Morley E.W. Am. J. Sc., 1887, v. 34, 333– 345.
8. Miller D.C. Rev. Mod. Phys., 1933, v. 5, 203–242. 9. Illingworth K.K. Phys. Rev., 1927, v. 3, 692–696. 10. Joos G. Ann. d. Physik, 1930, v. 7, 385. 11. Jaseja T.S. et al. Phys. Rev. A, 1964, v. 133, 1221. 12. Torr D.G. and Kolen P. In: Precision Measurements and Fun-
damental Constants, Taylor, B.N. and Phillips, W.D. eds. Natl. Bur. Stand. (U.S.), Spec. Pub., 1984, 617, 675. 13. Cahill R.T. The Roland DeWitte 1991 experiment. Progress in Physics, 2006, v. 3, 60–65.
14. Cahill R.T. A new light-speed anisotropy experiment: absolute motion and gravitational waves detected. Progress in Physics, 2006, v. 4, 73–92.
15. Mune´ra H.A. et al. In: Proceedings of SPIE, 2007, v. 6664, K1– K8, eds. Roychoudhuri C. et al.
16. Cahill R.T. Optical-ﬁber gravitational wave detector: dynamical 3-space turbulence detected. Progress in Physics, 2007, v. 4, 63–68.
17. Cahill R.T. and Stokes F. Correlated detection of sub-mHz gravitational waves by two optical-ﬁber interferometers. Progress in Physics, 2008, v. 2, 103–110.

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18. Krisher T.P., Maleki L., Lutes G.F., Primas L.E., Logan R.T., Anderson J.D. and Will C.M. Test of the isotropy of the oneway speed of light using hydrogen-maser frequency standards. Phys. Rev. D, 1990, v. 42, 731–734.
19. Cahill R.T. Resolving spacecraft Earth-ﬂyby anomalies with measured light speed anistropy. Progress in Physics, 2008, v. 4, 9–15.
20. Anderson J.D., Campbell J.K., Ekelund J.E., Ellis J. and Jordan J.F. Anomalous orbital-energy changes observed during spaceraft ﬂybys of Earth. Phys. Rev. Lett., 2008, v. 100, 091102.
21. Cahill R.T. Quantum foam, gravity and gravitational waves. In: Relativity, Gravitation, Cosmology: New Developments, Dvoeglazov V., ed., Nova Science Pub., New York, 2009.
22. Hicks W.M. Phil. Mag., 1902, v. 3, 9–42.
23. Hertz H. On the fundamental equations of electro-magnetics for bodies in motion. Wiedemann’s Ann., 1890, v. 41, 369; Electric waves. Collection of scientiﬁc papers. Dover Publ., New York, 1962.
24. Cahill R.T. Dynamical fractal 3-space and the generalised Schro¨dinger equation: equivalence principle and vorticity effects. Progress in Physics, 2006, v. 1, 27–34.
25. Cahill R.T. A quantum cosmology: no dark matter, dark energy nor accelerating Universe. arXiv: 0709.2909.
26. Cahill R.T. Unravelling the dark matter — dark energy paradigm. Apeiron, 2009, v. 16, no. 3, 323–375.
27. Cahill, R.T. The dynamical velocity superposition eﬀect in the quantum-foam theory of gravity. In: Relativity, Gravitation, Cosmology: New Developments, Dvoeglazov V., ed., Nova Science Pub., New York, 2009.
28. Pound R.V. and Rebka Jr. G.A. Phys. Rev. Lett., 1960, v. 4(7), 337–341.
29. Schreiber K.U., Velikoseltsev A., Rothacher M., Klu¨gel T., Stedman G.E. and Wilsthire D.L. Direct measurement of diurnal polar motion by ring laser gyroscopes. J. Geophys. Res., 2004, v. 109, B06405; arXiv: physics/0406156.
30. Smoot G.F., Gorenstein M.V. and Muller R.A. Phys. Rev. Lett., 1977, v. 39(14), 898.
31. Cahill R.T. 3-space inﬂow theory of gravity: boreholes, blackholes and the ﬁne structure constant. Progress in Physics, 2006, v. 2, 9–16.
32. Cahill R.T. Dark matter as a quantum foam in-ﬂow eﬀect. In: Trends in Dark Matter Research, ed. J. Val Blain, Nova Science Pub., NY, 2005, 95–140.
33. Vessot R.F.C. et al. Test of relativistic gravitation with a spaceborne hydrogen maser. Rev. Mod. Phys., 1980, v. 45, 2081.

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Geometry of Space-Time
Ulrich E. Bruchholz
Schillerstrasse 36, D-04808 Wurzen, Germany E-mail: Ulrich.Bruchholz@t-online.de; http://www.bruchholz-acoustics.de
The geometry of the space-time is deduced from gravitational and electromagnetic ﬁelds. We have to state that Rainich’s “already uniﬁed ﬁeld theory” is the ground work of the proposed theory. The latter is deduced independently on Rainich. Rainich’s analogies are brilliantly validated. His formulae are veriﬁed this way. Further reaching results and insights demonstrate that Rainich’s theory is viable. In ﬁnal result, we can formulate an enhanced equivalence principle. It is the equivalence of Newton’s force with the Lorentz force.

To the memory of John Archibald Wheeler, who foresaw this simple idea.
1 The predecessor

other manifolds.

The known source-free Einstein-Maxwell equations

=  41 Rik 

gikFabF ab


FiaFka ;

(1)

George Yuri Rainich already saw the analogies of the electromagnetic with the gravitational ﬁeld. Since Einstein’s equivalence principle implies a geometric approach of gravitation [1], electromagnetism has to be geometry too. Not enough, Rainich also saw that the electromagnetic ﬁeld tensor is performed from the congruences of two dual surfaces. It is the analogy of the curvature vector of the current path, performed from the main normal, see on generalized Frenet formulae in [2].
One can well pursue Rainich’s way in his papers from 1923 to 1924. First, he tried to ﬁnd a non-Riemannian geometry for the electromagnetic vacuum ﬁeld [3]. Later, he saw that Riemannian geometry is suﬃcient to describe electromagnetism [4, 5]. Rainich’s identities (also called algebraic Rainich conditions) are deduced without special techniques in [6]. Present paper provides a further derivation of Rainich’s identities, additionally identifying the concrete geometry.
Since a full geometric approach precludes sources, Rainich concluded a central role of singularities. However, it is deduced in [7] that this role is commonly overestimated. The singularities pass for a bar to the geometric approach. It is shown in [7] that formal singularities are in areas (according to observer’s coordinates), which are not locally imaged. The related boundaries specify the discrete values of the integration constants from ﬁeld equations [7].

= 0 F ia;a

;

(2)

+ + = 0 Fij;k

Fj k;i

Fki;j

(3)

involve a special kind of Riemannian geometry, what is explained as follows.
The Ricci main directions (written in terms according to Eisenhart [2]) follow from

det jRik + j gik = 0

(4)

with the solutions

= = + = = j1 j4

0 ; j2 j3

0

(5)

with

= = = = 02 R1aR1a R2aR2a R3aR3a R4aR4a ; (6)

what leads directly to Rainich’s identities

= = 14 RiaRka

k i

0

2

R R k

ba

ia

b

:

(7)

Characteristical are the two double-roots, that means: There are two dual surfaces of the congruences

ej1iej4k ej1kej4i and ej2iej3k ej2kej3i

2 The derivation
The ﬁrst precursor is to see in [8]. The derivation follows the steps according to the chapter “Geometric interpretation of the Ricci tensor — the Ricci main directions” in [2]. As well, we shall see that the space-time involves a vital diﬀerence to

with minimal and maximal mean Riemannian curvature. ej1 : : : ej4 are the vectors of an orthogonal quadruple (vierbein) in those “main surfaces”. At single roots we had 4 main directions. But we will see that the main surfaces are a spe-
Where  has a negative value, what has to do with the special signature of the space-time.

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ciality of the space-time. With the obtained solutions we get

9

gik = ej iej k + ej iej k + + ej! iej! k ej" iej" k ;

>>>>>>=

Rik 

=

ej iej k + ej iej k + + ej! iej! k + ej" iej" k :

>>>>>>;

(8)

If we set

cjik = cjki = Fabejiaejkb

(9)

follows from elementary calculations


9

 (cj !) + (cj") = 2 ; = cj = cj!" = cj! = cj " = 0 : ;

(10)

With it, the ﬁeld tensor

Fik = cj"(ej iej" k ej kej" i) +

+ cj !(ej iej! k ej kej! i)

(11)

is performed from the main surfaces. Rainich knew also these relations [4, 5].

3 Conclusions
Montesinos and Flores [9] deduce the electromagnetic energy-momentum tensor via Noether’s theorem [10]. That means, the Ricci tensor must have just the form according to Eqn. (1). Therefore, the geometry with the main surfaces is necessary for the space-time. Since the electromagnetic ﬁeld tensor is performed by the main surfaces, it is a curve parameter of the current path like the curvature vector (which is performed by the main normal, and is the geometric expression of both gravitation and accelerated motion), as Rainich already saw. We can formulate an enhanced equivalence principle this way. It is the equivalence of the Lorentz force with Newton’s force. Because the test body means a current point on the path, i.e. all forces to the test body come from curve parameters.
Montesinos and Flores [9] derived a symmetric energymomentum tensor from three diﬀerent theories, with the result that sources have to vanish in each case. That means:
1. The Maxwell theory is suﬃcient, because it runs as demonstrated in [7], even also regarding quantization. Non-Riemannian ansatzes are not needed;
2. Any ansatz with distributed charges or masses is false in principle. This error was helpful in classical theories before Einstein, which were separately handled. Now, such error turns up to be counterproductive.
It appears inviting to specify metrics ﬁrst via Eqn. (7) (see [6]), but this method has narrow limits. The electromagnetic integration constants (charge, magnetic momentum)

come from Maxwell’s equations. The geometric theory of ﬁelds [7] uniﬁes electromagnetism with gravitation natural way.
Submitted on June 23, 2009 / Accepted on July 17, 2009
References
1. Einstein A. Grundzu¨ge der Relativita¨tstheorie. A backtranslation from the Four Lectures on Theory of Relativity. Akademie-Verlag Berlin, Pergamon Press Oxford, Friedrich Vieweg & Sohn Braunschweig, 1969.
2. Eisenhart L.P. Riemannian geometry. Princeton University Press, Princeton, 1949.
3. Rainich G.Y. The electromagnetic ﬁeld and curvature. Proc. N.A.S., 1923, v. 9, 404–406.
4. Rainich G.Y. Electrodynamics in the general relativity theory. Proc. N.A.S., 1924, v. 10, 124–127.
5. Rainich G.Y. Second note on electrodynamics in the general relativity theory. Proc. N.A.S., 1924, v. 10, 294–298.
6. Caltenco Franca J.H., Lo´pez-Bonilla J.L., Pen˜a-Rivero R. The algebraic Rainich conditions. Progress in Physics, 2007, v. 3, 34–35.
7. Bruchholz U.E. Key notes on a geometric theory of ﬁelds. Progress in Physics, 2009, v. 2, 107–113.
8. Bruchholz U. Zur Berechnung stabiler elektromagnetischer Felder. Z. elektr. Inform.- u. Energietechnik, Leipzig, 1980, Bd. 10, 481–500.
9. Montesinos M. and Flores E. Symmetric energy-momentum tensor in Maxwell, Yang-Mills, and Proca theories obtained using only Noether’s theorem. arXiv: hep-th/0602190.
10. Noether E. Invariante Variationsprobleme. Nachr. d. Ko¨nig. Gesellsch. d. Wiss. zu Go¨ttingen, Math-phys. Klasse, 1918, 235–257 (English translation by M. A. Tavel in: Transport Theory and Statistical Physics, 1971, v. 1(3), 183–207; arXiv: physics/0503066).

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Derivation of Planck’s Constant from Maxwell’s Electrodynamics
Ulrich E. Bruchholz
Schillerstrasse 36, D-04808 Wurzen, Germany E-mail: Ulrich.Bruchholz@t-online.de; http://www.bruchholz-acoustics.de
Like Planck deduced the quantization of radiation energy from thermodynamics, the same is done from Maxwell’s theory. Only condition is the existence of a geometric boundary, as deduced from author’s Geometric theory of ﬁelds.

Let us go from Maxwell’s equations of the vacuum that culminate in wave equations for the electric potential

'=0

(1)

and

)=0

(2)

This volume integral would be impossible without the boundary, because the linear solution, being alone, is not physically meaningful for the inﬁnite extension.
We can write the last equation as

E=!

(8)

for the magnetic vector potential.

Take the wave solution from Eqn. (2), in which the vector

potential consists of a single component vertical to the prop-

agation direction


Ay = Ay ! (t x) ;

(3)

where c = 1 (normalization), ! is a constant (identical with

the circular frequency at the waves), x means the direction of the propagation, Ay is an arbitrary real function of ! (t x)
(independent on y; z).

The ﬁeld strengthes respectively ﬂow densities (which are

the same in the vacuum) become

Ey

=

@Ay @t

=


!Ay !

(t


x) ;

(4)

and

Bz =

@Ay @x

=


!Ay !

(t


x) ;

(5)

where Ay means the total derivative.

The energy density of the ﬁeld results in

= "
2


(Ey + Bz ) = ! " Ay ! (t


x) ;

(6)

(E means here energy), or

E = h;

(9)

because the latter volume integral has a constant value. The known fact that this value is always the same means also that only one solution exists with ! as a parameter.
Keep the calculation for the concrete value. This can be done only in numerical way, and might be a great challenge. The value of the above volume integral has to become =" . With it, the fundamental relation of Quantum Mechanics follows from classical ﬁelds.
Summarizingly, the derivation involves two predictions:
1. Photon has a geometric boundary. That may be the reason that photon behaves as a particle;
2. There is only one wave solution.

Submitted on June 23, 2009 / Accepted on July 23, 2009

References
1. Bruchholz U.E. Key notes on a geometric theory of ﬁelds. Progress in Physics, 2009, v. 2, 107–113.

where " means the vacuum permitivity.

The geometric theory of ﬁelds allows geometric bound-

aries from the non-linearities in the equations of this the-

ory [1]. If one assumes such a boundary, like those in station-

ary solutions of the non-linear equations, the included energy

becomes the volume integral within this boundary
ZZZ

 d (t x) dy dz =

ZZZ 


= !"

Ay ! (t x) d ! (t x) dy dz : (7)

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Changes in the Shape of Histograms Constructed from the Results of 239Pu Alpha-Activity Measurements Correlate with the Deviations of the Moon
from the Keplerian Orbit
Sergei N. Shapovalov , Ilya A. Rubinsteiny, Oleg A. Troshichev , and Simon E. Shnollz
St. Petersburg Arctic and Antarctic Institute, 38 Bering Str., St. Petersburg 199397, Russia ySkobeltsin’s Institute of Nuclear Physics, Moscow State University, Moscow 119991, Russia
zDepartment of Physics, Moscow State University, Moscow 119992, Russia zInst. of Theor. and Experim. Biophysics, Russian Acad. of Sci., Pushchino, Moscow Region, 142290, Russia
zPuschino State University, Prospect Nauki 3, Pushchino, Moscow Region, 142290, Russia E-mail: shapovalov@aari.nw.ru; shnoll@mail.ru
We have found that the shape of the histograms, constructed on the basis of the results of radioactivity measurements, changes in correlation with the distortions of the lunar Keplerian orbit (due to the gravitational inﬂuence of the Sun). Taking into account that the phenomenon of “macroscopic ﬂuctuations” (regular changes in the ﬁne structure of histograms constructed from the results of measurements of natural processes) does not depend on the nature of the process under study, one can consider the correlation of the histogram shape with the Moon’s deviations from the Keplerian orbit to be independent from the nature of the process the histograms were obtained on.

1 Introduction
In the last decades, the studies of solar-terrestrial relations, which were initiated by A. L. Chizhevsky [1, 2], rest upon the concept that these relations have an electromagnetic origin [3–9]. A supposition that the solar-terrestrial relations could be of gravitational nature — when the matter at issue are physico-chemical, chemical and biochemical processes — would raise objections, as the energy change upon gravitational disturbances is much less than that observed in the processes mentioned. As for correlations of physiological processes with tidal forces (see, for example, [10–12]), they can be explained on the basis of complex indirect mechanisms.
Nevertheless, there were reports [13–21] on a strong correlation between variation of some physical and biochemical processes and deviations of the Moon from the Keplerian orbit (evection, variation and annual inequality; see [30]). The conclusion was that gravitational disturbances should play an essential role in these phenomena. The processes that correlations were revealed for were very diﬀerent in their nature: there were ﬂuctuations of “computer time”, !'Pu -activity, the rate of a model chemical redox reaction, the content of haemoglobin in erythrocytes, and urea secretion.
There is no trivial explanation to the fact that physical and biochemical processes, which are little aﬀected by tidal forces, correlate with changes of the lunar orbit.
As shown for the processes of diverse nature, the spectrum of their amplitude ﬂuctuations (i.e., the shape of the corresponding histograms) correlates with a number of cosmophysical factors [22–28]. The change of energy in those processes (noise in electronic circuits, -decay, chemical reac-

tions) varies by tens orders of magnitude, yet the correlations are the same. Evidently, we deal with correlations of a nonenergy nature. So we can suggest that the correlations of various processes with the distortions of the lunar orbit reported in [13–21] have a non-energy nature as well.
Thereby we have checked if changes in the shape of histograms constructed from the results of !'Pu -activity measurements correlate with the deviations of the Moon from the Keplerian orbit. The measurements were carried out at NovoLazarevskaya station (Antarctida) and in Pushchino in 2003– 2008. Analysing regularities in the change of the histogram shape, we found periods corresponding to the periodical deviations of the Moon from the Keplerian orbit: variation (14.8 days) and evection (31.8 days). The correlations are analogous to those reported earlier [13–21], which suggests a common and very general nature of all these phenomena.
2 Materials and methods
The measurements of !'Pu -activity were performed at Novo-Lazarevskaya station (Antarctida) and in Pushchino in 2003–2008. -Activity was monitored continuously, with a second interval, using devices constructed by one of the authors (I. A. Rubinstein). The analysis of data consists in pairwise comparing of histograms constructed from the results of measurements. Histograms were constructed either for 60-point segments of one-second measurements (1-min histograms) or for 60-point segments of one-minute measurements (1-h histograms). All the operations of histogram construction and analysis, as well as calculation of intervals between similar histograms and plotting the corresponding dis-

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Fig. 1: Comparing 1-h histograms reveals periods equal to 350–354 h (in the region of variation, 14.8 days), 647 h (in the region of the
27-day period) and 762 h (in the region of evection, 31.8 days). In the ﬁgure, the number of similar histogram pairs (O-axis) is plotted versus the interval between similar histograms (N-axis, h).
tributions, were conducted with the aid of a computer program written by E. V. Pozharsky [22]. The decision of two histograms to be or not to be similar was made by an expert upon visual evaluation. A detailed description of all the procedures (measurements, histogram construction and analysis) can be found in [22].

3 Results
3.1 The shape of histograms changes with the periods of evection and variation

Figs. 1 and 2 show the results of our search for periodical changes in the shape of histograms constructed from the Antarctic data (Novo-Lazarevskaya station; since May 26, 2005 till the end of the year). We compared series of both 1-min and 1-h histograms in the regions of the putative periods: 762 6 h (a 31-day period, evection), 648 6 h (a 27-day period) and 355 6 h (a 15-day period, variation).
All the expected periods can be seen in Fig. 1. However, the period that corresponds to evection is, ceteris paribus, much more pronounced. To be sure that the periods revealed are not artefacts, we repeated the analysis many times with diﬀerent data. Fig. 2 shows the summary result of ﬁve other experiments, in which we compared 1-h histograms constructed from the data obtained on April–October, 2004.
Along with 1-h histograms, we also compared 1-min ones. Fig. 3 shows the results of this analysis, which was made in the region of evection period.
As can be seen in Fig. 3, the 60-fold increase in “resolution” does not change the character of the distribution: there is a sharp extremum, which corresponds to the evection pe-

Fig. 2: Determination of the evection period by comparing 1-h histograms constructed from the results of !'Pu -activity measurements on April–October, 2004 (a summary result of ﬁve experiments). Axes are deﬁned as in Fig. 1.
riod. It is very surprising. Evection is a rather slow process: its period equals to 31.8 days. Naturally, one minute (out of 45779!) is by no means enough for evection to manifest itself — the distortion of the Keplerian orbit will be negligible. So we believe that the clear periodicity in the alteration of the histogram shape cannot be explained by a slow change of the “eﬀecting force”.
3.2 “Palindrome eﬀects” in the evection periods
It seems that the apparently paradoxal narrowness of the extrema we see in the above ﬁgures has a relation to the sharp spatial anisotropy of our world [26–29]. Many observations

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Fig. 4: Evection phenomenon (periodical change of the extent of distortion of the lunar Keplerian orbit) on July–August, 2005. Evection maxima in 2005: May 26, June 26, July 27, August 28–29, September 29–30, November 1 and December 2. Evection minima in 2005: June 10, July 11–12, August 12–13, September 13–14, October 15–16 and November 16.

Fig. 3: Comparing 1-min histograms gave an evection period equal to 45779 min (31.79 days). Axes are deﬁned as in Fig. 1.
conﬁrmed this supposition — in particular, the experiments that made use of collimators, isolating narrowly directed beams of -particles [24–25]. This anisotropy is stable, which is evidenced by the high probability of a certain histogram shape to reappear every time the laboratory has the same orientation towards the sphere of ﬁxed stars. With the Earth rotating about its axis and moving along the circumsolar orbit, the laboratory will repeatedly pass through such points of the same star-related orientation. A manifestation of stable anisotropy of our space is the phenomenon of “palindromes”, which is the high probability of a series of “daytime” histograms to be similar to the inverse series of the “nighttime” ones. In the nighttime, the rotation of the Earth is co-directed with its movement along the sircumsolar orbit, this being the opposite in the daytime. As a result, the sequence of “star-orientation points” that the laboratory passes

through in the nighttime will be reversibly scanned by the laboratory in the daytime. Accordingly, series of daytime histograms were found to be opposite to the correspondent series of the nighttime ones [27, 28], with the “day-” and “nighttime” being accurately deﬁned as the local time since 6:00 to 18:00 (daytime) and since 18:00 to 6:00 of the next day (nighttime). Figuratively speaking, the rotating Earth consecutively reads the same text ﬁrst in the direct and then in the inverse order, and the result is the same — as in the phrase “step on no pets”.
As it turned out, the histogram series that correspond to the “direct” and “inverse” halfs of the evection cycle are also “palindromes”.
The periodical changes of the lunar Keplerian orbit in the evection cycles that correspond to the periods of our measurements are given in Fig. 4. According to this graph, we prepared series of 1-h histograms constructed from the results of !'Pu -activity measurements. The series were divided into the “odd” and “even” ones, corresponding to the descending and ascending halfs of the evection periods respectively (each half lasting 381.6 h or, more precisely, 22896 min). Then we compared the “odd” series to the “even” ones pairwise, with the even series being of two types: direct and inverse (with the direct and inverse sequence of histograms).
As shown in Fig. 5, there is a high probability of an “even” histogram to be similar to the “odd“ one of the same order number when the series of “odd” histograms is inverse. Without inversion, the similarity is much less probable. This is a typical palindrome.
4 Discussion
Thus, the shape of histograms constructed from the results of radioactivity measurements changes in correlation with the distortions of the lunar Keperian orbit caused by the gravita-

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Fig. 5: The palindrome eﬀect. When a series of consecutive 1-h histograms of the 1st (descending) half of the evection period is compared
to the corresponding series of the 2nd (ascending) half, the high probability of histograms of the same order number to be similar is observed only in the case of inversion of the 2nd histogram series. The ﬁgure shows a summary result of analysis of four diﬀerent sets of
data obtained in the period since May 26 to October 1, 2005 at Novo-Lazarevskaya station.

tional inﬂuence of the Sun. It changes in the same manner as the processes reported in [13–21]. Taking into account that the phenomenon of “macroscopic ﬂuctuations” (i.e., regular changes in the ﬁne structure of histograms constructed from the results of measurements of natural processes) does not depend on the nature of the process studied [26, 28], one can consider the correlation of the histogram shape with the deviations of the Moon from the Keplerian orbit to be independent of the process nature as well. Since gravitational forces would have no direct impact on physico-chemical and biological processes in terms of energy, the correlations revealed can be considered as resulting from gravitation-induced disturbances in the space geometry. These disturbances, changes of space curvature — to formulate in general, changes of the spacial-temporal scale — should equally manifest themselves in the processes of any nature. The data on strong correlations revealed for the ﬂuctuations of “computer time” [13– 21] might be an illustration of such alterations of the spacialtemporal scale.
The phenomena of half-day and half-year palindromes were explained by the repetition of a certain orientation of the Earth towards the Sun [27] and the sphere of ﬁxed stars [28] respectively. Adopting an analogous explanation to the palindrome with the period equal to that of evection (31.8 days) assumes a strong spatial anisotropy caused by the Moon.
Acknowledgements
S. E. Shnoll is thankful to M. N. Kondrashova for permanent discussions and spiritual support. The authors thank their colleagues for valuable discussions. We also appreciate help from the side of T. A. Zenchenko and K. I. Zenchenko, in conducting measurements and maintaining a computer bank of

experimental results. We are pleased to thank A. V. Agafonoﬀ for creative English translation. S. E. Shnoll is greatly indebted to D. Rabounski for deep understanding of our results and mental support.
Submitted on July 02, 2009 / Accepted on July 28, 2009
References
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2. Chizhevsky A.L. Terrestrial echo of solar storms. Mysl’ Press, Moscow, 1976 (in Russian).
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tuations of natural gravitational ﬁelds. Bioﬁzika, 1992, v. 7(3), 524–532 (in Russian).
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14. Gorshkov E.S., Shapovalov S.N., Sokolovsky V.V. and Troshichev O.A. On the gravitational conditionality of rate ﬂuctuations in the reaction of unithiol oxidation by nitrite ions. Bioﬁzika, 2000, v. 45(4), 631–635 (in Russian).
15. Shapovalov S.N., Gorshkov E.S., Troshichev O.A., Borisova T.D. and Frank-Kamenetsky A.V. Stochastic ﬂuctuations in the readings of measuring devices: eﬀects of cosmophysical inﬂuence? Bioﬁzika, 2001, v. 46(6), 819–822 (in Russian).
16. Shapovalov S.N. Principles of physical indication of cosmogeophysical ecological factors of non-electromagnetic nature. PhD Thesis, AANII and RGGMU Press, St. Petersburg, 2003 (in Russian).
17. Shapovalov S.N., Gorshkov E.S., Troshichev O.A., Borisova T.D., Frank-Kamenetsky A.V. Eﬀects of non-electromagnetic disturbances from the Sun in “computer time” instability. Biophysics, 2004, v. 49(1), S79–S84.
18. Sokolovsky V.V., Gorshkov E.S., Ivanov V.V., Shapovalov S.N. and Troshichev O.A. Relation of the regular gravitational ﬁeld variations to biochemical processes observed in vitro and in vivo. Biophysics, 2004, v. 49(1), S85–S91.
19. Shapovalov S.N., Gorshkov E.S., Troshichev O.A. Cosmophysical eﬀects observed in impulses of the microphotocolorimeter current. Biophysics, 2004, v. 49(1), S119–S122.
20. Troshichev O.A., Gorshkov E.S., Shapovalov S.N., Sokolovskii V.V. , Ivanov V.V., Vorobeitchikov V.M. Variation of the gravitational ﬁeld as a motive power for rhythmics of biochemical processes. Advances in Space Research, 2004, v. 34, 1619– 1624.
21. Shapovalov S.N., Troshichev O.A., Povazhny V.I., Sokolovsky V.V., Vorobeychikov V.M., Gorshkov E.S., Lozovsky V.T. and Alexandrova A.B. Cosmophysical regularities in the photoeffect experiment. Meteorologichesky Vestnik, 2008, v. 1(1), 1–23 (in Russian).
22. Shnoll S.E., Kolombet V.A., Pozharsky E.V., Zenchenko T.A., Zvereva I.M. and Konradov A.A. On the cosmophysical conditionality of “macroscopic ﬂuctuations”. Bioﬁzika, 1998, v. 43(5), 909–915 (in Russian).
23. Shnoll S.E., Rubinstein I.A., Zenchenko K.I., Zenchenko T.A., Udaltsova N.V., Konradov A.A., Shapovalov S.N., Makarevich A.V., Gorshkov E.S. and Troshichev O.A. Dependence of “macroscopic ﬂuctuations” on geographical coordinates (on the basis of materials of the Arctic (2000) and Antarctic (2001) expeditions). Bioﬁzika, 2003, v. 48(6), 1123–1131 (in Russian).
24. Shnoll S.E., Zenchenko K.I., Berulis I.I., Udaltsova N.V., Zhirkov S.S. and Rubinstein I.A. Dependence of “macroscopic ﬂuctuations” on cosmophisical factors. Space anisotropy. Bioﬁzika, 2004, v. 49(1), 132–139 (in Russian)

25. Shnoll S.E., Rubinshtejn I.A., Zenchenko K.I., Shlekhtarev V.A., Kaminsky A.V., Konradov A.A. and Udaltsova N.V. Experiments with rotating collimators cutting out pencil of alpha-particles at radioactive decay of Pu-239 evidence sharp anisotropy of space. Progress in Physics, 2004, v. 1, 81–84.
26. Shnoll S.E. Changes in ﬁne structure of stochastic distributions as a consequence of space-time ﬂuctuations. Progress in Physics, 2006, v. 2, 39–45.
27. Shnoll S.E., Pancheluga V.A. and Shnoll A.E. The palindrom eﬀect. Progress in Physics, 2008, v. 2, 151–153.
28. Shnoll S.E. The “scattering of the results of measurements” of processes of diverse nature is determined by the Earth’s motion in the inhomogeneous space-time continuum. The eﬀect of “half-year palindromes”. Progress in Physics, 2009, v. 1, 3–7.
29. Shnoll S.E. and Rubinstein I.A. “Regular changes in the ﬁne structure of histograms reaveled in the experiments with collimators which isolate beams of alpha-particles ﬂying at certain directions”. Progress in Physics, 2009, v. 2, 33–95.
30. Astronomical calendar (invariant part). Nauka Press, Moscow, 1981 (in Russian).

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Astrophysically Satisfactory Solutions to Einstein’s R-33 Gravitational Field Equations Exterior/Interior to Static Homogeneous Oblate Spheroidal Masses
Chifu Ebenezer Ndikilar
Physics Department, Gombe State University, P.M.B. 127, Gombe, Gombe State, Nigeria E-mail: ebenechifu@yahoo.com
In this article, we formulate solutions to Einstein’s geometrical ﬁeld equations derived using our new approach. Our ﬁeld equations exterior and interior to the mass distribution have only one unknown function determined by the mass or pressure distribution. Our obtained solutions yield the unknown function as generalizations of Newton’s gravitational scalar potential. Thus, our solution puts Einstein’s geometrical theory of gravity on same footing with Newton’s dynamical theory; with the dependence of the ﬁeld on one and only one unknown function comparable to Newton’s gravitational scalar potential. Our results in this article are of much signiﬁcance as the Sun and planets in the solar system are known to be more precisely oblate spheroidal in geometry. The oblate spheroidal geometries of these bodies have eﬀects on their gravitational ﬁelds and the motions of test particles and photons in these ﬁelds.

1 Introduction
After the publication of A. Einstein’s geometrical theory of gravitation in 1915/1916, the search for exact solutions to its inherent geometrical ﬁeld equations for various mass distributions began [1]. Four well known approaches have so far been proposed.
The ﬁrst approach is to seek a mapping under which the metric tensor assumed a simple form, such as the vanishing of the oﬀ-diagonal components.With suﬃciently clever assumptions of this sort, it is often possible to reduce the Einstein ﬁeld equations to a much simpler system of equations, even a single partial diﬀerential equation (as in the case of stationary axisymmetric vacuum solutions, which are characterised by the Ernst equation) or a system of ordinary diﬀerential equations (this led to the ﬁrst exact analytical solution — the famous Schwarzschild’s solution [2]). A special generalization of the Schwarzschild’s metric is the Kerr metric. This metric describes the geometry of space time around a rotating massive body.
The second method assumes that the metric tensor has symmetries-assumed forms of the Killing vectors. This led to the solution found by Weyl and Levi-Civita [3–6]. The third approach required that the metric tensor leads to a particular type of the classiﬁcations of Weyl and Riemann — Christoffel tensors. These are often stated interms of Petrov classiﬁcation of the possible symmetries of the Weyl tensor or the Segre classiﬁcation of the possible symmetries of the Ricci tensor. This leds to plane fronted wave solutions [3–6]. It is worth remarking that even after the symmetry reductions in the three methods above, the reduced system of equations is often diﬃcult to solve. The fourth approach is to seek Taylor series expansion of some initial value hyper surface, subject to consistent initial value data. This method has not proved

successful in generating solutions [3–6]. Recently [7–12], we introduced our own method and ap-
proach to formulation of exact analytical solutions as an extension of Schwarzschild’s method. In this article, we show how exact analytical solutions of order c (where c is the speed of light in vacuum) can be constructed in gravitational ﬁelds interior and exterior to static homogeneous oblate spheroids placed in empty space. For the sake of mathematical convenience we choose to use the 3rd (R!!) ﬁeld equation [7].

2 Exterior ﬁeld equation

The covariant metric tensor in the gravitational ﬁeld of a static

homogeneous oblate spheroid in oblate spheroidal coordina-

tes ; ;  has been obtained [7, 12] as


g =

1+

2
c

f (; )

;

(2.1)

g =

a

1 +
 

1+

2
c

f (; )


+

 (1
(1


+ )  )

;

(2.2)

 g

g =


a  1+  1


1+

2
c

f (; )


;

(2.3)

g

a

=

1 +
 

1+

2
c

f (; )


+

 (1
(1 +


 )  )

;

(2.4)

g!! = a (1 +  )(1  );

(2.5)

g = 0; otherwise;

(2.6)

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g =

2
1+ c


f (; )


g =

1  1+

D
  1  + 1+

1 + c f (; ) E

a 1 + c f (; )  [ (1  ) +  (1 +  )]

 g g  =

 1  1 +  1 +   a 1 + c f (; )  [ (1

D
1 1 + c f (; )
 ) +  (1 +  )]

E

g =

1+

1+

D
  1+ + 1 

1 + c f (; ) E

a 1 + c f (; )  [ (1  ) +  (1 +  )]


g!! = a 1 +  1 

g = 0 otherwise

(2.7) (2.8) (2.9) (2.10) (2.11) (2.12)

ggg; gg; g; gg g;

gg ; g; gg ; g;

gg g;

gg ; g; ggg g;g ; ggg!!g;g!!;

gg g g; g ; gg g!!g; g!!; ggg g;g;

gg g g;g ; gg g!!g;g!!;  gg g!!g; g!!;

ggg; gg; g; gg ; g; 2gg g ; gg ; g ;

gg!!; g!!; + 4gg g; + gg ; g; + 4gg g ; 2gg g; +

+ gg ; g; ggg g; g ; + ggg g; g ; ggg g; g; +

+ gg g g ;g ;  gg g g; g ; + gg g!!g ;g!!;

 gg g!!g; g!!; +  gg g g;g ; gg g g; g; +

+ gg g g; g ; g gg; g g; g; g g; g; g g; g;

g g ; g ; g g!!g!!; g g!!; g!!; + g g ; g ; + g g; g ;

g g; g ; +  gg g g; g ; g g g g; g ; +

+  gg g g;g ; +  gg g g;g ; +  g g g!!g ; g!!;

gg g g ; g ; + ggg g;g ; + ggg g;g ; + gg g!!g ; g!!; +

+  gg g g; g ; + g g g g ; +  g g g g ;g ;

 g g g!!g ; g!!; +  g g g!!g ;g!!; + g!!g; g!!; + g!!g ; g!!; +

+ 2g g!!g!!; g!!g ; g!!; g g!!g!!;

g!!g ; g!!; gg g!!g ;g!!;

g g g!!g ;g!!;  g g g!!g ;g!!; g g g!!g ; g!!;

 g g g!!g ; g!!;  gg g!!g; g!!;  gg g!!g; g!!; +

+ gg g; + 2g g; g; + 2g g; g; + 4g g g ; + 4g g ; g ; 2g g g;

2g g ; g; + 2g g!!; g!!; + 2g g; g; + 2g g g ; + 2g g ; g ; +

+ ggg g;g; + gg g g;g ; + gg g g; g; + gg g g; g; +

+ g g g g; g ; + g g g!!g; g!!; + g g g!!g ;g!!; gg g g;g ;

g g g g;g ; g g g g ;g ; gg g g; g ; + g g g g; g ; = 0

(2.18)

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and the contravariant metric tensor is as shown in formulas (2.7)–(2.12), where f (; ) is an unknown function deter-

mined by the mass distribution. From this covariant met-

ric tensor, we can then construct our ﬁeld equations for the

gravitational ﬁeld after formulating the Coeﬃcients of aﬃne

connection, Riemann Christoﬀel tensor, Ricci tensor and the

Einstein tensor [7–12]. After the above steps, it can be shown

that the exterior R33 ﬁeld equation in this gravitational ﬁeld

is given as;

R33

1 2

R

g33

=

0

:

(2.13)

or more explicitly interms of the aﬃne connections, Ricci tensor and covariant metric tensor as;

 !! 

 !! 

!!;

!!;

 !! 

!!

 !! 

! !!  ! !!

+

! !

!!

1 2

4

C

(2.14)
=0

with the symbols and numbers having their usual meaning and

R = g00R00 + g11R11 + 2g12R12 + g22R22 + g33R33 :
(2.15) Now, multiplying equation (2.13) by 2g33 and using the fact that g33g33 = 1 yields

2g33R33 R = 0 :

(2.16)

Writing the expression for the curvature scalar,R as in equation (2.15) gives;

g00R00

g11R11 2g12R12 g22R22 + g33R33 = 0 :

(2.17)

Writing the various terms of the ﬁeld equation (2.17) explicitly in terms of the metric tensor gives our ﬁeld equation explicitly as (2.18).
Now, we realize that our covariant metric tensor (2.1)– (2.6) can be written equally as

g (; ) = h (; ) + f (; ) ;

(2.19)

where h are the well known pure empty space components and f are the contributions due to the oblate spheroidal mass distribution. Consequently, as the mass distribution de-
  cays out; f 0 and hence g h. Therefore, the
metric tensor reduces to the pure empty space metric tensor
as the distribution of mass decays out. Also,

g (; ) = h (; ) + f  (; ) ;

(2.20)

where h are the well known pure empty space components and f are the contributions due to the oblate spheroidal mass distribution. Thus it can be shown that for this ﬁeld, the non zero metric components can be written as;

h00 = 1 ;

(2.21)

a2 2 + 2

h11 =

1 2 ;

(2.22)

a2 2 + 2

h22 =

1 + 2 ;

h33 = a2 1 + 2 1 + 2 ;

(2.23) (2.24)

f00

=

2
c2

f

;

f11 =

(1

a22 2 +

2)

: 
(n n=1

1

)

2n cn

fn

;

 f12 f21 =

(1

a2 2 +

2)

: 
(n n=1

1)

2n cn

fn

;

f22 =

(1

a22 2 +

2)

: 
(n n=1

1

)

2n cn

fn

;

(2.25) (2.26) (2.27) (2.28)

also,

h00

=

1
h00

;

(2.29)

h11

=

1
h11

;

(2.30)

h22

=

1
h22

;

(2.31)

h33

=

1
h33

;

f 00

=

: 
(n n=1

1)

2n cn

fn

;

(2.32) (2.33)

f 11 =

f11 (h11)2

+

0 (c

4) ;

(2.34)

 f 12

f 21 =

f12 h11h22

+ 0 (c

4) ;

(2.35)

f 22 =

f22 (h22)2

+

0 (c

4) :

(2.36)

To begin the explicit formulation of the R33 ﬁeld equation we note, ﬁrst of all, that all the terms of order c0 cancel
out identically since the empty space time metric tensor h independently satisﬁes the homogeneous R33 ﬁeld equation.
Therefore the lowest order of terms we expect in the exterior R33 ﬁeld equation is c 2. Hence in order to formulate the exterior R33 ﬁeld equation of order c 2, let us decompose our covariant metric tensor g into pure empty space part h (of order c0 only) and the nonempty space part f (of order c 2 or higher). Similarly, let our contravariant metric tensor g be decomposed into pure empty space part h (of order c0 only) and the nonempty space part f  (of order c 2 or higher). Substituting explicit expressions for equations

(2.19) and (2.20) into equation (2.18) and neglecting all terms of order c0, the exterior R33 ﬁeld equation can be written as
(2.37), where the coeﬃcients are given as (2.38)–(2.58).

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S (; ) f ; + S (; ) f; + S! (; ) f ; + S" (; ) f; + S# (; ) f; + + S$ (; ) f; + S% (; ) f; + S& (; ) f ; + S' (; ) f ; + S (; ) f ; + + S (; ) f  ; +S (; ) f ; +S! (; ) f; + S" (; ) f; + S# (; ) f ; + + S$ (; ) f ; + S% (; ) f  ; +S& (; ) f ; +S' (; ) f  + S  (; ) f  +
+ S  (; ) f = 0

S (; ) = 2hh S (; ) = h

S! (; ) = 4hh S" (; ) = h h S# (; ) = 2hh

S$ (; ) = h

S% (; ) = h; hh h ; hh!!h!!;

S& (; ) = hh ; +hh h!!h!!; + h h ; hh;

S' (; ) = hh h!!h!!; +  h h;

h h; hh ;

S (; ) = h h ; + h!!h!!;

S (; ) = h h ; + h!!h!!;

S (; ) = hh ;

S! (; ) = h ; h h!!h!!; +  h h h ; hh h;

S" (; ) = hh ; S# (; ) = h

h h; + 1 hh
2

h h ; + h h ;

h; +hh h!!h!!; + hh; h h

h!!h!!;
;

S$ (; ) =  hh h h;

S% (; ) = hh; h h ; h!!h!!;

S& (; ) = hh; h!!h!!;

S' (; ) = h h!!h!!; h h!!h!!; + h h h; h ; h h!!h; h!!;  h h h; h ; h ; h; 2h h;
h ; h ; 2h h ;

h!!; h!!;

S  (; ) = 4hh; +  hh h; (h ; + h ; ) + +  h h!!h ;h!!; + 2h!!h!!;  h h!!h ; + 2h; h; + 2h!!; h!!; + + 2h; h; hh h; h ;

S  (; ) =  hh!!h!!; + hh h; h ; hh h; h ;

 hh!!h; h!!; 2h!!h!!;

hh!!h!!; h; h; h!!; h!!;

h; h ; 2hh ; 2hh; :

(2.37)
(2.38) (2.39) (2.40) (2.41) (2.42) (2.43) (2.44) (2.45) (2.46) (2.47) (2.48) (2.49) (2.50) (2.51) (2.52) (2.53) (2.54) (2.55)
(2.56)
(2.57)
(2.58)

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K1 (; ) f + K2 (; ) f + K3 (; ) f  + K4 (; ) f + K5 (; ) f + K6 (; ) f = 0

(2.59)

2 1 2 1 2 + 2 2a42 2 + 2

K1 (; ) =

a2c2 (2 + 2) (1 2 + 2)

4 2 + 2 1 2 + 2 8 1 4

K2 (; ) =

a2c2 (2 + 2)2 (1 2 + 2)

2 1 + 2 K3 (; ) = a2c2 (2 + 2)

K4 (; ) =

8a22S1 (; ) + 2a22S9 (; ) c2 (1 2 + 2)2

8 1 4 1 2 2 a2c2 (2 + 2)2 (1 2 + 2)2 +

2a22S10 (; )

2a2S11 (; )

2a22S12 (; )

+ c2 (h11)2 (1 2 + 2) + c2h11h22 (1 2 + 2) c2 (h22)2 (1 2 + 2)

(2.60) (2.61) (2.62)
(2.63)

K5

(; )

=

a2c2

8 1 +
(2 + 2)2

2
(1

+ 2 2

+

2)2

+

a2c2

162 1 (2 + 2)2 (1

4 2

+ 2)

+

2 [S9 (; ) + c2

S13 (; )]

+

2a2 [ S8 (; ) S15 (; ) + S16 (; )]

2a2S17 (; )

2a22S17 (; )

+

c2 (h11)2 (1 2 + 2)

+ + c2h11h22 (1 2 + 2) c2 (h22)2 (1 2 + 2)

(2.64)

4a22 1 + 32 + 2 K6 (; ) = c2 (1 2 + 2)3

8 1 4 1 4 4 1022

82 1 2

a2c2 (2 + 2)2 (1 2 + 2)3 + a2c2 (2 + 2) (1 2 + 2)2 +

2a2  1 + 2 + 2 S8 (; ) + 22S9 (; )  1 2 2 S15 (; ) + 2 1 2 S16 (; )

+

c2 (1 2 + 2)2

2a22 (h11)2 ;1 S10 (; ) 2a22 2 1 + 2 S10 (; ) S19 (; ) 2a2 1 + 2 + 2 S11 (; )

c2 (h11)4 (1 2 + 2) +

c2 (h11)2 (1 2 + 2)

+

c2h11h22

2a2 [(h11h22) ;1 S11 (; )] 2a2 (h22)2 ;1 S12 (; )  (h22)2 ;2 S18 (; )

c2 (h11h22)2 (1 2 + 2) +

c2 (h22)4 (1 2 + 2)

+

2a2 1 2 2 S17 (; )  1 2 + 2 S20 (; )

+

2a2

D
S12

c2
(; )

(h11h22) (1 2 1 2

2

2 + 2)2 S18 (; )

E
S21 (; )

c2 (h22)2 (1 2 + 2)2

(2.65)

Substituting the explicit expressions for the nonempty space parts f and f  into equation (2.37), simplifying and

grouping like terms yields (2.59), where the terms consisting

it are (2.60)–(2.65).

Equation (2.59) is thus ﬁeld equation to the order

cou2r.

exact explicit We can now

R33 exterior
conveniently

formulate astrophysical solutions for the equation in the next

section; which are convergent in the exterior space time of a

homogeneous massive oblate spheroid placed in empty space.

3 Formulation of R-33 exterior solution

where Pn+ is a function to be determined for each value of n.
B C iStnaukdbiensptgeitnuindtitenongtctoshenets,piwdreoerpacotaisnoendthtfuhusenceftqaiucotantteihnatthtoethcoeneﬁﬃne=lcdi0eeniqstusaaotlfiionnenaarnolydn
bo0t,hwseidoebstaoifntthheeoebqtuaaintieodn equation. From the coeﬃcients of

0 = K1 (; ) P2+ () + K2 (; )

P1+ ()


+

+ K3 (; ) P0+ ()  + K4 (; ) P1+ () +

(3.3)

+ K5 (; ) P0+ ()  + K6 (; ) P0+ ()

or more explicitly

In the exterior oblate spheroidal space time [7]:

0 = a33 1 + 2 a24 P2+ () +

 0 and 1  1; 0 = constant

(3.1) + 2a33 1 + 2 2 P1+ ()  + a32 1 + 2 P1+ () +

Let us now seek a solution for the R33 ﬁeld equation (2.59)

in

the

form

of

the power seri: es
f (; ) =

Pn+

()

n:

(3.2)

n=0

+ a33

1 + 2

2

P0+ ()


+

1 + 2

1

2a22

2

a23 + 4a25

P0+ ()


+

+ 2a3 4 22 a44 a46 P0+ () :

(3.4)

Chifu E. N. Astrophysically Satisfactory Solutions to Einstein’s R-33 Gravitational Field Equations

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Equation (3.4) is the ﬁrst recurrence diﬀerential equation

for the unknown functions. All the other recurrence diﬀeren-

tial equations can thus follow, yielding inﬁnitely many recur-

rence diﬀerential equations that can be used to determine all

the unknown functions.

equaTthioenfo(3ll.o4w) idnegteprrmoifnoeusndP2p+oiinntstecramnsthouf sPb0+e manaddeP.1+F,irssitmly-,

ilarly the
amnidnePt1h+e.

other other

urneckunrorwenncefudnicﬀtieornesntPia3+l ;e:q:u:atiinontserwmisllodfePte0+r-

Secondly, we note that we have the freedom to

choose our arbitrary functions to satisfy the physical require-

ments or needs of any particular distribution or area of appli-

cation.

Let us now recall that for any gravitational ﬁeld [7, 13],

g00

1

+

2
c2

(3.5)

where is Newton’s gravitational scalar potential for the ﬁeld

under consideration. Thus we can then deduce that the un-

known function in our ﬁeld equation can be given approxi-

mately as

f (; ) + (; )

(3.6)

where + (; ) is Newton’s gravitational scalar potential ex-

terior to a homogeneous oblate spheroidal mass. Recently

[14], it has been shown that

+ (; ) = B0Q0 ( i) P0 ()+B2Q2 ( i) P2 () (3.7)

where Q0 and Q2 are the Legendre functions linearly independent to the Legendre polynomials P0 and P2 respectively; B0 and B2 are constants.
Let us now seek our exact analytical exterior solution
(3.4) to be as close as possible to the approximate exterior
solution (3.7). Now since the approximate solution possesses no term in the ﬁrst power of , let us choose

P0+ () = B0Q0 ( i) P0 + B2Q2 ( i)

(3.8)

and

P1+ ()  0 :

Hence, we can write P2+ in terms of P0+ as

P2+ () =

1 + 2 2 (1 + 2 a24)

P0+ () 

2

2

1 1

+

2 3a22 + 4a25 2 a23
2a32 a74 a76 + a3 a32 (1 + 2 a24)

1 P0+

P0+ ()

() :


(3.9)
(3.10)

We now remark that the ﬁrst three terms of our series so-

lution converge also remark that

everywhere our solution

in of

tohredeerxcte0rmioaryspbaecwe rtiitmteen.

We as

f (; ) = + (; ) + +0 (; )

(3.11)

where + (; ) is the corresponding
tional scalar potential given by (3.7) and

N+0ew(to;ni)ains

gravitathe pure

Etioinnstoefinoiradnerocr0g.eneral relativistic or post Newtonian correc-

Hence, we deduce that our exterior analytical solution is

of the general form
f (; ) = + (; ) +

+0 (; ) + : 

+2n (; ) : (3.12)

n=1

4 Formulation of interior R-33 ﬁeld equation and solution

For the interior space time, Einstein’s ﬁeld equations are well known to be given as;

R

1 2

R g

=

8G
c4

T

(4.1)

where T is the energy momentum tensor. Now, let us assume that the homogeneous mass distribu-
tion is a “perfect ﬂuid”. Thus, we can deﬁne the energy momentum tensor as

T = (0 + P0) uu P0g

(4.2)

where 0 is the proper mass density and P0 is the proper pres-
sure and u is the velocity four vector. Hence, the ﬁve non trivial interior ﬁeld equations can be written as;

R00

1 2

R

g00

=

8G
c4

[(0

+

P0)

u0u0

P0g00] ; (4.3)

R11

1 2

R g11

=

8G
c4

P0 g11

;

(4.4)

R12

1 2

R g12

=

8G
c4

P0 g12

;

(4.5)

R22

1 2

R g22

=

8G
c4

P0 g22

;

(4.6)

R33

1 2

R g33

=

8G
c4

P0 g33

:

(4.7)

Now, we formulate the solution of (4.7). For the sake of mathematical convenience, we assume in this article that the pressure is negligible compared to the mass density and hence

P0 0 :

(4.8)

g33gM33ul=tip1lywinegoebqtuaiantiporne(c4is.7e)lybays2ign3t3haensdecutsiionng2t;he fact that

g00R00 g11R11 g22R22 + g33R33 2g12R12 = 0 : (4.9)

Similarly, we obtain the interior equation explicitly as
K1 (; ) f + K2 (; ) f + K3 (; ) f  + + K4 (; ) f + K5 (; ) f + K6 (; ) f = 0 : (4.10)

78

Chifu E. N. Astrophysically Satisfactory Solutions to Einstein’s R-33 Gravitational Field Equations

October, 2009

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We now remark that, for the interior ﬁeld we are re-

quired to formulate interior solutions of (4.10) convergent in

the range

0  0 ;

1  1:

(4.11)

Let us thus seek a series solution of the form;
f (; ) = :  Zn () n:
n=0

(4.12)

B C uwshinergethZenfaacrtethuantknonwnn=fu0nicstiaolnisnetaorlbyeinddeetepremndineendt.seNt,owwe,
may equate coeﬃcients on both sides and hence obtain the
equations satisﬁed by Zn . We proceed similarly as in the case of the exterior solution to obtain recurrence diﬀerential
equations that determine the explicit expression for our exact analytical solution. Equating the coeﬃcients of 0, we obtain
the ﬁrst recurrence diﬀerential equation as

K1 (; ) Z0 ()  + K2 (; ) Z1 ()  + + K3 (; ) Z2 () + K4 (; ) Z0 ()  + +K5 (; ) Z1 () + K6 (; ) Z0 () = 0 :

(4.13)

In a similar manner, the other recurrence diﬀerential equations follow.
We can now proceed as in the previous section to choose the most astrophysically satisfactory solution to be as close as possible to the approximate solution. The gravitational scalar potential interior to a homogeneous oblate spheroid is well known [14] to be given as


(; ) = A0

1 2


A2P2()

3=2A2P2() 2; (4.14)

where P2 is Legendre’s polynomial of order 2 and A0, A2, are constants.
Since (4.14) converges for all values in the interval (4.11), it is very satisfactory for us to choose;

Z0 () = A0

1 2

A2P2()

(4.15)

and

Z1 ()  0 :

(4.16)

Thus the ﬁrst recurrence diﬀerential equation determines Z2 in terms of Z0 . Similarly, all the other recurrence diﬀerential equations will determine all the other functions in terms of Z0 . Hence we obtain our unique astrophysically most satisfactory interior solution. It is obvious that this unique solu-
tion will converge, precisely as the ﬁrst two terms. Moreover,
it is obvious that our unique solution reduces to the corre-
sponding pure Newtonian gravitational scalar potential in the
limit of the ﬁrst two terms. This solution may be written as

f (; ) = (; ) + 0 (; )

(4.17)

where (; ) is the corresponding Newtonian gravitational scalar potential given by (4.14) and 0 (; ) is the
pure instructively Einstenian (or general relativistic or post Newtonian correction) of order c0.
Proceeding exactly as above we may derive all the corresponding solutions of all the other non-trivial interior Einstein’s ﬁeld equations for the sake of mathematical completeness, comparison with those of the R33 equation and theoretical applications where and when necessary in Physics. It is clearly obvious how to extend the derivation of the interior Einstein ﬁeld equations above to include any given pres-
sure function P0 (; ), wherever and whenever necessary
and useful in physical theory.
5 Conclusions
Interestingly, the single dependent function f in our mathematically most simple and astrophysically most satisfactory solution turns out as the corresponding well known pure Newtonian exterior/interior gravitational scalar potential augmented by hitherto unknown pure Einsteinian (or general relativistic or post-Newtonian) gravitational scalar potential terms of orders c0, c 2, c 4, . . . Hence, this article has revealed a hitherto unknown sense in which the exterior/interior Einstein’s geometrical gravitational ﬁeld equations are obtained as a generalization or completion of Newton’s dynamical gravitational ﬁeld equations.
With the formulation of our mathematically most simple and astrophysically most satisfactory solutions in this article, the way is opened up for the formulation and solution of the general relativistic equations of motion for all test particles in the gravitational ﬁelds of all static homogeneous distributions of mass within oblate spheroidal regions in the universe. And precisely because these equations contain the pure Newtonian as well as post-Newtonian gravitational scalar potentials all their predictions shall be most naturally comparable to the corresponding predictions from the pure Newtonian theory. This is most satisfactory indeed.
It is now obvious how our work in this article may by emulated to (i) derive a mathematically most simple structure for all the metric tensors in the space times exterior or interior to any distribution of mass within any region having any of the 14 regular geometries in nature, (ii) formulate all the nontrivial Einstein geometrical gravitational ﬁeld equations and derive all their general solutions and (iii) derive astrophysically most satisfactory unique solutions for application to the motions of all test particles and comparison with corresponding pure Newtonian results and applications. Therefore our goal in this article has been completely achieved: to use the case of a spheroidal distribution of mass to show how the much vaunted Einstein’s geometrical gravitational ﬁeld equations may be solved exactly and analytically for any given distribution of mass within any region having any geometry.

Chifu E. N. Astrophysically Satisfactory Solutions to Einstein’s R-33 Gravitational Field Equations

79

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Finally, we conclude that at very long last — 93 years after the publication of the laws of General Relativity by Einstein in 1915 — we have found a method and process for (1) deriving a unique approximate astrophysically most satisfactory solutions for the space times exterior and interior to every distribution of mass within any region having any of the 14 regular geometries in nature, in terms of the corresponding pure Newton’s gravitational scalar potential, without even formulating the ﬁeld equation; and (2) systematically formulating and solving the geometrical gravitational ﬁeld equations in the space times of all distributions of mass in nature.
Acknowledgenent
The author is highly indebted to Prof. S. X. K. Howusu of the Physics Department, Kogi State University, Nigeria for his
4 immense inspiration, guidance; and for the use of his 
solution method to formulate the solution in this article.

10. Chifu E.N., Howusu S.X.K. and Lumbi L.W. Relativistic mechanics in gravitational ﬁelds exterior to rotating homogeneous mass distributions within regions of spherical geometry. Progress in Physics, 2009, v. 3, 18–23.
11. Chifu E.N and Howusu S.X.K. Solution of Einstein’s geometrical ﬁeld equations exterior to astrophysically real or hypothetical time varying distributions of mass within regions of spherical geometry. Progress in Physics, 2009 v. 3,45–48.
12. Chifu E.N., Usman A. and Meludu O.C. Orbits in homogeneous oblate spheroidal gravitational space time. Progress in Physics, 2008, v. 3, 49–53.
13. Weinberg S. Gravitation and cosmology. J. Wiley, New York, 1972, p. 175–188.
14. Howusu S.X.K. Gravitational ﬁelds of spheroidal bodiesextension of gravitational ﬁelds of spherical bodies. Galilean Electrodynamics, 2005, v. 16(5), 98–100.

Submitted on June 29, 2009 / Accepted on August 03, 2009

References
1. Bergmann P.G. Introduction to the Theory of Relativity. Prentice Hall, New Delhi, 1987.
2. Schwarzschild K. U¨ ber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie. Sitzungsberichte der Koniglich Preussischen Akademie der Wissenschaften, 1916, 189–196 (published in English as: Schwarzschild K. On the gravitational ﬁeld of a point mass according to Einstein’s theory. The Abraham Zelmanov Journal, 2008, v. 1, 10–19).
3. Finster F., Kamran N., Smoller J. and Yau S.T. Decay of solutions of the wave equation in the Kerr geometry. Communications in Mathematical Physics, 2006, v. 264, 465–503.
4. Anderson L., Elst V., Lim W.C. and Uggla C. Assymptotic silence of generic cosmological singularities. Physical Review Letters, 2001, v. 94, 51–101.
5. MacCallum M.A.H. Finding and using exact solutions of the Einstein equation. arxiv: 0314.4133.
6. Stephani H., Kramer D., MacCallum M.A.H., Hoenselars C., Herlt E. Exact solutions of Einstein’s ﬁeld equations. 2nd ed., Cambridge Univ. press, London, 2003.
7. Howusu S.X.K. The 210 astrophysical solutions plus 210 cosmological solutions of Einstein’s geometrical gravitational ﬁeld equations. Jos University Press, Jos, 2007.
8. Chifu E.N. and Howusu S.X.K. Gravitational radiation and propagation ﬁeld equation exterior to astrophysically real or hypothetical time varying distributions of mass within regions of spherical geometry. Physics Essays, 2009, v. 22(1), 73–77.
9. Chifu E.N. and Howusu S.X.K. Einstein’s equation of motion for a photon in ﬁelds exterior to astrophysically real or imaginary spherical mass distributions who’s tensor ﬁeld varies with azimuthal angle only. Journal of the Nigerian Association of Mathematical Physics, 2008, v. 13, 363–366.

80

Chifu E. N. Astrophysically Satisfactory Solutions to Einstein’s R-33 Gravitational Field Equations

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A New Finslerian Uniﬁed Field Theory of Physical Interactions
Indranu Suhendro
E-mail: spherical symmetry@yahoo.com
In this work, we shall present the foundational structure of a new uniﬁed ﬁeld theory of physical interactions in a geometric world-space endowed with a new kind of Finslerian metric. The intrinsic non-metricity in the structure of our world-geometry may have direct, genuine connection with quantum mechanics, which is yet to be fully explored at present. Building upon some of the previous works of the Author, our ultimate aim here is yet another quantum theory of gravity (in just four space-time dimensions). Our resulting new theory appears to present us with a novel Eulerian (intrinsically motiondependent) world-geometry in which the physical ﬁelds originate.

1 Introduction
This work is a complementary exposition to our several previous attempts at the geometrization of matter and physical ﬁelds, while each of them can be seen as an independent, selfcontained, coherent uniﬁed ﬁeld theory.
Our primary aim is to develop a new foundational worldgeometry based on the intuitive notion of a novel, fully naturalized kind of Finsler geometry, which extensively mimics the Eulerian description of the mechanics of continuous media with special emphasis on the world-velocity ﬁeld, in the sense that the whole space-time continuum itself is taken to be globally dynamic on both microscopic and macroscopic scales. In other words, the world-manifold itself, as a whole, is not merely an ambient four-dimensional geometric background, but an open (self-closed, yet unbounded), co-moving, self-organizing, self-projective entity, together with the individual particles (objects) encompassed by its structure.
2 Elementary construction of the new world-geometry

metric ﬁeld . As such, U" may encompass all possible metric-compatible (sub-)universes, especially those of the General Theory of Relativity. In this sense, U" may be viewed as a Meta-Universe, possibly without admitting any apparent boundary between its microscopic (interior) and macroscopic (exterior) mechanisms, as we shall see.
If we represent the metric-compatible part of the geometric basis of U" as g (x), then, following our uniﬁcation scenario, the total geometric basis of our generally non-metric manifold shall be given by

g (x; u) = g (x) + u g (x; u) = (g (x; u)) 

g (x; u) ; g (x; u) =

where u = @@NI g (x; u) is the world-velocity ﬁeld along the world-line

Zq

s (x; u) =

g (x; u) dx dx

Without initial recourse to the common structure of Finsler geometry, whose exposition can easily be found in the literature, we shall build the essential geometric world-space of our new theory somewhat from scratch.
We shall simply start with an intuitive vision of intrinsically motion-dependent objects, whose fuzzy Eulerian behavior, on the microscopic scale, is generated by the structure of the world-geometry in the ﬁrst place, and whose very presence, on the macroscopic scale, aﬀects the entire structure of the world-geometry. In this sense, the space-time continuum itself has a dynamic, non-metric character at heart, such that nothing whatsoever is intrinsically “ﬁxed”, including the deﬁning metric tensor itself, which evolves, as a structural entity of global coverage, in a self-closed (self-inclusive) yet unbounded (open) manner.
In the present theory, the Universe is indeed an evolving, holographic (self-projective) four-dimensional space-time continuum U" with local curvilinear coordinates x and an intrinsically fuzzy (quantum-like), possibly degenerate, non-

(with g (x; u) being the components of the generalized metric tensor to be subsequently given below), and where
are the components of the Kronecker delta. (Needless to say, the Einstein summation convention is applied throughout this work as usual.) Here the inner product is indicated by D: : : ; : : :E. We then have

@

@

@

g (x; u) =

g (x) + u

+ H u;

@x

@x

@x

where H denotes the gradient, that is, the covariant derivative. The components of the symmetric, bilinear metric tensor
g (x; u) for the given geometric basis are readily given by

g (x; u) = Dg (x; u) ; g (x; u)E
g  (x; u) g  (x; u) = :

As such, we obtain

g (x; u) = g (x) + 2 u^  +  (x; u)

:

Indranu Suhendro. A New Finslerian Uniﬁed Field Theory of Physical Interactions

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As usual, round brackets enclosing indices indicate symmetrization; subsequently, anti-symmetrization shall be indicated by square brackets. In the above relation,
u^ = Du; g (x)E and
2 (x; u) = g (x; u) u u

is the squared length of the world-velocity vector, which varies from point to point in our world-geometry. As we know, this squared length is equal to unity in metriccompatible Riemannian geometry.

The connection form of our world-geometry is obtained through the inner product


 (x; u) = g (x; u) ; @ g (x; u) :

@x

In an explicit manner, we see that


 (x; u) =  (x) + @

u +

@x

H u:

In accordance with our previous uniﬁed ﬁeld theories (see, for instance, [1–5]), the above expression must generally be asymmetric, with the torsion being given by the antisymmetric form


[

] (x; u) =


[

]

(x)

+

1 2

 @ @

x

+

1 2

H u

H u :


@

u+

@x

In contrast to the case of a Riemannian manifold (without background embedding), we have the following unique case:

H g (x; u)  @ g (x; u) @x

=

1 2

 H u

 (x; u) g (x; u) =

for which, additionally,  (x)  = 0. Consequently, the
covariant derivative of the world-metric tensor fails to vanish
in the present theory, as we obtain the following non-metric
expression:

H g (x; u) =

 H u:

At this point, in order to correspond with Finsler geometry in a manifest way, we shall write
H g (x; u) =  H u

and

g

(x; u)

=

1 2

@

@2 u@

u

2 (x; u)

in such a way that the following conditions are satisﬁed:

=

;

1 2


=

1 2

(

)

=

1 2

@
@ u

g

=

1 4

@u

@3 @u

@ u 2 (x; u) ;

(x; u) =

 u = 0 ;

u =0:

Once the velocity ﬁeld is known, the Hessian form of the

metric tensor enables us to write, in the momentum represen-

tation for a geometric object with mass m (initially at rest,

locally),

g

(x;

u)

=

1 2

m2

@

p

@2 @

p

2 (x; u) ;

p =mu

such that, with 2 (x; u) being expressed in parametric form,

physical geometry, that is, the existence of a geometric object

in space-time, is essentially always related to mass and its

energy content.

Taking into account the projective angular tensor given by

(x; u) = g (x; u)  (x; u)  (x; u) =

1 2 (x; u) u u ;
1 2 (x; u) u u ;

(x; u) u = 0 ;

where n is the number of dimensions of the geometric space
(in our case, of course, n = 4), in the customary Finslerian

way, it can easily be shown that


=

1
n


(x; u)

+  (x; u)

 +  (x; u) 1


+

;

= g (x; u)


=2 @
@u

ln

p
det

(g

(x;

u))

;

@ @u

ln

pdet

(g

(x;

u))

=

1 2

g

(x;

u)

@ @u

g (x; u)

for which, in our speciﬁc theory, we have, with 2 =

= g (x; u)

,

2

= n

(x; u)  +

+  (x; u)

1
2

 (x; u)

:

+

We may note that, along the world-line, for the intrinsic geodesic motion of a particle given by the parallelism
Du = (H u ) u = 0 ;
Ds

82

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the Finslerian condition

D g

(x; u) = 0

Ds

is always satisﬁed, along with the supplementary condition

D 2 (x; u) = 0 :
Ds
Consequently, we shall also have

D

(x; u) = 0 :

Ds

It is essential to note that, unlike in Weyl geometry, we
shall not expect to arrive at the much simpler gauge condition
H g (x; u) = g (x; u) A ( ). Instead, we shall always
employ the following alternative general form:

H g

(x;

u)

=

2

1 (x;

u)

ug

2 u^(

H )   u

where, as we can easily see, the diﬀeomorphic structure of the metric tensor for the condition of non-metricity of our worldgeometry is manifestly given by
u g  g (x; u) g (x) =

= 2 u^( ) + 2 (x; u)

3 Explicit physical (Eulerian) structure of the connection form

Having recognized the structural non-metric character of our

new world-geometry in the preceding section, we shall now

seek to outline the explicit physical structure of the connec-

tion form for the purpose of building a uniﬁed ﬁeld theory.

We ﬁrst note that the non-metric connection form of our

theory can always be given by the general expression


(x;

u)

=

1 2

g

(x;

u)

 @
@x

g

(x; u)


@
@x g

(x; u) + @
@x

g  (x; u)

+


+


[

] (x; u)

g (x; u)

g  (x; u)

 [

] (x; u) +


+ g  (x; u)

 [

] (x; u)

+

+

1 2

g

(x;

u)


H

g

(x; u)


H g (x; u) + H g  (x; u) :

Then, using the results given in the previous section, in direct relation to our previous metric-compatible uniﬁcation theory of gravity, electromagnetism, material spin, and the nuclear interaction [4], where the electromagnetic ﬁeld and

material spin are generated by the torsion ﬁeld, we readily

obtain


(x;

u)

=

1 2

g

(x;

u)


@

@ x

g

(x; u)


@
@ x g

(x; u) + @
@x

g  (x; u)

+

+

e
2 m c2

2 (x; u)

F 

u

F u

F u +


+ S g (x; u) g  (x; u) S + g  (x; u) S +

+

1 2

g

(x;

u)

(


H u

H u +

+  H u) :

Here it is interesting to note that even when = 0, which
gives a metric-compatible (“classical”) case, our connection form already explicitly depends on the world-velocity (in addition to position), hence the uniﬁed ﬁeld theory of physical interactions outlined in [4] can somehow already be considered as being a Finslerian one despite the fact that it is metriccompatible.
We recall, still from [4], that the electromagnetic ﬁeld F and the material spin ﬁeld S have a common geometric origin, which is the structural torsion of the space-time manifold, and are essentially given by the following expressions:

F = 2 m c2
e
S = S u


[

] u ;

S u ;

S u = 0; S = S[ ] ;


[

]

=

2

e m

c2

F

u + S ;

where m is the (rest) mass, e is the electric charge, and c is the

speed of light in vacuum, such that the physical ﬁelds are in-

trinsic to the space-time geometry itself, as manifest in gener-
alized geodesic equation of motion DDus = 0, which naturally
yields the general relativistic equation of motion of a charged,

massive particle in the gravitational ﬁeld


m c2 du +
ds

 u u = e F u ;


=

1 2

g


 @ g @ x


@g  @ x

+

@ g
@x

:

In other words, the physical ﬁelds other than gravity (chieﬂy, the electromagnetic ﬁeld) can also be represented as part of the internal structure of the free-fall of a particle. Just like gravity, being fully geometrized in our theory, these non-holonomic (vortical) ﬁelds are no longer external entities merely added into the world-picture in order to interact with

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gravity and the structure of space-time itself, thereby essentially fulﬁlling the geometrization program of physics as stated, for example, in [6].
Correspondingly, the nuclear (Yang-Mills) interaction is essentially given in our theory as an internal electromagnetic interaction by

Fi

= 2 !i


[

];

F

= m c2 F i
e

ui (i = 1; 2; 3) ;

where !i are the components of the tetrad (projective) ﬁeld relating the global space-time to the internal threedimensional space of the nuclear interaction.
In this direction, we may also deﬁne the extended electromagnetic ﬁeld, which explicitly depends on the worldvelocity, through

F~

(x; u) = 2 (x; u) F

= 2 2 (x; u) mc2
e


[

] u :

4 Substantial structure of covariant diﬀerentiation in U4
Given an arbitrary world-tensor T (x; u) at any point in our
Finslerian world-geometry, we have the following elementary substantial derivatives:

d d

T::::::

(x; u)

=

=

@
@x

T:::::: (x; u)

dx
d

+

@
@u

T:::::: (x; u)

@u ;
@

d
dx

T::::::

(x; u)

=

=

@
@x

T::::::

(x; u) +

@ @u

T:::::: (x; u)

@u
@x

;

where  is a global parameter. In this way, the substantial structure of covariant diﬀeren-
tiation in U4 shall be given by

H~  T:::::: (x; u) =

=

@
@ x

T::::::

(x; u) +

@
@ u&

+  (x; u) T:::::: (x; u) +

T:::::: (x; u)

@ @

u& x

+

 (x; u) T:::::: (x; u) + : : : +

+  (x; u) T:::::: (x; u)  (x; u) T :::::: (x; u)

 (x; u) T :::::: (x; u) : : :  (x; u) T:::::: (x; u)

along with the more regular (point-oriented) form

H T::::::

(x; u)

=

@

@
x

T::::::

(x; u)

+

+  (x; u) T:::::: (x; u) +  (x; u) T:::::: (x; u) + +

+  (x; u) T:::::: (x; u)  (x; u) T :::::: (x; u)

 (x; u) T :::::: (x; u)

 (x; u) T:::::: (x; u) :

Turning our attention to the world-metric tensor, we see that the expression

H~  g

(x;

u)

=

@
@ x

g

  (x; u) g (x; u)

(x; u) +

@
@ u

(g

(x;

u))

@ @

u x

 (x; u) g  (x; u)

may enable us to establish a rather indirect metricity-like condition. This can be done by invoking the condition

  (x; u) u = 0

and by setting

H~  g (x; u) = 0 :

Now, with the help of the already familiar relations

@
@ u g

(x; u) =

;

g

(x;

u)

@

@
u

g

(x;

u)

=

2
@

@
u

ln

Fdet

(g

(x;

u))

we shall again have

H H  g (x; u) =   u :

5 Generalized curvature forms

We are now equipped enough with the basic structural relations to investigate curvature forms in our theory. In doing so, we shall derive a set of generalized Bianchi identities corresponding to a peculiar class of ﬁeld equations, including some possible conservation laws (in rather special circumstances).
In a direct customary manner, we have the extended expression

H~  H~  H~  H~  T:::::: (x; u) =

= (H H H H) T:::::: (x; u) +

H +

@

@
u

T:::::: (x; u)

@ u @ x

H @
@ u

T:::::: (x; u)

@ @

u u

+

H +


@
@ u

T:::::: (x; u)

@ u 
 @ x

H


@
@ u

T:::::: (x; u)

@ @

u x


+

+@
@u

 @
@ u

T:::::: (x; u)

@ u 
 @ u @ x @ x

@ @u

 @
@ u

T:::::: (x; u)

@ u 
 @ u @ x @ x

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for which the essential part is

(H H

H H


)

::: T:::

(x; u)

=

= R

(x;

u)

::: T:::

(x; u)

+

+

R

(x;

u)

::: T:::

(x; u) + : : : +

+

 R 

(x;

u)

T::::::

(x; u)

R


(x;

u)

 ::: T:::

(x; u)

R


(x;

u)

::: T:::

(x; u)

:::

R


(x;

u)

::: T:::

(x;

u)

H 2

( ) 
 [ ] x; u


::: T:::

( ) x; u :

Here the world-curvature tensor, that is, the generalized, Eulerian Riemann tensor, is given by

R


(x;

u)

=

@
@ x

 (x; u)

@
@ x

 (x; u) +

+  (x; u)  (x; u)  (x; u)  (x; u)

for which the corresponding curvature form of mobility may simply be given by

R~


(x;

u)

=

@
@ x


(x; u)

+

@
@ u

 (x; u)

@
@ x

( )  x; u

@
@ u


( ) +  x; u

@u
@ x


@u
@ x

+  (x; u)  (x; u)

 (x; u)  (x; u) :

We can now write the following fundamental decomposition:
R  (x; u) = B  (x; u) + M  (x; u) + + N  (x; u) + U  (x; u) ;

( ) = B  x; u

@
@ x

( )  x; u

@
@ x

 (x; u) +

+  (x; u)  (x; u)

 (x; u)  (x; u) ;

H H ^ ^ M  (x; u) =  K  (x; u)

 K  (x; u) +

+ K (x; u) K (x; u) K (x; u) K (x; u) ;

H H ^ ^ N  (x; u) =  Q  (x; u)

 Q  (x; u) +

+ Q (x; u) Q (x; u) Q (x; u) Q (x; u) ;

U (x; u) = K(x; u) Q (x; u) K(x; u) Q (x; u) + + Q (x; u) K (x; u) Q (x; u) K (x; u) ;

where the Eulerian Levi-Civita connection, the Eulerian contorsion tensor, and the connection of non-metricity are re-

spectively given by


(x;

u)

=

1 2

( 

g

x;


)@
u @x

g

(x; u)

+ ( )  @ g

x; u

;

@x

( ) = ( ) 

K

x; u


[ ] x; u

@
g
@x

(x; u) +

( ) ( ) ( ) + ( ) ( )   g

x; u g

x; u

 [ ] x; u

 g

x; u

 [ ] x; u ;

H 
Q

(x;

u)

=

1 2


g


(x;

u)


( ) g

x; u

H +

( )  g

x; u ;

H g

(x; u) +

H^
such that represents covariant diﬀerentiation with respect

to the symmetric connection (x; u) alone. The curvature

tensor given by B (x; u) is, of course, the Eulerian Riemann-

Christoﬀel tensor, generalizing the one of the General Theory

of Relativity which depends on position alone. Of special interest, for the world-metric tensor, we note
that
H H H H (    ) g (x; u) = R  (x; u) + R  (x; u)

H 2 ( ) [] x; u  g (x; u)

where, with the usual
g  (x; u) R  (x; u). That

notation, is, more

R  (x; u) =
speciﬁcally, while

keeping in mind that


=

@
@ u

g

(x; u) =

we have
(H H H H) g (x; u) = R

+ R  (x; u) 2 [] (x; u)

;
 (x; u) +
H u :

As such, we have a genuine homothetic curvature given by
H (x; u) = R (x; u) =
= H^ Q (x; u) H^ Q (x; u) =

= @ Q (x; u) @ Q (x; u) ;

@x

@x

Q

(x; u) = Q

(x;

u)

=

1 2

g

(x; u) H

g

(x; u) =

= 2 Hu:

Upon setting


(x;

u)

=

1 2

H u;

we have


H (x; u) = 2 @  (x; u) @  (x; u)

@x

@x


2

(x; u) @ ln

 (x; u) @ ln


 :

@x

@x

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At this point, the generalized, Eulerian Ricci tensor is given in the form
R (x; u) = R  (x; u) = Z ( (x; u) ; K (x; u)) + + N (Q (x; u)) + X (K (x; u) ; Q (x; u)) ;

Z ( (x; u) ; K (x; u)) = B  (x; u) + M  (x; u) ; N (Q (x; u)) = N (x; u) ; X (K (x; u) ; Q (x; u)) = U  (x; u) ;

which admits the peculiar anti-symmetric part

R[

]

(x;

u)

=

1 2


@

@ x

+

1 2


@

@ x

Q

(x; u)

+ H^


[

] (x; u) +

K (x; u)

@ @x

K 

(x;


u)

+

@


Q (x; u) +

@x

+


[

] (x; u) K (x; u) +


[

] (x; u) Q (x; u) +

+


[

]

(x;

u)

Q


(x;

u)


[

]

(x;

u)

Q

(x;

u)

+

+

1 2

K (x; u) K

(x; u)

K (x; u) K (x; u) ;

where we have made use of the fact that K (x; u) =

=

[]
(x; u). Let us also keep in mind that the explicit phys-

[]

ical structure of the connection form forming our various cur-

vature expressions, as it relates to gravity, electromagnetism,

material spin, and the nuclear interaction, is given in Section 3

of this work, naturally following [4].

We can now obtain the complete Eulerian generalization

of the ﬁrst Bianchi identity as follows:

R  (x; u) + R  (x; u) + R   (x; u) =


=

2 g  (x; u)

@
@ x


[

]

(x;

u)

+

@ @x


+

@
@ x

[ ] (x; u)


2 g  (x; u)  (x; u) [] (x; u) +

[] (x; u) +


+

 (x; u)

[

(x; u) +
]

 (x; u)

[ ] (x; u)

+


H H + 2  [] (x; u) u + [ ] (x; u) u +


H +


[

]

(x;

u)

u

:

Similarly, after a somewhat lengthy calculation, we obtain, for the generalization of the second Bianchi identity,

H R  (x; u) + H R  (x; u) + H R  (x; u) = 


=2

[] (x; u) R

 (x; u) +


[] (x; u) R

 (x; u) +

+ [] (x; u) R  (x; u) +

+  (x; u) ((H ) Hu (H ) H u) +

+  (x; u) ((H ) Hu (H ) H u) +

+  (x; u) ((H ) Hu (H ) H u)

H  (x; u)


R (x; u) u + 2

 []

(x;

u)

u

H  (x; u)


R (x; u) u + 2

 []

(x;

u)

 u

H  (x; u)


R (x; u) u + 2

 []

(x;

u)

 u

+

+  (H u) H  (x; u) H  (x; u) +

+  (H u) H  (x; u) H  (x; u) +

+  (H u) H  (x; u) H  (x; u) ;

where

H  (x; u) H  (x; u) = R  (x; u) +

+  (x; u)  (x; u)   (x; u)  (x; u)

2


[

]

(x;

u)

 (x; u) :

By contraction, we may extract a physical density ﬁeld as follows:

J (x; u) =

=

H


1 2

R

(x; u) + R

(x; u)

1 2


R (x; u) ;

where R (x; u) = R  (x; u) are the components of the
generalized Ricci tensor of the second kind and
R (x; u) = R (x; u) = R (x; u) is the generalized Ricci
scalar. As we know, the Ricci tensor of the ﬁrst kind and the Ricci tensor of the second kind coincide only when the connection form is metric-compatible. The asymmetric, generally non-conservative world-entity given by

G

(x;

u)

=

1 2

R

(x; u) + R

(x; u)

1 2

R (x; u)

will therefore represent the generalized Einstein tensor, such that we may have a corresponding geometric object given by

C (x; u)  g (x; u) J (x; u) = = H G (x; u) H G  (x; u) g (x; u) :

6 Quantum gravity from the physical vacuum of U4
We are now in a position to derive a quantum mechanical wave equation from the underlying structure of our present theory. So far, our ﬁeld equations appear too complicated to handle for this particular purpose. It is quite enough that

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we know the structural content of the connection form, which

encompasses the geometrization of the known classical ﬁelds.

However, if we deal with a particular case, namely, that of

physical vacuum, we shall immediately be able to speak of

one type of emergent quantum gravity.

Assuming now that the world-geometry U4 is devoid of “ultimate physical substance” (that is, intrinsic material con-

ﬁnement on the most fundamental scale) other than, perhaps,

primordial radiation, the ﬁeld equation shall be given by

R (x; u) = 0

( ) = ( ) 0 for which, in general, R  x; u W  x; u

,

( ) where W x; u is the generalized Weyl conformal tensor. In

this way, all physical ﬁelds, including matter, are mere ap-

pearances in our geometric world-structure. Consequently,
( ) = 0 from R( ) x; u , the emergent picture of gravity is
readily given by the symmetric Eulerian Ricci tensor for the

composite structure of gravity, that is, explicitly,

B ( (x; u)) =


 M

K (x; u) +N

Q( ) +

+U

K (x; u) ; Q (

)

 ;

( ) = ( ) where we have written Q x; u Q , such that, in this
special consideration, gravity can essentially be thought of as
exterior electromagnetism as well as arising from the quan-
( ) = 0 tum fuzziness of the background non-metricity of the world-
geometry. In addition, from R[ ] x; u , we also have the following anti-symmetric counterpart:

R[

](

(x;

u)

;

K

(x;

u))

=

@

@ x

Q()

@ @x

Q

(

)


[

] (x; u) Q (

)+

+


[

]

(x;

u)

Q

(

)


[

]

(x;

u)

Q


(

);

Q

(

)

=

1 2

2

Hu:

Correspondingly, we shall set, for the “quantum poten-

tial”,

Q ( ) = @ ln 

@x

( ) =  such that the free, geodesic motion of a particle along the

fuzzy world-path s x; u 

in the empty U4 can

simultaneously be described by the pair of dynamical equa-

tions

Du = 0 ;

Ds
D

=

0

;

Ds

since, as we have previously seen, Q

 u = 0.

Immediately, we obtain the geometrically non-linear

wave equation

Fdet

1 (g

(x;

u))

@

@ x


g

(x; u)

F
det

(g

(x;

u))

@ @x


=

= (R ( (x; u) ; K (x; u)) + (Q ( ))) 

that is,


 2B

R_

(x;


u)


=

0

;

where

2B

=

F
det

1 (g

(x; u))

@ @x


g

(x;

u)

F
det

(g

(x;

u))

@

@ x


 is the covariant four-dimensional Beltrami wave operator and,
with the explicit dependence of on ,

_ R

(x;

u)

=

R

(

(x; u) ; K (x; u)) +

Q


is the emergent curvature scalar of our quantum ﬁeld, for

which

(Q (

))

=

_ N

Q


1  g
2

(x; u) @  @  ; @x @x

_ NQ

 =N Q

 +

+ U ( ) K x; u ; Q  g (x; u) H^ Q  :

In terms of the Eulerian Ricci scalar, which is now quan-

tized by the wave equation, we have a quantum gravitational

( ) wave equation with two quantized intrinsic sources, namely,
the torsional source M x; u , which combines the electro-

 = ( ( )) magnetic and material sources, and the quantum mechanical

source Q

Q x; u ,

2B B (x; u)  = M (x; u)  + Q  

thereby completing the quantum gravitational picture at an elementary stage.

7 Special analytic form of geodesic paths

Here we are interested in the derivation of the generalized

geodesic equation of motion such that our geodesic paths cor-

respond to the formal solution of the quantum gravitational

 =  ( ) wave equation in the preceding section. Indeed, owing to

the wave function

x; u , these geodesic paths shall be

conformal ones.
( ) = For our purpose, let x const: represent a family of

hypersurfaces
( ( )) = 0 persurface ,

in U4 for @

such @x

that
x

with
x

respect to , there

= uine unit normal velocity vector, given by n

a mobile hyexists a genddx , at some

= ( ) point whose extended path can be parametrized by   s ,

that is


=@
n  x;

(x)

@

(x)

@x

@x

g (x; u) n n = 0 :

The essential partial diﬀerential equation representing

any quantum gravitational hypersurface

can then

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simply be represented by the arbitrary parametric form


x;

@ @x

(x) =   = const such that

Z b
 (x; u)

 d


(x) d 0

a

d

where a and b are two points in .
Keeping in mind once again that u = 0 and that

u

=

1 2

@

@ u

2 (x; u)

@
@ x g

(x; u) =

=  (x; u) +  (x; u) +

 Hu

the generalized Euler-Lagrange equation corresponding to

our situation shall then be given by


d

@ 2 (x; u)

ds @ u

@ 2 (x; u) +
@x

+ @ 2 (x; u) @ u + b (x; u) = 0 ;

@u

@x

where the “external” term is given by

b

(x; u) = 4


[

] (x; u) u u :

As a matter of straightforward veriﬁcation, we have

du ds

 (x; u) u u = 0

A unique general solution to the above equation
ing to the quantum displacement ﬁeld =

corr,eswphoincdh-,

in our theory, generates the non-metric nature of the world-

manifold U4, can now be obtained as

Z Z


s (x; u) = s  = C1 + C2 exp H  ds ds

tangibility, refer a smoothly extensive coordinate patch P (x) to the four-dimensional tangent hyperplane M4 (y), whose
metric tensor  is Minkowskian, such that an ensemble of Minkowskian tangent hyperplanes, that is,

X
a

=

1;2;:::;N

M4(a)

(y)

cannot globally cover the curved manifold U4 without breaking analytic continuity (smoothness), at least up to the third order. Denoting the “invariant derivative” by
HA = EA (x; u) @ @x , this situation can then basically be
described by

g (x; u) = EA (x; u) EB (x; u) AB ;

EA

(x;

u)

=

@ @

yA
x

;

EA

(x;

u)

=

EA (x; u)

1;

yA = yA (x; u) ; x = x (y) ;

EA (x; u) EA (x; u) = ; EA (x; u) EB (x; u) = AB ;


(x;

u)

=

EA

(x;

u)

@

@ x

EA (x; u) =

H = EA (x; u) EB (x; u) BEA (x; u) :

Of fundamental importance in our uniﬁed ﬁeld theory are, of course, the torsion tensor given by


[

]

(x;

u)

=

1 2

EA

(x;

u)


@

@ x

EA (x; u)

@ @x

EA

(x;


u)

and the curvature tensor given by

R (x; u) =

=

EA

(x;

u)


@

@ x

@ @x

=

EA

(x;

u)


@

@ x

@ @x

@ @x

@ @x


EA

(x;


u)

=

@ @x

@ @x


EA

(x;


u)

:

where C1 and C2 are integration constants. This is such that,

at arbitrary world-points a and b, we have the conformal re-

lation (for C = C2)

Z
dsb = exp C H


 ds dsa ;

which sublimely corresponds to the case of our previous quantum theory of gravity [3].

Additionally, we can also see that

R (x; u) =

=

EA

(x;

u)


@

@ x

@ @x

@ @x

@ @x


EA (x; u) +

+


@

@ x

@ @x

@ @x

@ @x


g (x; u) :

Immediately, we obtain

8 Geometric structure of the electromagnetic potential
As another special consideration, let us now attempt to extensively describe the geometric structure of the electromagnetic potential in our theory.
Due to the degree of complicatedness of the detailed general coordinate transformations in U4, let us, for the sake of

R (x; u) = EA (x; u) EB (x; u) EC (x; u)

(HB HC HC HB) EA (x; u)

2

 (x; u)


[

] (x; u) :

Introducing a corresponding internal (“isotopic”) curvature form through
R AB (x; u) = EC (x; u) (HA HB HB HA) EC (x; u) ;

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we can write

R (x; u) = EA (x; u) EB (x; u) RAB (x; u)

2

 (x; u)


[

] (x; u) :

In physical terms, we therefore see that

R (x; u) = EA (x; u) EB (x; u) RAB (x; u)

2  (x; u) S

e mc2

2

(x;

u)

 (x; u) F

u ;

where the electromagnetic ﬁeld tensor can now be expressed by the extended form (given in Section 3)

F~

(x;

u)

=

2

mc2 e

2

(x;

u)


[

] (x; u) u ;

that is,

F~

(x;

u)

=

mc2 e

2

(x;

u)

@ @

u x

@u @x


EA (x; u) EB (x; u) (HB uA HB uA) :

d AB = d yA (x; u) d yB (x; u) d yB (x; u) d yA (x; u).
Upon using Gauss theorem, we therefore see that.

'

(x;

u)

=

1 2

mc e

ZZ


 (x; u) (HB HA


HA HB) EC (x; u) uC d AB:

In other words, we have

' (x; u) =

1 mc 2e

Z

Z


(x; u) R AB (x; u) u d AB

or, with d  = EA (x; u) EB (x; u) d AB,

' (x; u) =

1 mc2 Z 2e

Z
2 (x; u)


R

 (x; u) + 2  (x; u)


[


] (x; u)

u d 

;

which means that

' (x; u) =

1 mc2 Z 2e

Z
2(x; u)


An essential feature of the electromagnetic ﬁeld in our uniﬁed ﬁeld theory therefore manifests as a ﬁeld of vorticity, somewhat reminiscent of the case of ﬂuid dynamics, that is,
F~ (x; u) =

R  (x; u) + 2  (x; u) S (x; u) u d  

1Z Z 2

 (x; u) F  (x; u) u u d  :

=

2

mc e


(x; u)

!

EA (x; u) EB (x; u) AB ;

where the vorticity ﬁeld is given in two referential forms by

!

=

1 2


@ @

u x

@u @x


;

AB

=

1 2

(HB

uA

HA uB) :

For our regular Eulerian electromagnetic ﬁeld, we simply have

Combining the above expression with the geodesic equa-
tion of motion given by ddus =  (x; u) u u , we ﬁnally
obtain the integral equation of motion

' (x; u) =

1 mc2 Z 2e

Z
2 (x; u)


R  (x; u) + 2  (x; u) S (x; u) u d  

1Z 2

Z  du  ds


F

 (x; u) d 

;

F

=F

(x;

u)

=

2

mc e

!

EA (x; u) EB (x; u) AB :

After some algebraic (structural) factorization, a profound

physical solution to our most general Eulerian expression for

the electromagnetic ﬁeld can be obtained in integral form as

'

(x;

u)

=

mc e

I

C

(x;

u)


@

@ x

EA

(x;


u)

uA

dx

such that F~

(x;

u)

=

@ @x

'

(x; u)

@ @x ' (x; u), that

is, in order to preserve the customary gauge invariance, our

electromagnetic ﬁeld shall manifestly be a “pure curl”.

This structural form is, of course, given in the domain of a

vortical path C covered by a quasi-regular surface spanned

in two directions and essentially given by the form

which shows, for the ﬁrst time, the explicit dependence of the electromagnetic potential on world-velocity (as well as local acceleration), global curvature, and the material spin ﬁeld.
9 Closing remarks
In the foregoing presentation, we have created a new kind of Finsler space, from which we have built the foundation of a uniﬁed ﬁeld theory endowed with propagating torsion and curvature. Previously [1, 5], we have done it without the “luxury” of killing the metricity condition of Riemannian geometry; at present, the asymmetric connection form of our world-geometry, in addition to the metric and curvature, is a function of both position and world-velocity. Therefore,

Indranu Suhendro. A New Finslerian Uniﬁed Field Theory of Physical Interactions

89

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looking back on our previous works, we may conclude that, in particular, the theories outlined in [3,4], as a whole, appear to be a natural bridge between generalized Riemannian and Finslerian structures.
A very general presentation of my own version of the theory of non-linear connection has also been given in [3], where, in immediate relation to [4], the enveloping evolutive world-structure can be seen as some kind of conformal Finsler space with torsion. The union between [3] and [4] has indeed already given us the essence of a fully geometric quantum theory of gravity, with electromagnetism and the YangMills gauge ﬁeld included. The present work mainly serves to complement and enrich this purely geometric union.
Submitted on June 27, 2009 / Accepted on August 06, 2009
References
1. Suhendro I. A four-dimensional continuum theory of space-time and the classical physical ﬁelds. Progress in Physics, 2007, v. 4, 34–46.
2. Suhendro I. A new semi-symmetric uniﬁed ﬁeld theory of the classical ﬁelds of gravity and electromagnetism. Progress in Physics, 2007, v. 4, 47–62.
3. Suhendro I. A new conformal theory of semi-classical quantum general relativity. Progress in Physics, 2007, v. 4, 96–103.
4. Suhendro I. A uniﬁed ﬁeld theory of gravity, electromagnetism, and the Yang-Mills gauge ﬁeld. Progress in Physics, 2008, v. 1, 31–37.
5. Suhendro I. Spin-curvature and the uniﬁcation of ﬁelds in a twisted space. Svenska fysikarkivet, Stockholm, 2008.
6. Borissova L. and Rabounski D. Fields, vacuum, and the mirror Universe. 2nd edition, Svenska fysikarkivet, Stockholm, 2009, p. 26–29.

October, 2009

90

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Physical Consequences of Mathematical Principles
Eliahu Comay
Charactell Ltd., PO Box 39019, Tel-Aviv, 61390, Israel E-mail: elicomay@post.tau.ac.il
Physical consequences are derived from the following mathematical structures: the variational principle, Wigner’s classiﬁcations of the irreducible representations of the Poincare´ group and the duality invariance of the homogeneous Maxwell equations. The analysis is carried out within the validity domain of special relativity. Hierarchical relations between physical theories are used. Some new results are pointed out together with their comparison with experimental data. It is also predicted that a genuine Higgs particle will not be detected.

1 Introduction
Physics aims to describe processes which are observed in the real world. For this purpose, mathematical formulations of physical theories are constructed. Mathematical elements of a physical theory can be divided into three sets: elements that play a relative fundamental role and are regarded as cornerstones of the theory’s structure, elements used as a derivation tool and ﬁnal formulas that describe the behavior of a given system. This kind of classiﬁcation is used here for the convenience of the presentation. In particular, what is regarded here as a fundamental element may, in principle, be derived from more profound mathematical elements.
This work regards the following mathematical structures as cornerstones of the discussion. The variational principle and its relevant Lagrangian density; Wigner’s analysis of the irreducible representations of the Poincare´ group; the duality invariance of the homogeneous Maxwell equations. Some well known results of these elements are pointed out alongside others that are not very well known. Boldface numbers are used for marking the latter kind of results. It is shown that some of these results ﬁt experimental data whereas others are used as a prediction of yet unknown experimental data.
The discussion is carried out within a framework that is based on the following theoretical elements. First, Special Relativity is regarded as a covering theory and all expressions must be consistent with relativistic covariance. The De Broglie relation between the particle’s wave properties and its energy-momentum is used. Another issue is related to the hierarchical relations between physical theories. (A good discussion of this issue can be found in [1], pp. 1–6.) The following lines explain this issue in brief.
Every physical theory applies to a limited set of processes. For example, let us take the problem of moving bodies. It is well known that physical theories yield very good predictions for the motion of planets around the sun. On the other hand, nobody expects that a physical theory be able to predict the speciﬁc motion of an eagle ﬂying in the sky. This simple example proves that the validity of a physical theory should be

evaluated only with respect to a limited set of experiments.

The set of experiments which can be explained by a physical

theory is called its domain of validity. The relations between

domains of validity deﬁne hierarchical relations between the

corresponding theories. For example, given theories A; B

and A’s domain of validity is a subset of B’s domain of va-

lidity then B’s rank is higher than that of A.

An examination of Newtonian mechanics and relativistic

mechanics illustrates the notion of hierarchical relations be-

tween theories. Newtonian mechanics is good for low veloc-

ity experiments (because its predictions are consistent with

the error range of measurements). On the other hand, rel-

ativistic mechanics is good even for velocities that approach

the speed of light. Two conclusions can be derived from these

properties of the theories: First, relativistic mechanics has a

more profound basis because it is valid for all experiments where Newtonian mechanics holds and for many other exper-

iments where Newtonian mechanics fails. Another aspect of

the relations between Newtonian mechanics and relativistic

mechanics is that Newtonian mechanics imposes constraints

on the form of the low velocity limit of relativistic mechan-

ics. Indeed, the low velocity limit of relativistic mechanics

is (and must be) consistent with Newtonian formulas. Below, this kind of constraint is called constraint imposed by a lower rank theory. Some of the theoretical derivations included be-

low rely on this principle.

The Lorentz metric used is diagonal and its entries are

(1, 1, 1, 1). Greek indices run from 0 to 3. Expressions

are written in units where = c = 1. In this system of units

there is just one dimension. Here it is taken to be that of

length. Therefore, the dimension of a physical quantity is a power of length and is denoted by [Ln]. In particular, energy

and momentum take the dimension [L ]. The symbol Q;

denotes the partial derivative of the quantity Q with respect

to


x

.

An

upper

dot

denotes

a

diﬀerentiation

with

respect

to

time.

The second section discusses quantum mechanical conse-

quences of the variational principle. The Dirac equation is ex-

amined in the third section. The fourth section shows incon-

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sistencies of the Klein-Gordon (KG) and the Higgs equations. The ﬁfth section examines results obtained from Wigner’s classiﬁcation of the irreducible representations of the Poincare group. Consequences of the duality invariance of the homogeneous Maxwell equations together a regular charge-monopole theory are discussed in the sixth section. The seventh section contains concluding remarks.

2 The Variational Principle

This section is dedicated to the form of a quantum theory of a massive particle. Let us examine the pattern obtained in a

two slit interference experiment. Here one ﬁnds bright and

dark strips. A completely dark interference point indicates

that a full anti-phase destruction takes place there. Obviously,

this property should be obtained in every Lorentz frame of

reference. It follows that the phase must depend on a Lorentz

scalar.

The quantity which is suitable for this purpose is the ac-

tion of the system. Thus, let us examine a Lagrangian density

of the system and its action

Z

S = L( ; ;) d"x:

(1)

Now, if the Lagrangian density is a Lorentz scalar then also the action is a Lorentz scalar. Therefore, it is concluded that

1. A relativistically consistent quantum theory may be derived from a Lagrangian density which is a Lorentz scalar.

Another issue is related to the dimension of the quantities. The phase is an argument of an exponent. Therefore, it must be dimensionless. Thus, in the system of units used here the action is dimensionless and satisﬁes this requirement. It follows that
2. An acceptable Lagrangian density must have the di-
mension [L "].
This conclusion means that the wave function acquires a well deﬁned dimension.

Remark:

The foregoing arguments indicate that if one wishes to take an alternative way for constructing a relativistically selfconsistent quantum theory, then one must ﬁnd another physically meaningful quantity that is a dimensionless Lorentz scalar and is suitable for taking the role of the particle’s phase. Apparently, such a quantity does not exist. If this claim is correct then the variational principle is also a necessary condition for constructing a self-consistent relativistic quantum theory.

Another point is related to the independent variables x

of the wave function

(x)

(2)

which is a single set of four space-time coordinates. Therefore (2) cannot describe a composite particle, because such a particle requires, besides a description of the space-time location of its center of energy, additional coordinates for describing its internal structure. Therefore,
3. The wave function (x) describes an elementary
structureless pointlike particle.

This result is consistent with the nature of an elementary clas-

sical particle (see [2], pp. 46, 47). Below it is applied as a

useful criterion for evaluating experimental data.

The Lagrangian density is used here as the cornerstone

of the theory. Hence, the particle’s equations of motion are

the corresponding Euler-Lagrange equations (see [3], p. 14;

[4], p. 16)

@ @L

@x

@

@ @x

@L
@

=

0

:

(3)

On this basis it is concluded that

4. The particle’s equations of motion are the Euler-Lagrange equations derived from the Lagrangian density.

Obviously, diﬀerent kinds of Lagrangian density yield diﬀerent equations of motion. This point is discussed later.
Another issue is the consistency of a quantum theory of a massive particle with the classical theory, where the latter provides an example of constraints imposed by a lower rank theory. The classical limit of quantum mechanics is discussed in the literature (see [5], pp. 19–21 and elsewhere; [6], pp. 25– 27, 137–138).
In order to do that, the quantum theory should provide expressions for the energy and the momentum of the particle. As a matter of fact, having an appropriate expression for the energy at the system’s rest frame is enough. Indeed, a Lorentz boost guarantees that the theory provides appropriate expressions for the energy and momentum in any reference frame. Therefore, the following lines examine the construction of an expression for the energy of a massive quantum mechanical particle in its rest frame. For this end, let us take the Lagrangian density and construct the following second rank tensor (see [4], p. 19)

T

=

@L

@

@ @x

@ @x

Lg :

(4)

Now, density is a 0-component of a 4-vector and the same

is true for energy. Hence, energy density is a (0,0) compo-

nent of a second rank tensor. Moreover, like the dimension of
T the Lagrangian density, the dimension of  of (4) is [L "].
This is also the dimension of energy density. Now, in quan-

tum mechanics, the Hamiltonian is regarded as the energy op-
T erator. Thus, the entry  of (4) is regarded as an expression
for the Hamiltonian density

H=

_ @L
@_

L:

(5)

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It is explained below why an expression for density is re-
quired. Here, density properties can be readily taken from
electrodynamics (see [2], pp. 73–75). Density must have the
dimension [L !] and be a 0-component of a 4-vector satisfy-
ing the continuity equation

j; = 0 :

(6)

At this point, one may take either of the following alternatives:
H A. Use the Hamiltonian density together with the den-
sity expression and extract the Hamiltonian diﬀerential
operator H, operating on . The energy is an eigen-
value of this operator:

H =E ;

(7)

3 The Dirac ﬁeld

It is shown here that the Dirac ﬁeld satisﬁes the requirements derived above and that experimental data support the theory. The formulas are written in the standard notation [3,7].
The Dirac Lagrangian density is

L =  (i@ eA) m :

(11)

A variation with respect to  yields the corresponding Euler-Lagrange equation

(i@ eA) = m :

(12)

As stated in section 2, the dimension of a Lagrangian den-
sity is [L "]. Therefore, the dimension of is [L != ] and

the Dirac 4-current

j =  

(13)

Now the De Broglie relation

i

@ @t

=E

;

satisﬁes the required dimension and the continuity equation

(6) (see [7], p. 9). Thus, the density is the 0-component

(8) of (13)

Dirac = O :

(14)

yields the diﬀerential equation

i

@ @t

=H

:

(9)

At this point one can construct a Hilbert space that in-
cludes all eigenfunctions of the Hamiltonian H.
B. Use the expression for density as an inner product for and construct an orthonormal basis for the corresponding Hilbert space. Next construct the Hamiltonian ma-
trix. For the i; j functions of the Hilbert space basis,
the Hamiltonian matrix element is
Z
Hij = H( i; i;; j; j;) d!x : (10)

At this point, the Hamiltonian matrix is diagonalized and its energy eigenfunctions and eigenvalues are obtained.
Obviously, the mathematical structures of A and B are relevant to the same data. Therefore, both methods construct one and the same Hilbert space.
Equation (9) makes the following problem. As stated above, the Euler-Lagrange equation (3) is the system’s equation of motion. On the other hand, (9) is another diﬀerential equation. Hence, the following requirement should be satisﬁed.
5. Requirement 1: The ﬁrst order diﬀerential equation (9) should be consistent with the Euler-Lagrange equation of the theory (3).
The next two sections are devoted to two speciﬁc kinds of Lagrangian density of massive particles.

Substituting the Dirac Lagrangian density (11) into the general formula (5), one obtains the Dirac Hamiltonian density
H = O α ( ir eA) + m + eV : (15)
The density O can be factored out from (15) and the
expression enclosed within the square brackets is the Dirac Hamiltonian written as a diﬀerential operator. Its substitution into (9) yields the well known Dirac quantum mechanical equation

i

@ @t

=

α

(

ir

eA) + m + eV

:

(16)

It is also interesting to note that due to the linearity of the Dirac Lagrangian density (11) with respect to _ , the Dirac Hamiltonian density (15) as well as the Dirac Hamiltonian do not contain a derivative of with respect to time. Hence, (16)

is an explicit ﬁrst order diﬀerential equation. It is easily seen

that (16) agrees completely with the Euler-Lagrange equation

(12) of the Dirac ﬁeld. It follows that Requirement 1 which

is written near the end of section 2 is satisﬁed.

A Hilbert space can be constructed from the eigenfunc-

tions obtained as solutions of the Dirac equation (16). Here

the inner product of the Hilbert space is based on the density

of the Dirac function (14). The eigenfunctions of the Hamil-

tonian are used for building an orthonormal basis

Z
ij =

O
i

j d!x :

(17)

Now, the form of an energy eigenfunction is

(x; t) = e iEt(x):

(18)

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This form enables a construction of a Hilbert space based
on e iEt(x) (the Schro¨dinger picture) or on (x) (the Hei-

senberg picture). Here, in the Heisenberg picture, wave func-

tions of the Hilbert space are time independent.

As is well known, the non-relativistic limit of the Dirac

equation agrees with the Pauli equation of a spinning electron

(see [7], pp. 10–13). Hence, in accordance with the discus-

sion presented in the ﬁrst section, the Dirac relativistic quan-

tum mechanical equation is consistent with the constraint im-

posed by the lower rank theory of the non-relativistic quan-

tum mechanical equations. A related aspect of this constraint
is the density represented by the Dirac wave function (14). Indeed, in the non-relativistic limit of Dirac’s density, (14)

reduces to the product of the ”large” components of Dirac’s

(14) (see [7], pp. 10–13). Hence,
sity of the Pauli-Schro¨dinger equations

agOree. sTwhiisthagtrheeemdeenn-t

also proves the compatibility of the Hilbert space of the Pauli-

Schro¨dinger equations with that of the non-relativistic limit of

the Dirac equation.

Beside the satisfactory status of Dirac’s theory, his equa-

tion has an extraordinary success in describing experimental

results of electrons and muons in general and in atomic spec-

troscopy in particular. Moreover, experiments of very high

energy prove that quarks are spin-1/2 particles. In particular, high energy experimental data are consistent with the pointlike nature of electrons, muons and quarks (see [8], pp. 271,

272; [9], p. 149). Hence, the Dirac equation satisﬁes item 3

of section 2.

4 Lagrangian density of second order equations

This section discusses second order quantum equations of motion (denoted here by SOE) which are derived from a Lagrangian density. The presentation is analogous to that of the previous section where the Dirac equation is discussed. The analysis concentrates on terms containing the highest order derivatives. Thus, the speciﬁc form of terms containing lower order derivatives is not written explicitly and all kinds of these terms are denoted by the acronym for Low Order Terms LOT . Second order quantum diﬀerential equations are derived from Lagrangian densities of the following form:

L = ;;g + LOT:

(19)

This form of the Lagrangian density is used for the KG (see [3], p. 38) and the Higgs (see [4], p. 715) ﬁelds.
Applying the Euler-Lagrange variational principle to the
Lagrangian density (19) one obtains a second order diﬀeren-
tial equation that takes the following form

g@@ = LOT:

(20)

Here, unlike the case of the Dirac ﬁeld, the dimension of  is L . Hence, in order to satisfy dimensional requirements,

the expression for density must contain a derivative with respect to a coordinate. Thus, the 4-current takes the following form (see [3], p. 40; [10], p. 199)

j = i ( ; ;) + LOT

(21)

and the density is

 = i ( _ _ ) + LOT:

(22)

The left hand side of (21) is a 4-vector. Therefore,  of

SOE is a Lorentz scalar.
(5) Using the standard method , one ﬁnds that the Hamil-

tonian density takes the following form (see [3], p. 38; [10],

p. 198) H = _ _ + (r ) (r) + LOT:

(23)

An analysis of these expressions shows that, unlike the case of the Dirac equation, SOE theories encounter problems. Some of these problems are listed below.

a. One cannot obtain a diﬀerential operator representing

the Hamiltonian. Indeed, the highest order time deriva-

_ tive of the
spect to 

_SOE density
;  whereas

(22) is anti-symmetric with
the corresponding term of

rethe

Hamiltonian density (23) is symmetric with respect to

these functions (see [11], section 3, which discusses the

KG equation). Hence, in the case of SOE theories, one

cannot use method A of section 2 for constructing a

Hilbert space for the system.

b. The density associated with the wave function  is an indispensable element of the Hilbert space. The de-
pendence of the SOE density (22) on time-derivatives
proves that a SOE Hilbert space is built on functions of the four space-time coordinates x. Hence, SOE cannot use the Heisenberg picture where the functions of
= ( ) the Hilbert space are time independent H S t
(see [3], p. 7).

c. In the Schro¨dinger theory

represents density. It

follows that like the case of the Dirac ﬁeld, the dimen-
[ ] sion of this is L != . On the other hand, the di[ ] mension of the SOE function  is L  . Therefore,

the nonrelativistic limit of SOE theories is inconsistent

with the Schro¨dinger theoretical structure.

d. Unlike the Dirac Hamiltonian, which is independent
_ _ of time-derivatives of , the SOE Hamiltonian den-
sity has a term containing the bilinear product  .
Hence, it is not clear how a SOE analogue of the fun-
(9) damental quantum mechanical equation can be cre-
ated. Moreover, it should be proved that this ﬁrst order implicit nonlinear diﬀerential equation is consistent with the corresponding second order explicit diﬀer-
ential equation (20) of SOE, as stated by requirement
1 which is formulated near the end of section 2. With-
out substantiating the validity of the Hamiltonian, SOE

94

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theories violate a constraint imposed by a lower rank theory which is explained in the lines that precede (4).
e. Some SOE theories apply to real ﬁelds (see [3], p. 26; [4], p. 19 etc.). New problems arise for these kinds of physical objects. Indeed, density cannot be deﬁned for these particles (see [12], pp. 41–43). Moreover, a massive particle may be at rest. In this case its amplitude should be independent of time. But a real wave function has no phase. Therefore, in the case of a motionless real particle, the time-derivative of its wave function vanishes identically. For this reason, its physical behavior cannot be described by a diﬀerential equation with respect to time. Thus, a real SOE particle cannot be described by the SOE equation of motion (20) and it cannot have a Hamiltonian.
f. Another problem arises for a charged SOE particle. As
stated in item a above, this particle cannot have a diﬀer-
ential operator representing the Hamiltonian. Hence,
method A, discussed near (7)-(9), cannot be used for a
Hilbert space construction. Moreover, the inner product of a time-dependent Hilbert space is destroyed in the case of an external charge that approaches a charged
SOE particle (see [13], pp. 59–61). Hence, method B
does not hold either. It follows that a charged SOE particle has no Hamiltonian. Therefore, a charged SOE particle does not satisfy a constraint imposed by a lower rank theory.
This discussion points out theoretical diﬃculties of SOE ﬁelds. The experimental side responds accordingly. Point 3 of section 2 is useful for evaluating the data. Thus, a ﬁeld
(x) used in a Lagrangian density describes an elementary
point-like particle. It turns out that as of today, no scalar pointlike particle has been detected.
In the history of physics, the three -mesons have been regarded as KG particles and the electrically neutral  mem-
ber of this triplet was regarded as a Yukawa particle, namely, a real (pseudo) scalar KG particle. However, it has already
been established that -mesons are not elementary pointlike particles but composite particles made of qq and they occupy
a nonvanishing spatial volume. Thus, as of today, there is no experimental support for an SOE particle. The theoretical and experimental SOE problems mentioned above are regarded seriously here. On the basis of the foregoing analysis, it is predicted here that no genuine elementary SOE particle will be detected. A special case is the following statement: a genuine Higgs particle will not be detected.

this paper, which will certainly stand as one of the great intellectual achievements of our century” (see [16], p. 149). Wigner’s work shows that there are two physically relevant classes of irreducible representations of the Poincare´ group.
One class is characterized by a mass m > 0 and a spin s. The second class consists of cases where the self mass m = 0, the energy E > 0 and two values of helicity. (Helicity is
the projection of the particle’s spin in the direction of its mo-
mentum.) Two values of helicity s correspond to a spin s.
Thus, each massive particle makes a basis for a speciﬁc irreducible representation that is characterized by the pair of
values (m; s). A massless particle (like the photon) has a
zero self mass, a ﬁnite energy and two values of helicity (for a photon, the helicity is 1).
A result of this analysis is that a system that is stable for a long enough period of time is a basis for an irreducible representation of the Poincare´ group (see [15], pp. 48–50). Let us take a photon. Cosmic photons are detected by measuring devices on earth after traveling in space for a very very long time, compared to the duration of an electromagnetic interaction. Therefore, photons must belong to a unique irreducible representation of the Poincare´ group. This conclusion is inconsistent with the idea of Vector Meson Dominance (VMD). VMD regards the photon as a linear combination of a massless real photon and a massive vector meson. (For a presentation of VMD see [9], pp. 296–303; [17].)
The VMD idea has been suggested in order to explain experimental results of scattering of energetic photons on nucleons. The main points of the data are:
i. The overall charge of a proton is +e whereas the over-
all charge of a neutron vanishes. Therefore, charge constituents of a proton and a neutron are diﬀerent.
ii. In spite of the data of the previous item, interaction of a hard photon with a proton is nearly the same as its interaction with a neutron.
The theoretical analysis of Wigner’s work shows that VMD is unacceptable. Other inconsistencies of VMD with experimental data have also been published [18]. This state of affairs means that the currently accepted Standard Model has no theoretical explanation for the photon-nucleon interaction. This point is implicitly recognized by the PACS category of VMD which does not belong to a theoretical PACS class. Thus, on July 2009, VMD is included in the class of ”Other models for strong interactions”. Hence, the Standard Model does not provide a theoretical explanation for the scattering data of hard photons on nucleons.

5 Irreducible representations of the Poincare´ group

6 Duality transformations of electromagnetic ﬁelds

The profound signiﬁcance of Wigner’s analysis of the irreducible representations of the Poincare´ group (see [14]; [15], pp. 44–53; [16], pp. 143–150) is described by the following words: ”It is diﬃcult to overestimate the importance of

Electromagnetic ﬁelds travel in vacuum at the speed of light. Therefore, the associated particle, namely — the photon, is massless. For this reason, it cannot be examined in a frame where it is motionless. This result means that the argument of

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point e of section 4 does not hold for electromagnetic ﬁelds.

It follows that, unlike the wave function of a massive parti-

cle, electromagnetic ﬁelds can be described by a Lagrangian density that depends on real functions. This well known fact

is another aspect of the inherent diﬀerence between massive

and massless particles, which has been obtained by Wigner

and discussed in the previous section.

Thus, the system consists of electromagnetic ﬁelds whose

equations of motion (Maxwell equations) are derived from

a Lagrangian density and charge carrying massive particles

whose equation of motion (the Lorentz force) is derived from

a classical Lagrangian. Below, this theory is called ordinary

electrodynamics. All quantities are described by real func-

tions. The action of the system is (see [2], p. 75)

S=

Zp
m1

v dt

Z ZAje d"x

1
16

FF d"x ;

(24)

where the subscript e indicates that j is a current of electric

charges, A ﬁelds, and F

denotes  is the

the 4-potential of the electromagnetic corresponding ﬁelds tensor

F = A; A; :

(25)

The explicit form of this tensor is

0

1

BB@ 0 Ex

F  =

Ex Ey

0
Bz

Ey Bz
0

Ez By Bx

CCA :

(26)

Ez By Bx 0

These expressions enable one to derive Maxwell equations (see [2], pp. 78, 79 and 70, 71)

F ; = 4  je; F ; = 0 :

(27)

0 Here F  is the dual tensor of F 

1

F


= BB@

0
Bx By

Bx
0
Ez

By Ez
0

Bz Ey Ex

CCA :

(28)

Bz Ey Ex 0

These tensors satisfy the following relation

F  = 1 " F ;

(29)

2

where " is the completely antisymmetric unit tensor of
the fourth rank. The Lorentz force, which describes the motion of a charg-
ed particle, is obtained from a variation of the particle’s coordinates (see [2], pp. 49–51)

mae = eF v :

(30)

The foregoing expressions describe the well established theoretical structure of ordinary electrodynamics. Let us see the results of introducing duality transformations. Duality
transformations (also called duality rotations by =2) of elec-
tromagnetic ﬁelds take the following form (see [19], pp. 252, 551; [20], p. 1363)

E  B; B  E :

(31)

These transformations can be put into the following tensorial form

F   F ; F   F :

(32)

An examination of the homogeneous Maxwell equations

F ; = 0; F ; = 0;

(33)

proves that they are invariant under the duality transforma-

tions (32). On the other hand, an inequality is obtained for the inhomogeneous Maxwell equation

F ;

4  je:

(34)

This problem can be settled by the introduction of the notion of magnetic monopoles (called brieﬂy monopoles). Thus, duality transformations of the electromagnetic ﬁelds (32) are augmented by the following transformation that relates charges and monopoles

e  g; g  e ;

(35)

where g denotes the monopole strength.
Two things are established at this point:

1. The theoretical foundation of ordinary electrodynamics (24), and its equations of motion (27) and (30).

2. The mathematical form of duality transformations (32) and (35).

Now, a theory for a system of monopoles and electromagnetic

ﬁelds (called below monopole electrodynamics) is obtained

from the application of duality transformations to ordinary

electrodynamics. The action principle of this system is

Z p ZZ S =

m 1 v dt

Amjm d"x

1
16

Fm Fm d"x ; (36)

where the subscript (m) denotes that the quantities pertain to
monopole electrodynamics. Here the ﬁelds are derived from a 4-potential

Fm = Am; Am; ;

(37)

which is analogous to (25). Maxwell equations of monopole electrodynamics are

Fm

 ;

=

4jm;

Fm; = 0

(38)

96

Eliahu Comay. Physical Consequences of Mathematical Principles

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PROGRESS IN PHYSICS

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and the Lorentz force is

mam = gFm v :

(39)

Thus, we have two theories for two distinct systems: ordinary electrodynamics for a system of charges and ﬁelds and monopole electrodynamics for a system of monopoles and ﬁelds. The ﬁrst system does not contain monopoles and the second system does not contain charges. The problem is to ﬁnd the form of a uniﬁed theory that describes the motion of charges, monopoles and ﬁelds. Below, such a theory is called a charge-monopoly theory. The charge-monopole theory is a higher rank theory whose domain of validity includes those of ordinary electrodynamics and of monopole electrodynamics as well. On undertaking this assignment, one may examine two postulates:
1. Electromagnetic ﬁelds of ordinary electrodynamics are identical to electromagnetic ﬁelds of monopole electrodynamics.
2. The limit of the charge-monopole theory for a system that does not contain monopoles agrees with ordinary electrodynamics and limit of the charge-monopole theory for a system that does not contain charges agrees with monopole electrodynamics.
It turns out that these postulates are mutually contradictory. A charge-monopole theory that relies (implicitly) on the
ﬁrst postulate has been published by Dirac many years ago [21, 22]. (Ramiﬁcations of Dirac monopole theory can be found in the literature [20].) This theory shows the need to deﬁne physically unfavorable irregularities along strings. Moreover, the form of its limit that applies to a system of monopoles without charges is inconsistent with the theory of monopole electrodynamics, which is derived above from the duality transformations. Therefore, it does not satisfy the constraint imposed by a lower rank theory. The present experimental situation is that in spite of a long search, there is still no conﬁrmation of the existence of a Dirac monopole (see [23], p. 1209).
The second postulate was used for constructing a diﬀerent charge-monopole electrodynamics [24, 25]. This postulate guarantees that the constraints imposed by the two lower rank theories are satisﬁed. Moreover, this theory does not introduce new irregularities into electrodynamics. Thus, it is called below regular charge-monopole theory. The following statements describe important results of the regular charge-monopole theory: The theory can be derived from an action principle, whose limits take the form of (24) and (36), respectively. Charges do not interact with bound ﬁelds of monopoles; monopoles do not interact with bound ﬁelds of charges; radiation ﬁelds (namely, photons) of the systems are identical and charges as well as monopoles interact with them. Another result of this theory is that the size of an ele-
mentary monopole g is a free parameter. Hence, the theory is

relieved from the huge and unphysical Dirac’s monopole size
g = 34:25.
The regular charge-monopole theory is constructed on the basis of the second postulate. This point means that it is not guided by new experimental data. However, it turns out that it explains the important property of hard photon-nucleon interaction which is mentioned in the previous section. Indeed, just assume that quarks carry a monopole and postulate that
the elementary monopole unit g is much larger then the elec-
tric charge e (probably jgj ' 1). This property means that
photon-quark interaction depends mainly on monopoles and that the photon interaction with the quarks’ electric charge is a small perturbation. Therefore, the very similar results of photon-proton and photon-neutron scattering are explained. (Note also that all baryons have a core which carries three units of magnetic charge that attracts the three valence quarks. The overall magnetic charge of a hadron vanishes.) Other kinds of experimental support for the regular charge-monopole theory have been published elsewhere [26].
7 Concluding remarks
This work is based on the main assumption of theoretical physics which states that results derived from physically relevant mathematical structures are expected to ﬁt experimental data [27]. Three well known mathematical structures are used here: the variational principle, Wigner’s analysis of the irreducible representations of the Poincare´ group and duality transformations of electromagnetic ﬁelds.
The paper explains and uses three points which are either new or at least lack an adequate discussion in textbooks.
1. Constraints are imposed by a lower rank theory on properties of the corresponding limit of a higher rank theory (see a discussion in the Introduction).
2. The need to prove consistency between the EulerLagrange equation obtained from a Lagrangian density
and the quantum mechanical equation i @ =@t = H
which holds for the corresponding Hamiltonian.
3. The ﬁeld function (x) describes an elementary
pointlike particle (see the discussion near (2)).
Points 1 and 2 are useful for a theoretical evaluation of the acceptability of speciﬁc physical ideas. Point 3 is useful for ﬁnding an experimental support for these ideas.
The main results of the analysis presented in this work are as follows: Dirac equation is theoretically consistent and has an enormous experimental support. Second order quantum mechanical equations (like the Klein-Gordon and the Higgs equations) suﬀer from many theoretical problems and have no
experimental support. (-mesons are not pointlike, therefore,
they are not genuine Klein-Gordon particles.) Real ﬁelds cannot be used for a description of massive particles. The idea of Vector Meson Dominance is inconsistent with Wigner’s analysis of the irreducible representations of the Poincare´ group.

Eliahu Comay. Physical Consequences of Mathematical Principles

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Therefore, VMD is unacceptable and the Standard Model has no theoretical explanation for the data of a scattering process of an energetic photon on nucleon. Monopole theories that introduce irregularities along strings are inconsistent with point 1 of this section and have no experimental support. The regular charge monopole theory [24–26] is consistent with point 1 and has experimental support.
Submitted on August 02, 2009 / Accepted on August 06, 2009
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