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Proceedings of the International Conference on Two Cosmological Models

PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON
TWO COSMOLOGICAL MODELS
HELD AT THE UNIVERSIDAD IBEROAMERICANA IN 2010
M E X I C O 2 0 1 2
John A. Auping Coordinator

Primera edición: 2012
Universidad Iberoamaericana Prolongación Paseo de la Reforma 880 Lomas de Santa Fe, Mexico, D.F. John A. Auping jauping@iwm.com.mx Plaza y Valdés, S. A. de C. V. Plaza y Valdés, S.A. de C.V. Manuel María Contreras 73. Colonia San Rafael México, D.F. 06470. Teléfono: 50 97 20 70 editorial@plazayvaldes.com www.plazayvaldes.com Plaza y Valdés Editores Calle Murcia, 2. Colonia de los Ángeles Pozuelo de Alarcón 28223, Madrid, España Teléfono: 91 862 52 89 madrid@plazayvaldes.com www.plazayvaldes.es ISBN: 978-607-402-530-9 Impreso en México / Printed in Mexico

Contents*
*The pages have double numeration. The number on top is the book page number and the one below is the lecture page number.
Alfredo Sandoval, Preface.............................................................................................................................9
PART I, SPIRAL GALAXY ROTATIONAL VELOCITY1
Hans Ohanian, Gravitation and Space-time: Einstein´s contribution.....................................................13
John Auping, Putting the Standard ΛCDM Model and the Relativistic Models in Historical Context..........................................................................................................................35
Fred Cooperstock and S. Tieu, Relativistic Gravitational Dynamics and the Rotation Curves of Galaxies.......................................................................................................................................117
Octavio Valenzuela, Observational Constraints onGalaxy Dark Matter Halos2
John Moffat, Modified Gravity or Dark Matter?......................................................................................137
David Rodrigues, Patricio Letelier an Ilya Shapiro, Galaxy Rotation Curves Interpreted from General RelativitywithInfraredRenormalization Group Effects...............................................................151
PART II, GRAVITATIONAL DYNAMICS OF GALAXY CLUSTERS3
Hans Böhringer, The Concordant Cosmological Model and the Dark Matter HypothesisinExplaining GalaxyGlusterDynamics4..................................................................................163
Vladimir Ávila-Reese, The Standard Model5
Luisa Jaime, Leonardo Patiño and Marcelo Salgado Description and Results of a Robust Approach to F(R) Gravity.................................................................................................................181
Fred Cooperstock and S. Tieu, General Relativistic Dynamics of Galaxy Clusters..........................193
1 Lectures given on November 17th , 2010 2 This lecture was not made available for its publication in the Proceedings 3 Lectures given on November 18th , 2010 4 This lecture was not made available in its written form, so that a transcription of its oral presentation is published 5 This lecture was not made available for its publication in the Proceedings

Joel Brownstein, A Good Fit to the Missing Mass Problem in Galaxies and Clusters of Galaxies .........203 Philip Mannheim, Making the Case for Conformal Gravity....................................................................223 Hans Ohanian, Problems with Conformal Gravity6......................................................................................253 Tejinder Singh, The Effect of Inhomogeneities on the Average Cosmological Dynamics............................259 PART III, THE APPARENT ACCELERATION OF THE EXPANSION OF THE UNIVERSE7 Alain Blanchard, Cosmic Acceleration: What Do Data Actually Tell Us?................................................285 Alejandro Clocchiatti, Type 1a Supernovae and the Discovery of the Cosmic Acceleration ...............303 Philip Mannheim, Intinsically Quantum-Mechanical Gravity and the Cosmological Constant Problem.........................................................................................................................................................331 Thomas Buchert, Towards Physical Cosmology: Geometrical Interpretation of Dark Energy, Dark Matter and Inflation without Fundamental Sources .......................................................................341 David Wiltshire, Gravitational Energy as Dark Energy: Cosmic Structure and Apparent Acceleration....................................................................................................................................................361 Victor Toth, Cosmological Consecuences of Modified Gravity (MOG)...................................................385 Roberto Sussman, Non-spherical Voids: the Best of Alternative to Dark Energy?.................................399
6 Hans Ohanian’s comments on Philip Mannheim’s lecture, made available after the Conference. Philip Mannheim was invited to express his comments on these comments but preferred not to do so 7 Lectures given on November 19th , 2010

9
Preface
The International Conference on Two Cosmological Models was held at Universidad Iberoamericana in Mexico City, Mexico, from November 17th to 19th, 2010, as a forum devoted to the study and discussion of two important problems of modern cosmology. Of the more than 50 experts of the traditional ΛCDM model that were invited by the University to assist at the Conference, ﬁve accepted our invitation. Of the 11 representatives of alternative models that were invited, all agreed to participate. Two participants gave an overview of historical and relativistic aspects of the problem, without aligning with any model in particular. A total of 18 speakers from Brazil, Canada, Chile, France, Germany, India, Mexico, the Netherlands, New Zealand, and the USA gave a total of 20 lectures and participated in an open discussion on the following two topics: 1) The concept of dark matter as a possible explanation of the rotation velocity of galaxies and galaxy clusters in the context of Newtonian dynamics; and the alternative explanation through Einstein’s general relativity, without dark matter. 2) The concept of dark energy as a possible explanation of the apparent acceleration of the expansion of the universe; and the alternative explanation through Einstein’s gravitational theory, without dark energy.
The overall impact of the event was more than satisfactory. These Proceedings contain the lectures on the topics covered in the International Conference on Two Cosmological Models, except for two of them, who could not send us the written version of their lecture. One participant did not send us the written version of his lecture, but gave us permission to transcribe its verbal version. After the Conference, some participants made their lectures available in ArXiv, adding some references to more recent essays, published after the Conference, so that these Proceedings provide the reader with an update of the most current research on these very transcendental topics.
We would like to thank everyone who contributed to the success of the International Conference on Two Cosmological Models. Very special thanks are due to the invited speakers who addressed a very interesting and high quality set of talks and shared their deep knowledge and time with the participants.
We grately acknowledge Dr. John Auping-Birch and all the staﬀ of Universidad Iberoamericana for the warm hospitality, which was extended to all the participants. We specially thank Dr. Jos´e Morales-Orozco, Rector of Universidad Iberoamericana, Mexico City for sponsoring this international endeavor. We hope that these Proceedings will serve to foster the impressive growth of high precision cosmology and the discussion on the diﬀerent theoretical interpretations of its ﬁndings and, additionally, reinforce the existing ties between the Mexican researchers and scientists from all over the world.
Dr. Alfredo Sandoval Villalbazo Director of the Physics and Mathematics Department Universidad Iberoamericana Mexico, July 2012
1

PART I SPIRAL GALAXY ROTATIONAL VELOCITY

13
GRAVITATION AND SPACETIME: EINSTEIN’S CONTRIBUTION
Hans C. Ohanian Department of Physics, University of Vermont, Burlington, VT 05405-0125, USA
hohanian@uvm.edu
Abstract As I argued in Einstein’s Mistakes [1], most of Einstein’s great discoveries rest on conceptual mistakes which he used as stepping stones toward a ﬁnal, true, result. This lecture is a summary of the various mistakes that paved Einstein’s progress toward his theories of special and general relativity. It also includes a detailed discussion of an extra, previously unrecognized, mistake in the application of the equivalence principle, namely, that the gravitational redshift cannot be derived from this principle by Einstein’s 1911 argument, because the equivalence principle contains a contradiction that renders it invalid when applied in ﬂat spacetime.
1 Introduction
I want to begin this lecture by making a wish: I earnestly and passionately wish that the participants of this conference will make very many big mistakes. My wish is not malicious, because I believe it is by making great mistakes that we make great discoveries. As James Joyce said, “Errors are the portals of discovery.” And I am going to illustrate this maxim by showing you how Einstein’s great and wonderful mistakes led him to his great and wonderful discoveries in special and general relativity.
Although the modern view of space and time was not exclusively the work of Einstein, he made the most fundamental and most profound contributions, and for many years he was the dominant ﬁgure in relativistic physics, as well as the dominant ﬁgure in all of physics. Einstein became a celebrity, adored by the public, and even today, ﬁfty ﬁve years after his death, his celebrity status survives. If you google Einstein you get 155 million hits, which is only slightly below the number for Jesus.
Bernard Shaw compared Einstein to the great conquerors in history and called him a maker of universes. He said “Ptolemy made a universe, which lasted 1400 years. Newton, also, made a universe, which lasted 300 years. Einstein has made a universe, and I can’t tell you how long it will last” [2] .We are now in the 105th year of Einstein’s spacetime universe, and so far all is well. I don’t regard the theories that will be presented at this conference as an attempt to overthrow Einstein’s view of spacetime—these theories merely adjust and reﬁne Einstein’s work.
The only serious attempt at overthrowing Einstein’s spacetime is found in string theory, but so far, this has been a chaotic endeavor, without any sharply deﬁned target or any clearcut success. String theorists claim they will someday be able to predict everything, but so far they have predicted next to nothing. Last week, in Physics Today, I ﬁnally read a prediction made by a string theorist who had investigated neutrinos, and who announced “We showed that in no case could the theory generate light but not massless neutrinos. That work presents a clear
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example of a test of string theory” [3] . What this sentence means is a mystery to me; I have read it forward and backward and sideways, and I still am not sure what it means. As best I can tell, this string theorist seems to announce that his theory can generate only neutrinos of mass zero, which, of course, disagrees with the observational fact that neutrinos have mass. And he regards this as progress?
Because Einstein played such a preeminent role in relativity, every development in relativity has tended to be credited to him. The physicist and historian Jagdish Mehra thought this was a result of “the sociology of science, the question of the cat and the cream. Einstein was the big cat of relativity, and the whole saucer of its cream belonged to him by right and by legend, or so most people assume!” [4].
But legends are often wrong. Not all of the contributions to relativity came from Einstein, and even those contributions that originated from Einstein were often in need of corrections, improvements, and emendations. In this lecture, I will dissect the mistakes that Einstein made in the seminal papers that led to the development of special relativity and general relativity in 1905 and 1911-1916, and I will show how his great and surprising mistakes led to his great and surprising discoveries.
2 Special relativity
The groundwork for special relativity had already been laid by Einstein’s predecessors, especially Hendrik Lorentz and Henri Poincare. Einstein did not admit to that in his ﬁrst paper on relativity, but he admitted it later, saying that by 1905 relativity “was in the air” [5] . Einstein’s 1905 paper contains several fundamental contributions. Not all of them were new; his statement of the principle of relativity had been anticipated by Poincare and his coordinate transformations were a rederivation of the Lorentz transformations obtained by Lorentz a year earlier, which Einstein had not noticed. And some of the contributions in the 1905 paper involved serious mistakes. Despite, and perhaps because of, these mistakes, this paper launched the theory of relativity by laying down a general program for how to implement the principle of relativity for all laws of physics by means of the Lorentz transformations.
Einstein begins his 1905 paper with a discussion of synchronization of clocks. This discussion is phrased in deceptively simple language, although it deals with a profound physical and philosophical question. He asks, What does it mean to say “that a train arrives here at 7 o’clock?” And he answers that it means that “the pointing of the small hand of his watch to 7 and the arrival of the train are simultaneous events” [6] . That much is simple. But things get more complicated when we want to synchronize clocks or events at diﬀerent locations.To achieve synchronization at distant locations, say, Ciudad de M´exico and M´erida, we need some special synchronization procedure, and Einstein decided to adopt a procedure of sending light signals back and forth between the two locations. If the light signal leaves here at noon, and comes back in 6 milliseconds, then it must have reached M´erida at noon plus 3 milliseconds, which tells our colleagues at the Universidad de Yucat´an how to synchronize their clock with ours.
Einstein became obsessed with this synchronization procedure, and he believed it was the solution to the puzzle of the invariance of the speed of light, a problem he had wondered about since his teenage years. Due to the emphasis he gave to this procedure in his 1905 paper, it became known as the “Einstein procedure” for synchronization. But it did not originate with Einstein. It actually was a procedure that had been adopted in the 1850s, when the ﬁrst long-distance telegraph lines were laid in the US, and astronomers decided to use back-and-forth telegraph signals to synchronize distant clocks, for use in accurate determinations of geographical longitude. Throughout the second half of the 19th century, this method was widely used for transcontinental
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geographical surveys, such as the surveys of India and of Russia; and, when the ﬁrst transoceanic telegraph lines were laid, it was also used for intercontinental longitude determinations. In the ﬁrst years of the 20th century, the method was used for a new longitude determination of the Paris Observatory relative to Greenwich, and maybe that was where Einstein heard about this method and decided to imitate it.
Einstein argues that if we adopt this synchronization procedure for the clocks at the starting and ending points of a racetrack, it becomes meaningless to measure the separate one-way speeds of light forward and backward along the racetrack, because such measurements would be logically circular. Einstein therefore claims that the equality of one-way speeds of light is a stipulation, beyond the reach of actual experiment.
His logic is impeccable, but his physics is atrocious. Einstein forgets that there might be other ways to synchronize clocks, such as transport of clocks from one place to the other. Before telegraphy became available, such clock transport was widely used by astronomers in the 19 century for determinations of geographical longitude. To synchronize clocks at diﬀerent stations, astronomers transported chronometers between the stations, often using several chronometers, to improve the accuracy by averaging. For instance, to determine the longitude diﬀerence between Greenwich and Valentia (on the west coast of Ireland), George Airy, then the Astronomer Royal, transported 30 chronometers from Greenwich to Valentia, and then he repeated this times, for an even better average [7].
And Einstein forgets that there might be other ways to measure the one-way speed of light without synchronization of clocks. As far back as the17th and 18th centuries, the Danish astronomer Olaf Roemer and the English astronomer James Bradley measured the one-way speed of light without making use of any clock synchronization. Roemer (1676) exploited the time delay in the observed eclipses of the satellites of Jupiter, and Bradley (1727) exploited the aberration of starlight. Both methods hinge, in essence, on comparing the speed of light with the speed of the Earth in its orbit.
Why or how Einstein overlooked these two well-known historical determinations of the speed of light is a puzzle. Roemer’s and Bradley’s determinations of the speed of light were well known in the 19th century, and they were even discussed in introductory physics textbooks [8]. Maybe Einstein decided to dismiss these astronomical methods out of hand because, in practice, they could not achieve the high precision attained by terrestrial methods, ﬁrst by Armand Fizeau in 1849 and then by Abraham Michelson in a series of measurements in the 1880s and 1890s. Michelson continued to improve these measurements, and for his work he came to be called the “master of light” (in Spanish, “el sen˜or de la luz,” which rather sounds like a religious title). But in his paper Einstein was discussing questions of principle, not questions of practice, and the practical limits of the precision of synchronization were not an issue.
Maybe Einstein deleted Roemer and Bradley from his mind because they were an inconvenient truth, awkward to ﬁt into his own way of thinking about the speed of light and the synchronization problem. During the months of intense, feverish thinking that preceded the writing of the relativity paper Einstein was in the grip of a mystical, intuitive, and irrational inspiration. The historian Peter Galison described how Einstein, during an afternoon walk in the hills around Berne, suddenly arrived at the idea that the solution of the problem of a universal speed of light rested in the synchronization [9]. And the biographer Abraham Pais thought that perhaps this intense inspirational experience “was so overwhelming that it seared his mind and partially blocked out reﬂections and information that had been with him earlier” [10].
In the absence of Roemer and Bradley, Einstein drew the mistaken conclusion that the oneway speed of light must be established by stipulation, that is, by ﬁddling with the synchronization of clocks in such a way as to produce equal one-way speeds in opposite directions. Instead of promulgating the universal, constant value of the one-way speed of light as a stipulation, or a
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dogma, Einstein should have treated it as simple postulate, that is, a proposed law or principle of physics to be tested and conﬁrmed by experiment, which is what he did, quite correctly with the round-trip speed of light.
Einstein’s stipulation was a mistake, but it was a fruitful mistake, because it permitted him to rush forward with the development of relativity and bypass the troublesome question of how to obtain conclusive experimental conﬁrmation of the universal value of the one-way speed of light.
Here we see a crucial diﬀerence between Newton’s and Einstein’s approach to physics. Newton declared “I make no hypotheses.” Whether that is quite true of all his work is debatable, but at least such was his announced intention. In contrast, Einstein made hypotheses whenever he could get away with it, and sometimes he even made hypotheses that were preempted and or even contradicted by known experimental and observational facts. He was a dogmatist, and he had the arrogance and stubbornness that goes with that. We see an example of this in his stipulation about the one-way speed of light and his cavalier rejection of Roemer’s and Bradley’s methods. We see another example in Einstein’s hypothesis of quanta of light, which he based on Wien’s law for the blackbody spectrum, in willful and deliberate deﬁance of Planck’s law. By 1905, Planck’s law was experimentally well established and known to be correct, and Einstein must have been aware that his hypothesis of thermal radiation as a gas of quanta, treated by classical statistics, was in conﬂict with Planck’s law. He never mentioned this inconvenient truth, and he insisted on a hypothesis that—until the introduction of quantum statistics twenty years later—was in contradiction with the experimental evidence. And we see the most glaring example of Einstein’s dogmatism in his rejection of the probabilistic interpretation of quantum mechanics and his famous declaration that “God does not play dice.”
Newton believed, or pretended to believe, in a “bottom-up” approach to physics, based on experimental and observational facts from which, by a process of generalization, or induction, the laws of physics could be extracted. Einstein believed in a “top-down” approach to physics, by inspirational formulation of hypotheses from which consequences could be derived. As John Auping points out [11], Einstein’s method is deductive, not inductive, and this is clearly revealed in a newspaper article Einstein wrote in 1919:
The truly great advances in our understanding of nature originated in a way almost diametrically opposed to induction. The intuitive grasp of the essentials of a large ensemble of facts leads the researcher to the formulation of one or more hypothetical fundamental laws. From the fundamental law (system of axioms) he draws his conclusions as completely as possible in a purely logical-deductive manner [12].
Einstein believed he had the intuitive insight to perceive the truth, and he, and only he, knew what hypotheses to make. He had the conﬁdence of a visionary and of a fanatic, and he did not hesitate to go against experimental and observational facts when these did not ﬁt his theories.
In the second section of his paper, Einstein exploits the constant one-way speed of light as the basis for a derivation of the transformation of space and time coordinates between two inertial reference frames. His treatment of this derivation is correct, but astoundingly clumsy. Even in abbreviated form presented in the printed publication, the calculations go on and on for ﬁve and a half tedious pages; and, as shown in the recent book on Kinematics by Alberto Mart´ınez [13], if he had spelled out all the details of his calculations, it would have taken him thirty pages of print. The trouble was that Einstein failed to see that there is a simple physical argument to establish that lengths transverse to the velocity will not expand or contract: if two meter sticks in relative motion are oriented transversely, then, by symmetry of the physical situation, a contraction of the second relative to the ﬁrst requires an equal contraction of the ﬁrst relative to the second, which is contradictory, and rules out any contraction (or expansion) [14]. Today, this argument is familiar to all students of relativity. If the transverse lengths are unchanged,
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Figure 1: Minkowsky’s derivation of the Lorentz transformation
they can be ignored, and it then suﬃces to examine the transformations of the time coordinate and the longitudinal length coordinate, which makes the calculation much simpler.
How much simpler can be seen by taking a look at the derivation of the transformation equations presented by Hermann Minkowski in a lecture to the Congress of German Natural Scientists and Physicians in 1908. Minkowski had been one of Einstein’s math professors at the Polytechnic in Zurich, and he had a rather poor opinion of Einstein—he called him a “lazy dog.” When Einstein published his famous ﬁve papers in 1905, Minkowski was astounded; he said “I really would not have believed him capable of it” [15]. Minkowski then took the invariance of the speed of light from Einstein’s relativity paper and produced a new, graphical derivation of the transformation equations. It consists of one simple diagram (Fig. 1), from which, in half a minute, you can deduce the transformation equations. In fact the derivation from this diagram is so simple that Minkowski did not even bother to spell it out in the published version of his lecture. I imagine that at the Congress he merely drew this diagram on a blackboard and said one or two dozen words about it. If you want to try to derive the transformation equations, here are the two dozen words you need to know: First draw the line OA’ along the t’ axis, then the line A’B’ tangent to the hyperbola t2 − x2 = 1 at A’. The rest of the diagram is self-evident.
The diﬀerence between Einstein’s ﬁve and a half messy pages of calculations and this simple diagram reveals the diﬀerence between a master mathematician such as Minkowski and his pupil. But, as Einsteins achievements show, you don’t need to be a brilliant mathematician to become a great physicist. In fact, Einstein had little mathematical talent, and he had no real interest in mathematics. He thought in images, not in words or formulas. In an interview with a psychologist, he said: “I rarely think in words at all. A thought comes, and I may express it in words afterwards” [16]. In his autobiography, he excused his mathematical deﬁciencies saying, “I saw that mathematics was split up into many specialties, each of which could absorb the short lifespan granted to us.Thus I saw myself in the position of Buridan’s ass, which was unable to decide on a particular bundle of hay” [17]. What he does not mention in this autobiography is that throughout his life he relied heavily on friends and on assistants to do his mathematics for
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him. At the Polytechnic and for his early investigations of general relativity, he relied on his friend Marcel Grossman, who taught him the basics of Riemannian geometry. Later, when his career blossomed with his appointment in Berlin, he hired a personal mathematical assistant, and in later years he always had an assistant available; these came and went, a total of ten or so altogether, and they did all the calculations that Einstein found too tedious.
Besides improving on Einstein’s mathematical presentation, Minkowski made a crucial contribution to our understanding of spacetime by recognizing that Einstein’s theory of relativity implies a complete uniﬁcation of space and time. The Lorentz transformation equations show that what is time and what is space depends on the reference frame, and that transformations of reference frame mix space and time. As Minkowski expressed it, “space by itself, and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality” [18] .
This uniﬁcation of space and time also caught the attention of philosophers, and of the public, and of poets and writers who delighted to insert esoteric references to uniﬁed space and time in their writings. Carlos Fuentes included a well-tuned phrase of this kind in one of his books: “Habr´a solo la unidad total, olvidada, sin nombre y sin hombre que la nombre: fundidos espacio y tiempo, materia y energ´ıa” [19] (I can try to translate this, but it doesn’t quite have the same ring in English, and it loses the charming musicality that Fuentes gave it: “There will be only total unity, forgotten, without a name and without anyone to give it a name: space and time are merged, and so are matter and energy”) .
Minkowski deserves credit for his perception of the uniﬁcation of space and time as well as for the uniﬁcation of electric and magnetic ﬁelds, for which he invented the 16-component electromagnetic ﬁeld tensor. But we must not forget that Einstein laid the groundwork for that, and without Einstein’s 1905 paper, Minkowski would never have arrived at a uniﬁcation of space and time. Half in serious, half in jest, Mi√nkowsky also contributed a zany formula relating the units of space and time, 3 × 105 km = −1 sec . This incorporates the speed of light, 300000 km per sec, and it incorporates the notion that, in a purely formal way, time can be treated as a mathematically imaginary fourth dimension of space. Minkowski called this his mystic formula. He would undoubtedly have made many more valuable contributions to special and to general relativity, but he died prematurely, in 1909.
Einstein was not pleased with Minkowski’s mathematical approach. He called Minkowski’s tensor formalism “unnecessary erudition,” and said that “ever since the mathematicians have thrown themselves on the theory of relativity I can’t understand it any more” [20]. At ﬁrst he remained stubbornly opposed to the use of tensor formalism in the treatment of relativity, and he did not adopt the use of tensors until ﬁve or six years later, when he began to work on general relativity.
From his transformation formulas for the coordinates of inertial reference frames, Einstein deduced the time dilation and the length contraction: a moving clock runs slow, and a moving body, such as a meter stick, is contracted. He suggested that this implies that a clock on the equator of the Earth, moving with the Earth’s rotation, will tick slower than a clock at the pole. This was an unfortunate choice of example, because the gravitational time dilation of the clock at the pole, which Einstein was to discover a few years later, actually compensates for the time dilation of the clock located at the equator. But Einstein must be given credit for taking the time dilation of moving clocks seriously, whereas Lorentz, who had deduced the same transformation laws a year earlier, attached no real physical signiﬁcance to the change of clock rates implied by his transformation equations.
In his deduction of time dilation and length contraction Einstein commits a serious sin of omission: he never gives us any physical explanation of time dilation and length contraction. He treats these as purely abstract, mathematical consequences of the Lorentz transformations. The
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latter, in turn, are consequences of the requirement of a constant, universal speed of light in all reference frames. This creates the impression of a teleological approach to physics: Are we to believe that clocks run slow and meter sticks contract because they want to keep the speed of light constant?
For nonphysicists, these weird, counterintuitive eﬀects of relativity proved an unsurmountable obstacle, because Einstein asks us to believe these eﬀects without giving us any mechanical, intuitive explanation. Even physicists supportive of relativity, such as Arnold Sommerfeld, found this a bit much too much to swallow. After reading Einstein’s derivations, Sommerfeld complained to Lorentz: “As ingenious as they are, it seems to me that there is something almost unhealthy in their nonconstructive and unvisualizable dogmatics. An Englishman would hardly have put forth such a theory. . . I hope you will be able to breathe some life into this ingenious conceptual framework” [21].
The fact is that both time dilation and length contraction have simple physical explanations in terms of the laws of mechanics and electrodynamics (although we really need quantum mechanics for the analysis of the oscillations and sizes of atoms). It is not at all diﬃcult to prove from the laws of relativistic dynamics that clocks, such as atomic clocks, slow down, and solid bodies, such as crystal lattices, contract when moving at high speed [22].
The other surprise in the Lorentz transformations is the relativity of synchronization—but even that has a simple explanation in term of clock transport [23]. Clocks in a moving reference frame are desynchronized relative to our, stationary, reference frame, because slow clock transport in the moving reference frame is not slow in our reference frame, and the extra time dilation accumulated by the transported clock leads to a breakdown of synchronization from the viewpoint of our reference frame. A simple calculation conﬁrms that the transport process accumulates the correct time delay: Suppose that one clock remains at the origin of the moving reference frame and therefore moves at constant speed V , and the other clock is slowly transported with speed V + δV until it is separated from the origin by a displacement ∆x (measured in our reference frame). If this process takes a time ∆t, the two clocks will display a time diﬀerence

τ2 − τ1 = 1 − (V + δV )2/c2∆t − 1 − V 2/c2∆t

−V (δV ∆t)/c2 −V ∆x/c2

=

1 − V 2/c2

1 − V 2/c2

(1)

Thus, after the transport is completed, at any given instant of t time, the transported clock displays a time shift ∆t = −(V ∆x/c2)/ 1 − V 2/c2 relative to the clock at the origin, which agrees exactly with the time shift calculated from the Lorentz transformation equation for t .
What these physical explanations of the length contraction and time dilation tell us is that there is nothing magical or teleological about the Lorentz transformations. The Minkowski metric indicates a new geometry for spacetime, but this geometry is not produced by magic, but by the laws of physics. It is not some kind of abstract construct contrived to keep the speed of light constant, as Einstein would have us believe. Poincare once confronted Einstein at one of the Solvay Conferences and demanded to know what mechanics Einstein was using to reach his conclusions about time dilation and length contraction. Einstein dismissed the question; he said “no mechanics” [24]. Poincar´e couldn’t believe that a physicist would say something like that, and he didn’t bother to reply.
After dealing with the time dilation and the length contraction, Einstein’s 1905 paper deals with the transformation formulas for electric and magnetic ﬁelds, transformation formulas for energy in e.m. waves, aberration, and Doppler shift. The results for electric and magnetic ﬁelds had already been obtained by Lorentz, a year earlier; the other results were new.
And then, in the ﬁnal section of his paper, Einstein formulates new, relativistic equations of motion for a moving charge—called an electron—being accelerated by electric and magnetic

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ﬁelds. He correctly calculates the so-called longitudinal mass which indicates the eﬀective inertial resistance for acceleration in a direction parallel to the velocity. But he makes a bad mistake in the “transverse mass,” for acceleration in a direction perpendicular to the velocity, for which he obtains the mistaken formula

m

T ransverse mass = 1 − v2/c2

(2)

Not all of Einstein’s mistakes were fruitful, and this was simply a silly mistake, of no redeeming artistic value. In a footnote added years later to a reprint of his paper, Einstein sheepishly admitted that this mistake was not “advantageous.” What makes this mistake all the more surprising is that the expressions for transverse and longitudinal mass were well known in the physics literature. Lorentz had used them in his paper on relativistic electrodynamics a year earlier, where he expressed the mass of the electron in terms of its electrostatic self energy and stated the correct velocity dependence for the transverse mass [25],

e2

1

m2 = 6πc2R 1 − v2/c2

(3)

But Einstein had failed to notice Lorentz’s paper. Einstein’s mistake was immediately spotted by Max Planck, who read Einstein’s paper soon after it was published (maybe even before it was published, because Planck was editor of Annalen der Physik , where had Einstein sent his paper for publication). Planck thought Einstein’s paper very impressive, and he sympathized with the top-down approach, perhaps because in his own work on quantization of black-body radiation he had taken the same top-down approach—he had postulated the quantization of energy and deduced the radiation law, just as Einstein had assumed a universal speed of light and deduced the Lorentz transformations and their consequences. Einstein’s adoption of the speed of light as a universal constant also pleased Planck, who was very interested in universal constants and their role in physics. Planck then proceeded to remodel and correct Einstein’s treatment of dynamics. With an elegant Lagrangian formulation of relativistic mechanics, Planck recalculated Einstein’s transverse mass, and he found and published the correct formula for the transverse mass [26]. Unfortunately, today hardly anybody remembers this contribution of Planck to relativity—it was the ﬁrst of many papers on relativity by authors other than Einstein to appear after 1905.

3 EQUIVALENCE PRINCIPLE AND REDSHIFT
I now will turn to general relativity and dissect the main mistakes in Einstein’s development of general relativity. Like the development of special relativity, Einstein’s development of general relativity rests on several great mistakes. These mistakes served him as stepping stones to his ﬁnal wonderful discovery of curved spacetime.
For Einstein, the key to general relativity was the principle of equivalence of acceleration and gravitation which he discovered in 1907. As he later described it in a lecture at Kyoto University: “I was sitting in my chair at the patent oﬃce at Berne. Suddenly I had an idea: when a person is in free fall, he does not feel his own weight. I was amazed. This simple thought experiment made a deep impression on me. It led me to a theory of gravitation” [27]. His sister Maja claimed that what triggered Einstein’s sudden idea was the fatal accident of a roofer who slipped from one of the roofs near Einstein’s apartment. That story sounds too good to be true, but the roofs of Berne are old and slippery, and unfortunate accidents are plausible.

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Einstein saw in this equivalence principle a relativity of acceleration. He thought that physicists placed in an accelerated reference frame cannot be sure whether the eﬀects they feel and observe are due to the pseudoforces associated with the acceleration or to the presence of a constant gravitational ﬁeld. As he described it in one of his lectures, imagine that two physicists wake up from a drugged sleep and ﬁnd themselves in a closed box, in which they observe that bodies released in midair fall to the ﬂoor with a universal acceleration g. What can the physicists conclude from this? One of them concludes that the box sits on the surface of a planet, whose gravitational attraction produces the acceleration g. The other concludes that the box is nowhere near any planet, and is instead being accelerated by some external propulsion mechanism. And Einstein asks, “Is there any criterion by which the two physicists can decide who is right?” and he answers, “We know of no such criterion” [28].
With the wisdom of a hundred years of hindsight, I can give you several such criteria. One obvious violation of the equivalence of acceleration and gravitation is found in the tidal forces generated by gravitational ﬁelds, which are absent in a (linearly) accelerated reference frame. These tidal forces arise from gradients in the gravitational ﬁeld. In Newtonian gravitation these gradients can be zero only under exceptional circumstances; in Einstein’s theory of general relativity, they can never be zero (they correspond to the components of the Riemann tensor, some of which must be diﬀerent from zero if the spacetime is curved). It might be argued that such tidal forces become small and insigniﬁcant when the region accessible to the experimenter is made extremely small, but even in very a small region, tidal eﬀects remain observable with sensitive equipment. For example, the GOCE satellite of the European Space Agency, in orbit and in free fall, makes high-precision measurements of the tidal gravitational ﬁeld of the Earth with a diﬀerential-accelerometer only a few cm across. In fact, the GOCE satellite measures components of the Riemann tensor (R0k0l) that is, it measures the curvature of spacetime.
Another example is provided by the Stanford Gravity Probe B experiment completed in 20042006, which used gyroscopes in a satellite orbiting the Earth to detect the precession caused by a general-relativistic coupling between the spins of the gyroscopes and the Earth. In this experiment, Francis Everitt and his fellow experimenters had to go to extraordinary lengths to avoid interference from tidal forces. They had to manufacture their gyroscopes as perfectly round spheres, to within ±10−6cm . Even a small deviation from roundness would have permitted the tidal forces to exert torques on the gyroscopes and generate a much larger precession than what the experimenters were looking for, so this experiment would have detected tidal forces rather than spin-spin coupling. Note that the tidal-torque precession is independent of the size of the gyroscopes. We cannot eliminate the tidal-torque precession by using smaller gyroscopes—other things being equal, the precession rate depends on the shape of the gyro, but not on the size.
Besides the obvious troubles with tidal eﬀects, the equivalence principle has a fundamental inability to deal correctly with the propagation of light—the principle works, more or less, for slow-moving particles, but it mishandles fast-moving particles and light, unless we adopt curved spacetime. For instance, we might try to calculate the deﬂection of light in a gravitational ﬁeld by beginning with the deﬂection that occurs in an accelerating elevator. Obviously, in such an elevator, a ray of light moving from one side to the other will deﬂect downward by some amount relative to the elevator. But, quantitatively, the calculated amount is wrong—the deﬂection calculated in the elevator is only half as large as the actual result in a gravitational ﬁeld calculated from Einstein’s general relativity (this is the infamous factor of two by which Einstein’s ﬁrst deﬂection calculations diﬀered from his ﬁnal result). Thus, the equivalence principle fails by a factor of two, and not because of any tidal eﬀect.
The equivalence principle is capricious and unreliable. The equivalence principle works when it works and doesn’t when it doesn’t—you have to apply it with caution. Some violations of the principle of equivalence were already mentioned by Arthur Eddington in 1923 in his The
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Figure 2: An elevator accelerating upward with two clocks for testing the frequency shift of a lightwave traveling a distance h from ﬂoor to ceiling.
Mathematical Theory of Relativity, the ﬁrst textbook on general relativity. Eddington gave a dismissive opinion about the equivalence principle:
”It is essentially a hypothesis to be tested by experiment as opportunity oﬀers. Moreover, it is to be regarded as a suggestion, rather than a dogma admitting no exceptions. Clearly there must be some phenomena. . . which discriminate between a ﬂat world and a curved world; otherwise we could have no knowledge of world curvature. For these the Principle of Equivalence breaks down. . . The Principle of Equivalence oﬀers a suggestion for trial, which may be expected to succeed sometimes, and fail sometimes” [29].
Unaware of these troubles, in 1911 Einstein applied the equivalence principle to a Gedankenexperiment involving a perfectly uniform gravitational ﬁeld, in which tidal eﬀects are not an issue. [30] He considered a light source located on the surface of the Earth or some other gravitating body, and asked what happens to the frequency of a light wave emitted upward. To ﬁnd out, he replaced the gravitational ﬁeld by a reference frame accelerating upward and examined the propagation of light in this reference frame. This Gedankenexperiment is known to all students of relativity, and the accelerating reference frame is usually visualized as an elevator (see Fig. 2). While the light wave travels from the ﬂoor to the ceiling, the upward acceleration of the elevator increases the speed of the clock at the ceiling relative to the initial speed of the clock at the ﬂoor by about gh/c where g the acceleration of the elevator (equal to the acceleration of gravity the elevator is intended to mimic), h its height, and c the speed of the light wave. Upon arrival at the ceiling, the light wave will then suﬀer a Doppler shift ∆ν = −(gh/c2)ν . Relying on the equivalence principle, Einstein therefore concluded that a light wave propagating upward in a gravitational ﬁeld should suﬀer the same frequency shift. This is the gravitational redshift, also called the gravitational time dilation, because we interpret it as a slowing of clocks in a gravitational ﬁeld. As a corollary of this gravitational time dilation, Einstein concluded that the speed of light is lower in a gravitational ﬁeld and that light rays suﬀer a deﬂection when passing near a mass.
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There is something puzzling about this deduction of the gravitational redshift. The calculation relies on what seems to be no more than simple Newtonian physics, that is, the Doppler-shift formula and the equivalence of gravitational and inertial eﬀects, both of which seem innocuous and trivial. In Newtonian physics, it is immediately evident that a constant gravitational ﬁeld can be replaced by an accelerated reference frame; the gravitational force in the former is identical to the acceleration pseudoforce in the latter. And yet something is ﬁshy about Einstein’s calculation, because, according to Newtonian physics, in a static gravitational ﬁeld, a propagating light wave cannot acquire a frequency shift, not even if the speed of the wave is somehow altered by the gravitational potential. An alteration of speed would mean that the gravitational ﬁeld behaves like an optical medium with a position-dependent index of refraction—in such a medium the wavelength of a light wave changes, but the frequency remains constant. This is why we have to interpret the gravitational redshift as a time dilation of the clocks, rather than as an inherent frequency change of the propagating wave. But nowhere in his calculation does Einstein seem to introduce anything about any time dilation, so why does a time dilation of clocks emerge in the end, as if by a trick of magic? Where does Einstein trick us?
The answer to this puzzle is that Einstein actually introduced the special-relativistic time dilation implicitly, by a sleight of hand. He used a mixture of Newtonian and relativistic physics—he took the equivalence principle from Newtonian physics and he took the postulate of a constant, universal speed of light from relativistic physics. He assumed that each wave pulse emitted from the ﬂoor of the elevator has the same speed as the preceding pulse, whereas in Newtonian physics we would have to assume that the wave is carried along by the medium in the elevator, and that the wave speed is constant relative to the ﬂoor, but increases relative to the inertial reference frame within which the elevator accelerates. The constant speed of light hinges on the special-relativistic time dilation of clocks or, more precisely, it hinges on the relativity of synchronization, which is a consequence of this time dilation (as shown by the clock-transport argument in part I of this lecture).
We can understand the crucial role of relativistic physics in the derivation of the redshift more clearly if we replace the light wave by a sound wave or by an evenly-spaced sequence of BB pellets ﬁred from a BB gun. Each pulse of sound or each BB pellet has the same speed relative to the elevator, but between emission of one pulse and the next, the elevator increases its speed, and therefore, relative to the inertial reference frame within which the elevator accelerates, each pulse of sound or each BB has a slightly higher speed than the preceding one. By taking into account the Newtonian addition law for velocities, we then readily ﬁnd that the frequency shift is reduced to zero—there is no frequency shift at all [31]. And, by the equivalence principle, we would then conclude that there is no frequency shift for a sequence of sound or pellet signals in a gravitational ﬁeld.
This makes it clear that to obtain a redshift we need to consider relativistic corrections to the Newtonian calculation (if we need more prompting, the factor 1/c2 of in Einstein’s redshift formula actually gives us a strong clue that relativistic eﬀects play a role). And, indeed, if instead of the Newtonian addition law for velocity, we use the special-relativistic velocity-addition law, then the Gedankenexperiment with sound pulses or BB pellets yields the expected redshift, exactly as in Einstein’s calculation with light waves. For the sake of simplicity, let’s assume that the speed u of the sound pulses or BB pellets is much larger than the increment of speed of the elevator during the travel time of the signal. The ﬁrst pulse is launched at time t = 0 and reaches the ceiling at time t1 h/u [32]. The next pulse is launched at time t = ∆τ1 and reaches the ceiling at time t2 = ∆τ1 + h/(u − g∆τ1) where is the speed of the elevator at launch and is the pulse speed relative to our original reference frame, according to the relativistic combination law for velocities, u = (u + g∆τ1)/(1 + ug∆τ1/c2) [33]. Ignoring terms of order g2, we then ﬁnd
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h gh

t2 = ∆τ1 + u + c2 ∆τ1

(4)

The time diﬀerence between the arrivals times of the two pulses is therefore

gh

∆τ2 = t2 − t1 = (1 + c2 )∆τ1

(5)

This redshift formula for the pulse periods agrees with Einstein’s redshift formula for the fre-

quencies.

The preceding calculation establishes an important point about the derivation of the grav-

itational redshift: it arises from an inconsistent mix of Newtonian physics for the equivalence

principle and the relativistic addition law for velocity, applied either to light or to sound pulses

or pellets, or any other signaling method. The gravitational redshift cannot be derived from

purely Newtonian physics [34].

Nor can the gravitational redshift be derived from purely relativistic physics. Although

relativistic physics yields the desired expression for the redshift in the accelerating elevator,

this leaves a gap in the derivation—in relativistic physics the equivalence principle cannot be

deduced from the laws of mechanics, as it can in Newtonian physics. The trouble is that the

relativistic generalization of the gravitational force always includes a dependence on the particle

velocity, which leads to diﬀerent accelerations for particles of diﬀerent velocities. For instance,

Lev Okun found that the simplest relativistic generalization of the gravitational force has a

strong dependence on the direction of the velocity [35]. For an ultrarelativistic particle moving

in a tangential direction, the force has a component in the radial direction and also a component

in the tangential direction, whereas for a low-speed particle the force is, of course, purely radial.

This discrepancy indicates that the equivalence principle is not valid.

More generally, it is easy to prove that the equivalence principle can never be valid in a

relativistic theory of gravitation in ﬂat spacetime. The proof is by contradiction. Suppose

the equivalence principle is valid, so all particles have the same gravitational acceleration at

any given point, and light propagates with its standard speed relative to these freely falling

particles. Then the redshift found in an accelerated reference frame requires a corresponding

redshift in a gravitational ﬁeld, which tells us that clocks in a gravitational ﬁeld run slow.

Furthermore, the equivalence principle tells us that the local speed of light has its standard value c0 = 2.99... × 1010 cm/s when measured by the local, slow, clocks. Following Einstein, we can then conclude that the speed of light must be lower than c0 when measured by “normal” clocks, that is, the speed of light must decrease in a gravitational ﬁeld. This leads to a contradiction

when we consider an ultrarelativistic particle, of initial speed almost equal to c0 falling radially downward in the gravitational ﬁeld of a mass, into regions of stronger and stronger ﬁelds. Since

this particle must obey the decreasing speed limit set by the speed of light, the particle must

decelerate, whereas the equivalence principle demands that the particle must accelerate, like a

slow-moving particle. This logical contradiction proves that the equivalence principle cannot be

valid.

And this raises a troublesome question: Did Einstein fail to notice the logical contradiction

between his equivalence principle and the decreased speed of light? Or did he notice, but preferred

to keep silent? If Einstein’s nondisclosure of the conﬂict between his picture of a gas of quanta

and Planck’s law provides a precedent, is would seem that sometimes he preferred to remain

silent.

General relativity sidesteps this contradiction in the equivalence principle by exploiting curved

spacetime. The distinction between locally measured distances and times in the curved space-

time vs. coordinate distances and times permits us to have a constant local speed of light and

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nevertheless a decreasing coordinate speed—we can have our cake and eat it too. Thus, the equivalence principle (except for tidal forces) is consistent with general relativity and curved spacetime, though not with special relativity and ﬂat spacetime [36] .
Can anything be salvaged from the ruins of Einstein’s derivation of the gravitational redshift via the equivalence principle? We cannot salvage a quantitative, exact result for the redshift, but we can salvage a qualitative, order-of-magnitude result. Einstein was not entitled to assume that the equivalence principle applies to light or to fast-moving particles—he had no evidence for that, and, as shown above, any attempt to stipulate that the equivalence principle is valid for fast-moving particles leads to a contradiction. He was entitled to assume that it applies approximately to slow-moving particles, but, because of the essential role of relativistic corrections in the derivation of the redshift in an accelerating reference frame, he should have anticipated the possible existence of similar small relativistic corrections in the equivalence principle, which might increase or decrease the redshift. Thus, at most, Einstein could legitimately claim that the gravitational redshift is of the order of magnitude ∆ν = −(gh/c2)ν . The sign of the redshift wouldhave been left undetermined by such an order-of-magnitude estimate, so Einstein could not have been sure whether light slows down or speeds up in a gravitational ﬁeld.
The absence of a legitimate derivation of the redshift from the equivalence principle weakens the physical motivation that Einstein sought to give his theory of gravitation. But this would still have left him with enough clues to pursue the development of gravitational theory by the same path he followed from 1911 onward, although he would have had to admit that his calculation of the redshift and his subsequent calculation of the deﬂection of light were only order-of-magnitude estimates. Since his ﬁrst calculation of the deﬂection was actually in error by a factor of two, this would not have made much diﬀerence to the historical developments that led to the ﬁrst, failed, German attempt at a measurement of the light deﬂection in 1912 and, later, to the successful British attempt in 1919.
Of course, the defects in Einstein’s derivation of the redshift do not mean that his formula for the gravitational redshift is wrong—these defects merely mean that the formula cannot be derived by the seductively simple argument proposed by Einstein in 1911, an argument uncritically imitated in just about every textbook on relativistic gravitation, introductory or advanced [38].
The inconsistent mix of Newtonian and relativistic physics that Einstein used in his derivations of the gravitational redshift was one of his great mistakes. Einstein was as obsessive and dogmatic about the equivalence principle as he was about his stipulation for the one-way speed of light, and he failed to see that the equivalence principle is self-evident only when it involves no more than the equality of rates of free fall of small test masses with low speeds, that is, when it reduces merely to the equality of gravitational and inertial masses.
Years ago, the Irish physicist J. L. Synge said “The Principle of Equivalence performed the essential oﬃce of midwife at the birth of general relativity,” and he added, “ I suggest that the midwife be now buried with appropriate honours and the facts of absolute space-time be faced” [39], by which Synge really meant the facts of absolute curved spacetime. I think Synge was right, and it would be best to forget about the equivalence principle, except in a historical context. Late in his life Einstein said about Mach’s principle “Actually, one should no longer speak of Mach’s principle at all” [40]. We can say the same about the equivalence principle.
4 GENERAL RELATIVITY
As used by Einstein in the derivation of the redshift, the equivalence principle was a mistake, but it was his greatest and most wonderful mistake, and it led him to his greatest, most wonderful discovery—curved spacetime. We now understand that the time dilation actually implies that
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spacetime is curved—the time part of space-time is “shorter” in a gravitational ﬁeld than outside of the ﬁeld. We can have a redshift in a static gravitational ﬁeld only if we have a curved spacetime. But when Einstein published his paper on the equivalence principle in 1911, he did not yet understand that. He was aware that his redshift result means that clocks at a deeper gravitational potential run slower, but he vaguely speculated that that this arose, somehow, from a reduced speed of light—he was not yet ready to imagine a curved spacetime. As Emilio Segr´e liked to say, “When the mind is not prepared, the eye does not recognize” [41].
Einstein ﬁnally reached the conclusion that spacetime must be curved in 1912, by a diﬀerent path. In the spring of that year, while still at the University of Prague, he published a paper [42] exploring his idea that the speed of light in a gravitational ﬁeld is slower by an amount depending on the gravitational potential, c = (1 + Φ/c20) × c0. He had deduced this expression from the gravitational time dilation, by assuming that the speed has the standard value c0 = 2.99...×1010 cm/s when measured by the local, slow, clocks and is therefore lower when measured by “normal” clocks (his expression for the speed is actually wrong; the actual dependence of the speed of light on the potential is twice as strong, that is, a factor of 2 is missing). At the end of this paper, in a short appendix added at the last moment in the proofs, he draws attention to a curious fact about the equation of motion of a particle that he had derived from his relationship between the speed of light and the gravitational potential: the equation of motion can be expressed by a principle of extremum action, involving the variable speed of light,

δ

(1 + Φ/c20)c20dt2 − dx2 − dy2 − dz2 = 0

(6)

This equation coincides precisely with the equation for a geodesic in a curved spacetime, with a metric tensor g00 = (1 + Φ/c20) [43]. The last sentence of the appendix Einstein says that he suspects that his equation has a much deeper meaning and that it reveals how the equations of motion are to be constructed in general.
Here Einstein fell just short of recognizing that his equation is a geodesic equation, but this revelation came to him soon thereafter. He remembered that he had seen such equations for the extremum length of geodesics in some lectures on curved spaces he had attended at the Polytechnic. This gave him the missing link, and Einstein suddenly saw the real meaning of gravitation: there are no gravitational forces; there is only a curved spacetime in which particles move on the straightest possible wordlines. As he later reported in his Kyoto lecture: “. . . I suddenly realized that Gauss’s theory of [curved] surfaces holds the key for unlocking this mystery. I suddenly remembered that Gauss’s theory was contained in the geometry course given by Geiser when I was a student. I realized the foundations of geometry have physical signiﬁcance” [44].
And from that time on he relentlessly pursued the goal of a geometrical theory of gravitation. Years later, after his success with general relativity brought him worldwide fame, one of his Zurich friends told him he had found the certainty of Einstein’s convictions about gravitation almost frightening: “Your conﬁdence , the conﬁdence of your thinking, . . . , at the time you when were with us, is for me a tremendous psychological experience. You were so certain, that your certainty had for me an overwhelming eﬀect” [45]. This absolute certainty in Einstein’s convictions is also revealed by the famous comment he made to a student when he received the ﬁrst reports of the observational conﬁrmation of his prediction for the deﬂection of light. He said: “I always knew the theory was right.” The student asked him, What if the measurements had contradicted your theory? And Einstein gave the grand reply: “Then I would have felt sorry for the Dear Lord. My theory is right anyway” [46].
Einstein’s path from his introduction of curved spacetime in 1912 to his ﬁnal theory of general relativity in 1915 was a lengthy and arduous eﬀort. Einstein had little knowledge of the

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mathematics of curved spaces; there were few books from which he could learn this material, and those that were available were written in the usual impenetrable jargon of mathematicians, and with awkward notation. He begged for help from his friend Marcel Grossmann: “Grossmann you must help me or I’ll go crazy” [47]. But even Grossmann was not suﬃciently familiar with Riemannian geometry and did not understand the critical relationship between the diﬀerential identities of the Riemann tensor and the conservation laws in the equations that he and Einstein tried to construct. This led to a series of proposals for theories that all had to be quickly abandoned. As Einstein himself admitted later: “The series of my publications on gravity is a chain of wrong turns” [48]. Planck advised Einstein to give up the attempt, “As an older friend,” he said, “I must advise against it. . . In the ﬁrst place you won’t succeed; and even if you succeed, nobody will believe you” [49].
In the end, in 1915, Einstein succeeded brilliantly. And in the next year, he published a long and careful exposition of his theory in Annalen der Physik in 1916. The paper consists of an introduction that tries to lay the physical foundation for the theory, followed by a careful exposition of the essential aspects of Riemannian geometry, then the new ﬁeld equations for the gravitational ﬁeld, and ﬁnally applications to light deﬂection and the perihelion precession of Mercury.
Oddly, the discussion of the physical foundations of the theory does not mention the 1912 argument that initially led him to the idea of a curved spacetime. And, equally oddly, this argument has been rarely used in later textbooks on general relativity. The exception is an early book of by Tullio Levi-Civita (1923), an Italian mathematician and expert on Riemannian geometry, who gave a simple argument based on that of Einstein, but leading directly from the Newtonian potential to curved spacetime [50]. Levi-Civita simply wrote the extremum principle for a particle moving in a Newtonian gravitational ﬁeld as

δ c2(1 − v2/2c2 + Φ/c2)dt = 0

(7)

where the integrand diﬀers from the usual Lagrangian for motion in a gravitational ﬁeld only by an irrelevant minus sign and factor of c2. Taking advantage of the small magnitude of velocity
and Newtonian potential compared with c, he rewrote this approximately as

δ c2 (1 − v2/2c2 + 2Φ/c2)dt = δ c2 (1 − 2Φ/c2)dt2 − dl2/c2 = 0

(8)

which is essentially the same as Einstein’s extremum principle, and corresponds to motion in a curved spacetime geometry, with a metric tensor whose 00 component depends on the Newtonian potential, and therefore depends on space (this is always a curved spacetime geometry, except when g00 α x2 ).
In his 1916 paper, Einstein does not use the straightforward and conclusive argument of 1912. Instead, he tries to use the equivalence principle. He considers a rotating turntable, like a Merry-go-round, and he claims that the geometry of this turntable is a curved space, because meter sticks laid along the circumference contract by the usual length contraction of special relativity, whereas meter sticks laid along the radius do not. This makes the ratio of the measured circumference larger than the measured radius, that is, circumference > 2πr . But this is simply a misconception: the meter sticks at diﬀerent locations around the circumference are instantaneously in diﬀerent inertial reference frames, and it is evidently incorrect to add lengths measured in such diﬀerent reference frames. The apparent curved geometry is an artifact resulting from a bad choice of measurement procedure. If we use meter sticks at rest, ﬂoating just above the circumference of the turntable, we will of course ﬁnd that the geometry is ﬂat. The transformation from coordinates at rest to coordinates in rotation is merely a coordinate

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transformation, which cannot change the geometry, that is, the curvature tensor is zero, no matter what coordinate system we use to calculate it.
But Einstein, and quite a few of his followers, accepted this Merry-go-round argument and concluded that the centripetal acceleration of the turntable produces a curved geometry, and if so, by the equivalence principle, the gravitational ﬁeld of a mass should also produce a curved geometry. Einstein would have done better to motivate the curved spacetime geometry by the extremum principle that ﬁrst gave him the key to curved spacetime in 1912. Or he could have used the gravitational time dilation that he had extracted from his elevator Gedankenexperiment.
Einstein called his theory “general relativity” for two reasons: he thought that the behavior of physical systems in an accelerated reference frame is indistinguishable from their behavior in an unaccelerated reference frame placed in a gravitational ﬁeld; and he thought that by writing the equations of physics in a form that was valid in all conceivable coordinate systems he was giving them some kind of new relativity, more general than the relativity of the special theory. He called this the covariance principle of the equations of physics. The ﬁrst of these reasons was an outright mistake—acceleration and gravity can be distinguished by suitable experiments, as I have already pointed out. And the second of these reasons reﬂects a misconception about the meaning of relativity. Validity of all laws of physics in all coordinate systems is not a principle of relativity—it is a triviality. It’s like saying that an elephant remains an elephant when you express its height in meters instead of centimeters. The mathematician Erich Kretschmann soon pointed out to Einstein that, of course, all laws of physics can be expressed in all conceivable coordinates, even Newton’s laws can be expressed in all coordinates, although they then look very messy (for instance, in rotating coordinates, they acquire centrifugal and Coriolis terms).
The real meaning of Einstein’s covariance principle was hidden in a tacit assumption that Einstein failed to state, but took for granted: not only can the laws of physics be expressed in all conceivable coordinates, but when you express the laws in the special coordinates that correspond to a local geodesic reference frame —that is, a freely falling reference frame [51] —at and near one point, then the laws reduce to those of special relativity. The real content of Einstein’s covariance principle lies in this added condition. Obviously, the added condition imposes severe restrictions on how matter can and cannot couple to the gravitational ﬁeld; in fact, it completely determines the details of these couplings.
The added condition makes Einstein’s covariance principle into a principle of gauge invariance, analogous to the gauge invariance of electrodynamics. I don’t want to burden you with the mathematics of gauge invariance, but here is a simple physical explanation. Suppose we place a small Faraday cage at some point in an electric ﬁeld. Then in the interior of the cage, the electric ﬁeld will be reduced to zero, but the electric potential associated with the external electric ﬁeld will remain diﬀerent from zero. However, any experiment we perform inside the cage will be totally uncoupled from the external electric ﬁeld, and will be totally unaﬀected by the potential. The potential in the cage is diﬀerent from zero, but it is merely a physically irrelevant additive constant—this is the principle of gauge invariance for the potential. (It has among its consequences the conservation of electric charge, as shown by a neat, elementary, argument of Wigner’s [52].
In a completely analogous manner, suppose we consider a small laboratory in free fall in a gravitational ﬁeld. The laboratory plays the role of a Faraday cage for gravitation. Within the laboratory, the gravitational ﬁeld disappears (more or less), and there only remains the gravitational potential—or, more precisely, the metric tensor that plays the role of potential in Einstein’s theory. Einstein’s covariance principle, with the additional tacit assumption included, tells us that the laws of physics in the freely falling laboratory are the same as those of special relativity, and therefore any experiment we perform in the laboratory will be uncoupled from the external gravitational ﬁeld, and will be totally unaﬀected by the gravitational potential. Thus,
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the covariance principle is a principle of gauge invariance. In essence, this invariance principle

can be regarded as a general mathematical implementation of the equivalence principle, for all

particles (whether fast-moving or slow-moving), for light, and for all the laws of physics [53].

The modern view of covariance is somewhat broader and permits theories of gravitation that

go beyond Einstein’s. Instead of insisting that the laws of physics in a freely falling reference

frame are exactly those of special relativity, we are today willing to accept that they diﬀer

by terms involving the Riemann tensor, that is, we regard nonminimal couplings as consistent

with covariance. Some of the theories to be presented at our conference rely on this broader

interpretation of covariance. Vladimir Fock, a Soviet expert on relativity, severely criticized

Einstein for calling his theory general relativity; he claimed that in general relativity there can

be no relativity, and said “However paradoxical it may seem, Einstein himself. . . showed such a

lack of understanding when he named his theory” [54]. John Wheeler proposed to replace the

name of the theory by “geometrodynamics,” in analogy with electrodynamics. But the name

general relativity has stuck, and nothing can now be done about that. It is probably best to

ignore this unfortunate choice of name and remember that many parents give totally silly names

to their oﬀspring—and custom grants them the right to do so.

The gauge invariance arising from the principle of covariance has broad physical consequences:

it implies conservation laws for energy momentum and other “conservation” laws, and in Ein-

stein’s theory it implies the exact equality of inertial mass and gravitational mass, even for

systems that contain substantial amounts of gravitational self-energy [55] . The gauge invari-

ance of Einstein’s theory also compels gravitational waves to have spin 2, and only spin 2. I

regard this as the most fundamental physical consequence of covariance. And the argument can

be turned around: if we assume that the carrier of gravitational interactions is a spin-2 ﬁeld,

without any spin 1 or spin 0 components, then it must be a tensor ﬁeld with gauge invariance,

and from this we can conclude that the ﬁeld equations must have covariance and must coincide

with the Einstein equations. If Einstein had not discovered his theory in 1915, it would have

been discovered sometime around 1930, when gauge transformations and their implications for

the spin content of ﬁelds came to be understood.

It is ironical that in his mathematical calculations with his ﬁeld equations in the 1916 paper,

Einstein proceeded without general covariance. Although he repeatedly aﬃrmed the general

covariance principle, he found it inconvenient to adhere to this principle, and he wrote his

equations in a form that is not generally covariant. Today we write these equations in the form

Rµν −

1 2

Rgµν

=

−8πGTµν ,

and

we

call

them

the

“Ein√stein’s

equations,”

but

Einstein

didn’t

write

them that way. Instead, he adopted the condition −g = 1 and wrote his equations in a more

convenient,

simpliﬁed

form

∂Γαµν /∂xα

+ Γαµβ Γβνα

=

−κ(Tµν

−

1 2

gµν

T

)

.

But√this

form

of

the

ﬁeld

equations is not generally covariant, because it is not generally true that −g = 1. It’s a case

of Do as I say but don’t do as I do.

Actually,

the

equations

Rµν

−

1 2

Rgµν

=

−8πGTµν

were

ﬁrst

obtained

by

David

Hilbert,

the

well-known G¨ottingen mathematician. And what is more, he announced these equations in a

lecture at G¨ottingen a few days before Einstein. But he then made the mistake of setting R = 0,

because he was interested only in the gravitational ﬁelds associated with electric and magnetic

ﬁelds. By this mistake, Hilbert left the resurrection of this term to Einstein, a few days later.

John Auping has aptly called these alternating forward and backward steps in the approach to the

ﬁeld equations the pas de deux of Einstein and Hilbert [56]. This erratic historical development

brings to mind the words that Kepler used about his own road to discovery, “the roads that lead

man to knowledge are as wondrous as that knowledge itself” [57].

From a broader perspective, we can see that Einstein’s 1916 paper on general relativity suﬀers

from much the same problems as his 1905 paper on special relativity. The physical foundations

are shaky and riddled with mistakes, but by his amazing intuition, Einstein arrives at correct,

17

30
or almost correct, ﬁnal results despite of that. He somehow managed to ﬁnd his way through the fog of his own confusion and reach his goal despite his mistakes.
The crucial mistakes in both papers were based on sudden, inspirational ideas, to which Einstein took an obsessive liking, and which he elevated into dogmas. These were big mistakes, but they also were wonderful mistakes that led Einstein to astounding discoveries. We can say of Einstein what Arthur Koestler said of Kepler: “The measure of Kepler’s genius is the intensity of his contradictions, and the use he made of them” [56]. To which I will add, To err is human, and to err greatly is divine. . . at least sometimes.
References
[1] H. C. Ohanian, Einstein’s Mistakes (W. W. Norton and Co, New York, 2008). For brevity, this book will be designated by E.M. hereafter.
[2] E.M., p. 337. [3] G. Kane, “String theory and the real world,” Physics Today, November 2010, p. 39. [4] E.M., p. 166. [5] L. S. Feuer, Einstein and the generations of science (Transaction Publishers, New Brunswick,
NJ, 1982), p. xiii. [6] A. Einstein, Annalen der Physik 17, 891 (1905). [7] S. T. S. Lecky, Wrinkles in Practical Navigation (Philip and Son, London, 1942), p. 44. [8] For instance, A. Ganot’s E´l´ements de Physique, published in more than a dozen editions, and
also available in a German translation. In this book, Roemer’s method is discussed in Section 505 (fourteenth edition). [9] P. Galison, Einstein’s Clocks, Poincar´e’s maps (W. W. Norton and Co, New York, 2003), p. 291. [10] A. Pais, ‘Subtle is the Lord. . . ’ (Clarendon Press, Oxford, 1994), p. 173. [11] J. Auping Birch, Una revisi´on de las teor´ıas sobre el Origen y la Evoluci´on del Universo (Universidad Iberoamericana, 2009), p. 141. [12] A. Einstein, Berliner Tageblatt, December 25, 1919 (my translation). [13] A. A. Mart´ınez, Kinematics: the lost origins of Einstein’s relativity (Johns Hopkins Press, Baltimore, 2009). [14] A symmetry argument of this kind was stated by Wolfgang Pauli in his book Theory of Relativity (Pergamon Press, London, 1958), originally published as an article in the Mathematical Encyclopedia in 1921, and maybe it was known earlier. [15] E.M., p. 17. [16] E.M., p. 332. [17] E.M., p. 2.
18

31
[18] H. Minkowski, “Space and Time,” Address delivered at the 80th Assembly of German Natural Scientists and Physicians, Cologne, September 1908.
[19] C. Fuentes, La muerte de Artemio Cruz (Penguin Books, New York, 1996), p. 255.
[20] E.M., p. 162.
[21] E.M., p. 97.
[22] W. F. G. Swann, Rev. Mod. Phys. 13, 197 (1941) and J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press, Cambridge, 1987).
[23] D. Bohm, The Special Theory of Relativity (Routledge, London, 1996), p. 33. Bohm’s argument is stated in terms of clock frequencies, which is an unnecessary complication.
[24] E.M., p. 281.
[25] H. A. Lorentz, Proceedings of the Academy of Sciences of Amsterdam 6 (1904).
[26] M. Planck, Verh. d. Deutsch. Phys. Ges. 8, 136 (1906).
[27] E.M., p. 183.
[28] E.M., p. 185.
[29] Arthur Eddington, The Mathematical Theory of Relativity (Cambridge University Press, Cambridge, 1963), pp. 40, 41.
[30] Annalen der Physik, 35, 818 (1911).
[31] This absence of redshift is immediately obvious if we consider that, in the instantaneous reference frame of the elevator at launch, each successive sound or pellet signal has exactly the same initial conditions, and therefore necessarily has the same travel time to the ceiling. The absence of redshift in Newtonian physics was ﬁrst noticed by P. Florides and published in the announcement of a lecture [International Journal of Modern Physics A, 20, 2759 (2002)]. However, the announcement gives no details and does not say what conclusions Florides drew from his discovery. Florides has informed me that he expects to publish more about this topic shortly, but has not yet done so.
[32] This assumes that the speed u is large (u2 >> gh ) so the pellets are not signiﬁcantly decelerated during their upward motion and, furthermore, the displacement gt2/2 of the ceiling during the travel time can be ignored.
[33] Note that the crucial term in this expression for u is the denominator, which represents the deviation from Newtonian physics. The term g∆τ1 in the numerator of u and the same term in the denominator of t2 can usually be neglected; in any case, these terms approximately cancel. In Newtonian physics, which corresponds to c2 tending to inﬁnity, these terms cancel exactly, and the redshift then disappears.
[34] There is another simple derivation of the gravitational time dilation that focuses directly on the role of relativity and dispenses with the imbroglio of signal transmissions. To compare the rate of a clock 2 with the rate of a clock 1, simply drop an auxiliary clock from 2 to 1. Since this auxiliary clock is in free fall, the equivalence principle tells us that it is unaﬀected by the gravitational ﬁeld, so we can presume its rate is constant and we can use this clock as a standard of frequency. When the auxiliary clock arrives at clock 1, it will have a speed
19

32
√ v = 2gh . Hence clock 1 will have a time dilation factor relative to the auxiliary clock and also relative to clock 2. This is the usual gravitational time dilation factor 1 − v2/c2 1 − gh/c2 for the lower clock relative to the upper. The argument can be easily extended to nonuniform gravitational ﬁelds; see H. C. Ohanian and R. Ruﬃni, Gravitation and Spacetime (W. W. Norton and Co, New York, 1994), pp. 167, 168. [35] L. B. Okun, “The Concept of Mass,” Physics Today, January 1989, p. 31. [36] A simple calculation shows that in general relativity, in the Schwarzschild geometry, the coordinate speed of a freel√y falling particle decelerates monotonically if the initial, asymptotic, speed is larger than c/ 3. A particle of a lower speed ﬁrst accelerates, and then decelerates when it approaches the Schwarzschild radius (H. C. Ohanian, “Reversed gravitational acceleration of high-speed particles,” to be published). [37] And, I have to confess, also in my own textbook, H. C. Ohanian and R. Ruﬃni, op. cit. This mistake will be corrected in the next edition, currently in progress. [38] For instance, T. Padmanabhan, Gravitation, Foundations and Frontiers (Cambridge University Press, Cambridge, 2010), pp. 125-127, gives a very thorough, exact treatment (by Rindler coordinates) showing how the redshift is generated by acceleration of a reference frame. But he then misuses the equivalence principle to obtain the redshift in a gravitational ﬁeld. In this he commits the same mistake as Einstein: he proves that the equivalence principle is valid for slow-moving particles, and then takes it for granted that the same is true for fast-moving particles or light. [39] J. L. Synge, Relativity, the General Theory (North Holland, Amsterdam, 1971), p. IX. [40] E.M., p. 249. [41] E.M., p. 189. [42] Annalen der Physik, 38, 443 (1912). [43] If Φ = gx then this is the Rindler metric, and not curved. [44] A. Pais, op. cit., pp. 211, 212. [45] E.M., p. 256. [46] E.M., p. 255. [47] E.M., p. 192. [48] E.M., p. 198. [49] E.M., p. 205. [50] T. Levi-Civita, The Absolute Diﬀerential Calculus (Dover, New York, 1977), p. 293; see also W. Rindler, Essential Relativity (Springer, New York, 1977), p. 123. [51] Geodesic here means Γµαβ = 0 , that is, geodesic in regard to the Einstein tensor ﬁeld. [52] E. Wigner, Symmetries and Reﬂections (Indiana University Press, Bloomington, 1967), pp. 11, 12.
20

33
[53] Invariance can also be stated as a requirement that the laws have the same “form” in all coordinates, so they are general tensor equations and the tensor character is achieved by means of the gravitational ﬁeld tensor, and no other tensor (that is, there is no other auxiliary prior geometry, or absolute object).
[54] V. Fock, The Theory of Space, Time, and Gravitation (McMillan Co, New York, 1964), p. 8.
[55] H. C. Ohanian, arXiv:1010.5557[gr-qc], October 2010. There is an intriguing exception to the equality of inertial and gravitationsl mass: the inertial mass of a wormhole is twice its gravitational mass, something that would be nice to test if we ever ﬁnd a wormhole.
[56] J. Auping Birch, op. cit., p. 162n. [57] E.M., p. 331. [58] E.M., p. ix.
21

35
Putting the standard ΛCDM model and the relativistic models in historical context
John A. Auping
Department of Physics and Mathematics, Universidad Iberoamericana, Mexico City, Mexico. E-mail: jauping@iwm.com.mx
Abstract. The historical origin of the idea of the apparently missing matter, later labeled as dark matter, is shown to be the use of Newtonian mathematics in trying to explain the rotation velocity of stars in spiral galaxies and galaxies in galaxy clusters. Much attention is paid to Thieu and Cooperstock, who, in using relativistic dynamics, have shown that dark matter can be disposed of as a myth. Dark energy is shown to be an artifact originally hypothesized to explain the apparent recent acceleration of the expansion velocity of the Universe. This apparent acceleration is shown to be an optical illusion that can be disposed of as a myth, once Einstein’s gravitational theory is taken seriously. Much attention is paid to Buchert’s and Wiltshire’s work on explaining the apparent acceleration of the expansion velocity of the Universe without dark energy. Keywords: dark matter, dark energy, rotation velocity of galaxies and galaxy clusters, expansion velocity of the Universe, Newtonian gravitational dynamics, relativistic gravitational dynamics.
Contents
Part 1.- Origin of the speculation on non-baryonic dark matter.......................................................36 1.1.- Dark matter in spiral galaxies..................................................................................................36 1.2.- Dark matter in galaxy clusters.................................................................................................41 1.3.- The Press-Schechter theorem and dark matter......................................................................43 1.4.- The location of dark matter......................................................................................................44
Part 2.- Cooperstock & Tieu’s relativistic approach to galacxy gravitational dynamics................47
Part 3.- Brownstein & Moffat’s relativistic approach to gravitational galaxy dynamics................55
Part 4.- Relativistic dynamics of galaxy clusters without cold dark matter....................................58 4.1.- Cooperstock & Tieu1.5.- Modified Newtonian Dynamics........................................................58 4.2.- Brownstein & Moffat1.5.- Modified Newtonian Dynamics.......................................................62
Part 5.- Concluding remarks from the point of view of philosophy of science...............................66
Part 6.- The origin of the speculation on dark energy.......................................................................68 6.1.- Dark energy and the acceleration of the expansion of the Universe......................................61 6.2.- Baysesian probability calculus................................................................................................70 6.3.- Dark energy and the gravitational dynamics of galaxy clusters..............................................74 6.4- Dark energy and the cosmic microwave background radiation...............................................76
Part 7.- General relativity refutes the speculation about dark energy.............................................80 7.1.- How averaging parameters in a non-homogeneous Universe produces the backreaction.......................................................................................82 7.2.- Clocks run at different rates in voids and walls.......................................................................91 7.3.- The new relativistic Buchert-Wiltshire paradigm.....................................................................92 7.4.- Conclusion............................................................................................................................114

36
Part 1.- Origin of the speculation on non-baryonic dark matter
1.1.- Dark matter in spiral galaxies The amount of non-baryonic dark matter is established through the discrepancy between the
observed visible mass –which is baryonic— and the total mass calculated from certain effects generated by gravitational fields. The first person alerting to the supposed missing mass in galaxies and galaxy clusters, in 1933, was Fritz Zwicky (1898-1974), a Swiss astronomer working in Pasadena, California. By comparing the redshift of individual galaxies that belong to a cluster with the redshift of the entire cluster, he was able to establish the proper velocity of a galaxy. Thus he could prove that the orbital velocity of galaxies in a cluster is higher than expected if one would only take into account the mass of visible matter (stars and ionized gas) from the point of view of Newtonian gravitational dynamics.1 Along this same line of reasoning, Rubin y Coyne speculated that the method of establishing the peculiar velocities of galaxies within a cluster through their redshift, serves to reveal “the relative distribution of dark and luminous matter” and “indicates the existence of large amounts of (dark) matter.”2
Zwicky also established that one can determine the total mass of a galaxy through the observation of the curvature of light coming from a star or galaxy that is located behind the Sun or a galaxy cluster. This method, derived from the theory of general relativity, served originally to corroborate this theory. Now that it has been corroborated, one proceeds in the opposite order and the curvature of light allows us to calculate the total mass of a galaxy that is located between a luminous object and the Earth.3 According to Zwicky, the observation of these effects of gravitational lensing provides us with the most simple and most exact determination of the masses of galaxies.4
The arcs of galaxies curved by gravitational lensing: the Abell 2218 cluster
1 Fritz Zwicky, “Die Rotverschiebung von extragalaktischen Nebeln”, Helvetica Physica Acta, vol. 6 (1933): 110127; “On the Masses of Nebulae and of Clusters of Nebulae”, in: The Astrophysical Journal, v. 86 (1937): 217 -46 2 Vera Rubin & George Coyne, eds. Large Scale Motions in the Universe (1988): 262, 101-102
3 The curvature is   0.5 g l / c2 radians where l is the distance travelled by light through a gravitational field and g is the gravitational acceleration. The factor g depends directly on the mass of the object that causes the
bending of the light. See George Gamov, En el país de las maravillas. Relatividad y cuantos (1958): 96 4 Fritz Zwicky, “On the Masses of Nebulae and of Clusters of Nebulae”, in: The Astrophysical Journal, vol. 86 (1937): 238. Zwicky denominates “nebulae” what we know today to be galaxies.

37
Observations have been made of the orbital velocities of stars in spiral galaxies that seem to reveal the existence of a halo of non-baryonic dark matter which extends further than the visible disk of the galaxy, on the condition that we interpret these velocities from the point of view of Newtonian gravitational laws. These laws predict that the acceleration diminishes with the inverse square of the distance of the central mass and that the orbital velocity diminishes with the inverse root of the distance. In a series of 10 important publications, in The Astrophysical Journal from 1977 to 1985, Vera Rubin and her team observed about 60 spiral galaxies (20 of type Sa, 20 of type Sb and 20 of type Sc (note 5) and reported that the orbital velocity was almost constant, independent of the distance of the center of the galaxy.6 It is important to distinguish between the astrophysical observations of Vera Rubin and her team and the interpretations they made of these observations. Two cosmographic observations are beyond any doubt: 1.- First observation.- In the solar system, the orbital velocity of the planets diminishes with the inverse root of the distance ( v  1/ r ), so the velocity diminishes as the distance increases:
Graph.- First observation: orbital velocity and distance from the Sun in the solar system7
2.- Second observation: the rotational velocity of stars of a spiral galaxy, increases rapidly at short distance from the galaxy center, then stops diminishing with distance, and remains more or less constant (the curve flattens out). However, the visible mass diminishes rapidly as one moves away from the galaxy center.
5 Spiral galaxies type Sa have a big center, with the arms close to each other; galaxies type Sb, a smaller center with distinguishable arms; and the type Sc galaxies, an ever smaller center with arms quite separate from each other. 6 Vera Rubin & Kent Ford et al. “Extended rotation curves of high-luminosity spiral galaxies. I The angle between the rotation axis of the nucleus and the outer disk of NGC 3672,” The Astrophysical Journal, vol. 217 (1977): L1L4; “II The anemic Sa galaxy NGC 4378,” ibidem, vol. 224 (1978): 782-795; “III. The spiral galaxy NGC 7217,” ibidem, vol. 226 (1978): 770-776; “IV. Systematic dynamical properties,” ibidem, vol. 225 (1978): L107-L111; “V. NGC 1961, The most massive spiral known,” ibidem, vol. 225 (1979): 35-39; “Rotational properties of 21 Sc galaxies with a large range of luminosities and radii, from NGC 4605 (R=4 lpc) to UGC 2885 (R=122 kpc)” ibidem, vol. 238 (1980): 471-487; “Rotation and mass of the inner 5 kiloparsecs of the SO galaxy NGC 3115,” ibidem, vol. 239 (1980): 50-53; “Rotational properties of 23 Sb galaxies,” ibidem, vol. 261 (1982): 439-456; “Rotation velocities of 16 Sa galaxies and a comparison of Sa, Sb, and Sc rotation properties,” ibidem, vol. 289 (1985): 81-104 7 Original drawing by Vera Rubin in Scientific American (1983): 90

38 Image.- Photo of the spiral galaxy M31 in which Rubin drew the flat rotation curve 8
Graph.- Second observation: orbital velocity and distance of the center in 9 type Sc galaxies 9
These two observations of real facts in spiral galaxies are interpreted by Vera Rubin and her team from the point of view of a cosmological model with Newtonian gravitational dynamics:
8 Malcolm Longair, Galaxy Formation (2008): 67 9 Original graphs by Vera Rubin in Scientific American (1983): 93

39
1.- First part of the interpretation. Rubin and her team start from the assumption that in spiral galaxies the gravitational dynamics operating are Newtonian. According to Newton the gravitational acceleration diminishes with the inverse square of the distance, and orbital velocity diminishes with the inverse root of the distance, as is explained in the next mathematical box.
MATHEMATICAL BOX 1, ORBITAL VELOCITY IN THE CONTEXT OF NEWTONIAN GRAVITATIONAL DYNAMICS
The mathematical reasoning proper of a Newtonian model is the following. According to Newton’s second law of movement, the acceleration a is:
F  ma  GMm/ r 2 a  GM / r 2 (1) and the acceleration of a body in orbit around a big central mass is:
a  v2 / r (2)
From (1) and (2) we deduce: GM / r 2  v2 / r (3)
And from (3) we deduce the orbital or rotational velocity: v  GM (4) r
which means that the orbital velocity is proportional to the root of the mass and inversely proportional to the root of the distance:
v  M (5) r
Since it is reasonable to assume that the mass of the galaxy is concentrated at its center and diminishes if one moves away from the center, one would expect, according to equation (5) that the rotational velocity v would rapidly decrease if one moves away from the center, since in that assumption with the total mass M (within the range r ) gradually stopping to inrease and r increasing linearly, M / r would diminish rapidly. The surprise is that one observes the contrary: the orbital velocity, at a certain distance from the center and beyond, remains constant even though we move away from the center. The only way to explain this strange phenomenon is the assumption that the mass, instead of gradually stopping to increase, actually increases linearly with radius, up to a certain, far away distance. For example, at twice the distance from the center, we would have twice the mass. That would explain why the rotational velocity remains constant with distance:
if M  r  v  0 (6)
So, within the context of Newtonian gravitational dynamics there is no other option left, but to assume the existence of a supposed halo of non-baryonic dark matter.
2.- Second part of the interpretation. The second part of the interpretation of Rubin and her team is the following: given Newtonian gravitational dynamics, an apparent incompatibility arises

40
between the visible galaxy mass (stars and gas), observed through the radiation in some frequency of the electromagnetic specter, and the observed constancy of the orbital velocity. This is interpreted as requiring enormous quantities of additional and invisible mass, that increase linearly with distance from the center, as can be seen in the next graph. Given the orbital velocity in these dynamics and given the fact the gravitational constant G and the orbital velocity v are constant, that is more or less independent from the distance from the galaxy center r , the only way, within the context of these Newtonian dynamics, to resolve this problem is the speculation that the total mass M contained in the sphere with radius r increases linearly with the radius. 3.- Third part of the interpretation. Given this speculation of the total galaxy mass increasing linearly with the radius and given the fact that the visible mass decreases rapidly with the distance from the galaxy center, it follows logically that ‘dark matter’, which is supposed not to interact with light, and increases with distance from the galaxy center, which implies that this dark matter is not associated to the visible matter. As a consequence of that speculation, the mass luminosity rate of the galaxy ( M / L ) increases dramatically, when one moves away from the galaxy center. Graph.- Linear correlation between mass and distance to the galaxy center in Sa and Sc galaxies 10
It is important to point out that in this interpretation, NO observations corroborate the speculations. Dark matter is not observed, precisely because it is supposed not to interact with light, as Rubin points out in a Scientific American article of 1983: “All attempts to detect a halo by its visual, infrared, radio or X-ray radiation have failed. (..) In sum, the only requirement for the halo is the presence of matter in any cold, dark form that meets the M / L constraint.”11 The whole conjecture about the dark matter halo depends on the truth of the theoretical assumption that orbital velocities in spiral galaxies can be explained by Newtonian gravitational dynamics. This assumption allows for the speculation that the total galaxy mass increases linearly with distance from the center, as seen in the graph above.
10 Original drawing by Vera Rubin in Scientific American (1983): 95 11 Vera Rubin, “Dark Matter in Spiral Galaxies”, en: Scientific American vol. 248 (1983): 98

41
Once the idea of a halo of exotic dark matter was published in the Scientific American in 1983, many cosmologists started making references to the speculation about a halo of exotic dark matter in spiral galaxies, dissociated from visible matter, as though it was a scientific fact. For example, in 1994, Kolb & Turner reproduced some flat rotation curves and affirmed that: “Rotation curve measurements indicate that virtually all spiral galaxies have a dark, diffuse ‘halo’ associated with them which contributes at least 3 to 10 times the mass of the ‘visible matter’ (stars and the like).”12 In 2002, Hawking attributed to a halo of exotic dark matter, the fact that the stars at the edge of spiral galaxies like the Milky Way, NGC 3198 or NGC 9646, are maintained in their orbits and not thrown into the outer space, and presented this fact as the “most convincing” proof until now in favor of the existence of exotic dark matter.13 Not all cosmologists reflected on the Newtonian assumptions of these speculations, but some did. For example, in his The Cosmic Century of 2006, Longair reproduced the original M31 spiral galaxy image with the flat rotation curve drawn by Rubin, and commented on the Newtonian dynamics underlying these speculations. 14 It is worthwhile quoting Longair at some length:
“Vera Rubin and her colleagues pioneered systematic studies of the rotation curves of galaxies (...). [I]n the outer regions of galaxies, the velocity curves are generally remarkably flat, vrot  constant (...). The significance of this result can be appreciated from a simple Newtonian calculation. If the galaxy is taken to be spherical and the mass within the radius r is M , the circular rotational velocity at distance r is found by equating the inward gravitational acceleration ( GM / r 2 ), to the centripetal acceleration ( vrot2 / r ), and so vrot  (GM / r)1/ 2 . Thus, if vrot is constant, it follows that M  r , so that the total mass within radius r increases linearly with the distance from the centre. This result contrasts strongly with the variation of the surface brightness of spiral galaxies, which decrease much more rapidly with distance from the centre than as r 2 .”15
Peebles too noted that Newtonian gravitational dynamics may not be applicable in the cases of galaxy clusters and spiral galaxies: “[d]iscovering the nature of the dark matter, or explaining why the Newtonian mechanics used to infer its existence has been misapplied, has to be counted as one of the most exciting and immediate opportunities in cosmology today.”16 He did not follow up, however, on his own doubts:17
1.2.- Dark matter in galaxy clusters The speculation about the existence of non-baryonic dark matter extends to galaxy clusters. In
the next mathematical box (number 5), I explain how total galaxy cluster mass is estimated in the standard CDM model, following a procedure that is based on Newtonian gravitational dynamics.
12 Edward Kolb & Michael Turner, The Early Universe (1994): 17-18 13 Stephen Hawking, El Universo en una Cáscara de Nuez (2005): 186 14 Malcolm Longair, Galaxy Formation (2008): 66-69 y Malcolm Longair, The Cosmic Century (2006): 248-253 15 Malcolm Longair, The Cosmic Century (2006): 248-249, bold characters are mine. 16 James Peebles, “Dark Matter”, in. Principles of Physical Cosmology (1993): 417 17 James Peebles, “Dark Matter”, in: Principles of Physical Cosmology (1993): 417-456

42

MATHEMATICAL BOX 2, ESTIMATING TOTAL GALAXY CLUSTER MASS IN THE CONTEXT OF NEWTONIAN GRAVITATIONAL DYNAMICS

The equation for kinetic energy is derived directly from Newton’s second law of movement:18

K  1 mv2 (7) 2
and if we assume that the distribution of the velocity is isotropic, in the three

directions of the system of coordinates and we assume also spherical symmetry in the

galaxy cluster, we obtain:

K



3M 2

vr 2

(8)

where vr is the average radial velocity. Let us assume also the validity of the virial theorem, that supposes Newtonian gravitational dynamics:19

K



1 2

Ug

(9)

and we obtain the equation for potential gravitational energy, derived from Newtonian physics:20

(47)

Ug

 GM1M 2 R

(10)

where R is the weighed average of the distance between objects with mass M . From

equations (9) and (10), we obtain:

K



1 2

GM

2

/

Rcl

(11)

From equations (8) and (11), we obtain (with Longair in, Galay Formation)

3 v2 M

Rcl (note 21)

(12)

G

where M is the galaxy cluster mass; v the average rotational velocity of one

galaxy; and Rcl the average distance between galaxies. For that reason,

(50) vrot 

1 GM 3 Rcl

(13)

The important point to make here is that the estimate of the galaxy cluster mass22 is based on Newtonian gravitational dynamics and, for that reason, overestimates the total galaxy cluster mass in various orders of magnitude, just as is the case with spiral galaxies. In estimating the

18 See Appendix II, equations 81-91, in: John Auping, El Origen y la Evolución del Universo (2009): 543-545 19 Section C1 of Appendix VIII in: John Auping, El Origen y la Evolución del Universo (2009): 736 20 See Appendix II, equation 101, in: John Auping, El Origen y la Evolución del Universo (2009): 547 21 See Malcolm Longair, Galaxy Formation (2008): 66 22 See mathematical box 2

43

baryonic mass of these clusters on the basis of their luminosity,23 and subtract this baryonic mass from the total mass,24 as obtained from Newtonian rotational velocity, modern cosmology obtains its estimate of the total non-baryonic galaxy cluster mass, which is various times the baryonic mass. In a recent survey of galaxy clusters, Hans Böhringer established that the proportions of non-baryonic dark matter and baryonic visible matter are 85% and 15%, respectively and that the 15% corresponding to baryonic visible matter is distributed between stars, 2%, and gas, 13%, in big clusters; and 5% and 10%, respectively in small clusters.25

1.3.- The Press-Schechter theorem and dark matter Some cosmologists have told me that they think that Press-Schechter theorem, dating from
1974, proves the existence of dark matter.26 I donot agree. This theorem pretends to establish the number N of objects with different masses ( N1(M1), N2 (M 2 ), N3 (M3 ) , etc.), per volume of
space (for example, Mpc3 ), produced by an original cloud of particles with certain initial mass
(both baryonic and non-baryonic) starting to collapse because of its initial inhomogeneities or perturbations. These collapses repeat themselves at different scales, in a more or less hierarchical form, for example, on a bigger scale, the number of galaxies in a galaxy clusters and on a smaller scale, the number of stars in a galaxy.

MATHEMATICAL BOX 3. THE PRESS-SCHECHTER FUNCTION

The Press-Schechter equation gives us the number of objects N with certain mass M as a function of the critical mass M  , related to the cause of the collapse, and of time:27

(51) N (M )



1 2

1  

n  3

 M2

M

 

M



(3n) / 6  

exp

  

 

M M

(3n) / 3    

(14)

where “all the time dependence of N(M ) has been absorbed into the variation of

M  with cosmic epoch” and  is “the mean density of the background model”, 28 n is

the value of the spectral index, and the critical mass of reference M  is defined as:29

(52)

M



M

 0



t t0

 4 /(3n)  

(15)

(as

in

Malcolm

Longair’s,

Galaxy

Formation),

where

M

 0

is

the

value

of

M 

at the

present time t0 .

23 See mathematical box 2 24 See mathematical boxes 4 & 5 25 Hans Böhringer of the Max-Planck-Institute für extraterrestrische Physik, in “Galaxy clusters as cosmological probes”, lecture given at the Universidad Iberoamericana, April 16th, 2008 26 See William Press & Paul Schechter, “Formation of galaxies and clusters of galaxies by self-similar gravitational condensation,” en: The Astrophysical Journal, vol. 187 (1974): 425-438 y la synthesis en Malcolm Longair, “The Press-Schechter Mass Function,” en Galaxy Formation, 2nd ed. (2008): 482-489 27 Malcolm Longair, Galaxy Formation (2008):484, equation (16.25) 28 Malcolm Longair, Galaxy Formation (2008):484 29 Malcolm Longair, Galaxy Formation (2008):483-484, equation (16.22)

44

In the Press-Schechter function we find the term for the critical mass M  , which the object

must have in order to collapse, and this mass M  is a function of the observed mass of the object

at

the

present moment

M

 0

,

on

the

one

hasnd,

and the

time

that has

passed

since

the

original

cloud started collapsing, on the other hand. The value of this critical mass at the present time

(

M

 0

)

is

determined

with

the

laws

of

Newtonian

gravitation

analyzed

before,

which

implies

that

this mass is overestimated in various orders of magnitude. The Press-Schechter function depends

also of the estimate of the mean mass density  of the cosmological model that is used, and the

validity of which is assumed. Originally, Press and Schechter used the Einstein-de Sitter model

with 0  1;   0 , but it is also possible to use the standard model CDM with

0  1;   0.7 or any other model. Both the term for the critical mass that assumes the validity of the Newtonian gravitational dynamics, and the term for the mass density, are model-

dependent, and imply previous estimates of non-baryonic dark matter. For that reason, the Press-

Schechter formalism does not prove, but rather assumes that the largest part of galaxy cluster mass is non-baryonic and is therefore compatible with that assumption.

Besides, a critical analysis of the Press-Schechter theorem by Monaco,30 reveals that this formalism, from the point of view of astrophysics is quite wrong but yields apparently good results: “there is a simple, effective and wrong way to describe the cosmological mass function. Wrong of course does not refer to the results but to the whole procedure.”31

1.4.- The location of dark matter In order to investigate the location of dark matter in clusters, advantage has been taken of the
special circumstances that occur when galaxies or galaxy clusters collide and cross each other. The fact that in the case of a collision of galaxies or galaxy clusters, the stars do not collide, but the gas does, implies that the heated gas is separated from the stars. In the case of the galaxy cluster 1E0657-558 also known as the Bullet Cluster, Clowe and his team corroborated the fact that there are two clusters in collision, seen from aside.32

The clouds of hot plasma of each cluster collide and get mixed up and reduce their relative velocity, but the stars of the galaxies do not collide physically, so that the visible plasma and the galaxies are spatially separated. The separation of galaxies and plasma permits estimating the proportions of visible baryonic matter of both on the basis of their respective luminosity. By observing the effect of weak gravitational lensing —a slight distortion of the elliptic form of the galaxies—, which is more accentuated where galaxies are (with relatively little visible matter), than the plasma regions (with more visible matter), the hypothesis is corroborated that the location of the dark matter is in and around the galaxies, generating the effect of the observed gravitational lensing. The variations of gravitational lensing “are in agreement with the galaxy

30 Pierluigi Monaco, “Dynamics in the Cosmological Mass Function (or, why does the Press & Schechter Function work?”, in: Giuliano Giuricin & Marino Mezzetti eds., Observational Cosmology: The Development of Galaxy Systems (1999): 186-197 31 Ibidem, pág. 187 32 Douglas Clowe et al., “A direct empirical proof of the existence of dark matter”, arXiv:astro-ph/0608407, reproduced in: Astrophysical Journal Letters (2006). Also, idem, “Colliding clusters shed light on dark matter,” in: Scientific American (agosto 22, 2006)

45
positions and offset from the gas.”33, and the dark matter is associated to the visible matter of the galaxies. This is not a case, therefore, of ‘pure’ dark matter, dissociated from the visible matter. Clowe’s team does not speculate about the character of the dark matter the existence of which they believe to have corroborated.34
Image.- Dark matter (blue) is associated to the galaxies and dissociated from the plasma (pink)
The conclusions of the analysis of the Bullet Cluster 1E0657-558, first realized by Clowe and his team, in 2006, then replicated by Bradac35 and her team in the case of another merger of clusters, catalogued as MACS J0025.4-1222, and indirectly corroborated by Massey and his team, who used the observed distortion of the form of half a million galaxies to reconstruct, the distribution of the total intermediate mass that causes the distortion by lensing, are the following:
1) “[T]he baryons follow the distribution of dark matter even on large scales.”36 2) There are large amounts of dark matter in these galaxy clusters. Hans Böhringer estimates
the proportions of non-baryonic dark matter and baryonic visible matter at 85% and 15% (stars, 2%, and gas, 13%, in big clusters), respectively.37 3) The statistical methods to measure the amount of mass through weak gravitational lensing, which is at the basis of these investigations and its conclusions, is a mixture of relativistic –as far as the fact of gravitational lensing is concerned— and, in the words of Clowe, “Newtonian gravity”,38 or in the words of Massey, “Newtonian” 39 gravitational dynamics, so that this proof of the existence of dark matter rests on the validity of Newtonian gravitational dynamics in these cases.
33 Douglas Clowe et al., “Catching a bullet: direct evidence for the existence of dark matter,” arXiv:astroph/0611496, p. 4 34 Dennis Zaritsky, a member of Clowe’s team, admits that one does not know what this dark matter is. Quoted in “Colliding Clusters Shed Light on Dark Matter”, Scientific American (August 22, 2006) 35 Marusa Bradac et al., “Revealing the properties of dark matter in the merging cluster MACS J0025.4-1222, en: arXiv:0806.2320 36 Richard Massey et al., “Dark matter maps reveal cosmic scaffolding,” in: Nature online, January 2008, p. 5. The dark matter is made visible by the grey contours in the first image and the grey spots in the three other ones. 37 Hans Böhringer of the Max-Planck-Institute für extraterrestrische Physik, in “Galaxy clusters as cosmological probes”, lecture given at the Universidad Iberoamericana, April 16th, 2008 38 Douglas Clowe et al., “Catching a bullet: direct evidence for the existence of dark matter,” arXiv:astroph/0611496, p. 3 39 Richard Massey et al., “Probing Dark Matter and Dark Energy with Space-Based Weak Lensing”, arXiv: astroph/0403229, p. 4, see also idem., “Dark matter maps reveal cosmic scaffolding, ” in: Nature online

46
1.5.- Modified Newtonian Dynamics Another possible hypothesis to explain the discrepancy between the observed and expected
Newtonian dynamics of galaxy rotation velocity is the Modified Newtonian Dynamics (MOND) without dark matter as developed by Mordechai Milgrom, an astrophysicist from Israel, in different publications, beginning in 1983.40 Milgrom maintains Newton’s second law of movement, but modifies it for very slow accelerations, such as those that are common at great distances from galaxy centers.41 The problem of this solution is the arbitrary division between ‘high’ and ‘low’ accelerations, on the one hand, and the arbitrary modification of the Newtonian dynamics, on the other hand, since it does not conform to known physical laws. Milgrom is very conscious of this fact, when formulating the following dilemma: “Dark matter is the only explanation that astronomers can conjure up for the various mass discrepancies, if we cleave to the accepted laws of physics. But if we accept a departure from these standard laws, we might do away with dark matter.”42
Mario Livio agrees with Milgrom that there are only two possible solutions to the problem, but prefers dark matter: “There are only two ways to explain the high speeds of these clouds. (...) [E]ither Newton’s law of gravitation breaks down in the circumstances prevailing in the outskirts of galaxies, or the high orbital speeds are caused by the gravitational attraction of invisible matter. (...) Astronomers have been forced to accept the second possibility: galaxies must contain large amounts of dark matter.”43 It is remarkable that Milgrom and Livio and many others assume that the accepted laws of physics applicable in these cases are necessarily Newtonian.
40Mordechai Milgrom, ”Do Modified Newtonian Dynamics Follow from the Cold Dark Matter Paradigm?”, in: Astrophysical Journal (may 2002)
41 Milgrom suggests that slow accelerations produce a orbital velocity independent of distance: a  a0 & a0  1.2 *1010 ms2 & F  a 2 / a0  a  GMa0 / r which, with a  v2 / r yields v  (GM a0 )1/ 4
42 Mordechai Milgrom, “Does Dark Matter Really Exist?”, in: Scientific American (agosto de 2002), p. 44 43 Mario Livio The Accelerating Universe (2000): 90

47
Part 2.- Cooperstock & Tieu’s relativistic approach to galaxy gravitational dynamics
To understand why the assumption of Newtonian gravitational dynamics is not necessary, we have to reformulate the essence of the problem:
Graph.- The observed and expected Newtonian dynamics of galaxy rotation velocity
A few cosmologists grasped the opportunity referred to by Longair and Peebles, as to the Newtonian dynamics at the basis of the dark matter speculation, notably two teams from Canada, Brownstein and Moffat, on the one hand, and Cooperstock and Tieu, on the other hand, who offered, one team independently of the other, twelve years after Peeble’s proposal, an orthodox solution, along the lines of Einstein’s general relativity, that makes the speculations about dark matter superfluous. According to Cooperstock and Tieu, galactic dynamics present a non-linear, relativistic problem. Eddington had mentioned this non-linearity for a system that is variable in time, and the authors extend it to non-linear, but stationary (non-time dependent) problems, as in galactic dynamics:
“In dismissing general relativity in favor of Newtonian gravitational theory for the study of galactic dynamics, insufficient attention has been paid to the fact that the stars that compose the galaxies are essentially in motion under gravity alone (‘gravitationally bound’). It has been known since the time of Eddington that the gravitationally bound problem in general relativity is an intrinsically non-linear problem even when the conditions are such tat the field is weak and the motions are non-relativistic, at least in the time-dependent case. Most significantly, we found that under these conditions, the general relativistic analysis of the problem is also non-linear for the stationary (non-time-dependent) case at hand. Thus the intrinsically linear Newtonian-based approach used to this point has been inadequate for the description of galactic dynamics (...). We … demonstrate that via general relativity, the generating potentials producing the observed flattened galactic rotation curves are necessarily linked to the mass density distributions of the flattened disks [of ordinary baryonic matter], obviating any necessity for dark matter halos in the total galactic composition.”44
Cooperstock, colaborating first with Tieu and then with Carrick, has analyzed a total of 7 spiral galaxies from the relativistic point of view:
44 Cooperstock & Tieu, General Relativity Resolves Galactic Rotation Without Exotic Dark Matter (2005): arXiv:astro-ph/0507619, ps. 2-3

48

1) In their first publication of 2005, the authors analyzed four spiral galaxies (the Via Lactea, and NGC 3031, NGC 3198, and NGC 7331)45
2) In December 2010 they added another three spiral galaxies proving the same point (NGC 2841, NGC 2903 and NGC 5033)46

Cooperstock, Tieu and Carrick conceive the sprial galaxies as systems that are analogous to “fluids rotating uniformly without pressure and symmetric around the axis of rotation,”47 and explained the rotational dynamics by the gravitational attraction exercised by baryonic matter, within the known form of the visible disk, in relativistic gravitational dynamics (see the next mathematical box).

MATHEMATICAL BOX 4. SPIRAL GALAXY MASS IN RELATIVISTIC, NONLINEAR GRAVITATIONAL DYNAMICS ACCORDING TO COOPERSTOCK

Cooperstock & Tieu start from the line element of anobject in ‘free fall’ in general relativity, adapted to the polar, cylindrical coordinates r y z :
ds2  evw (u dz 2  dr 2 )  r 2ewd  ew (c dt  N d)2 (16)
where u , v , w , and N are coefficients whose value is a function of the coordinates r and z . For various reasons, explained by the authors48 one may simplify this equation equating u  1 and w  0 :
ds2  ev ( dz 2  dr 2 )  r 2d  (c dt  N d)2 (17)

We obtain the relation between angular velocity  , and tangential velocity V and the coefficient N (using     (r, z)t ):49

 Nc r2

(18)

and (14) V   r (19)

so that, by (18) and (19):

V  N c (20) r

The authors use Einstein’s field equations for N and  in a weak field with a cloud of particles in rotational motion, not subject to pressure neither to friction:50

45Fred Cooperstock & Steven Tieu, General Relativity Resolves Galactic Rotation Without Exotic Dark Matter (2005): arXiv:astro-ph/0507619 46 J. D. Carrick and F. I. Cooperstock, General relativistic dynamics applied to the rotation curves of galaxias, arXiv:1101.3224, December 2010 47 Fred Cooperstock & Steven Tieu, “General Relativity Resolves Galactic Rotation Without Exotic Dark Matter” (2005): arXiv:astro-ph/0507619, p. 4 48 Fred Cooperstock & Steven Tieu, “Galactic Dynamics via General Relativity”, en: International Journal of Modern Physics vol. 22 (2007): 4-5 49 Fred Cooperstock & Steven Tieu, “Galactic Dynamics via General Relativity”, in International Journal of Modern Physics vol. 22 (2007): 4 50 Fred Cooperstock & Steven Tieu, “Galactic Dynamics via General Relativity”, in: International Journal of Modern Physics vol. 22 (2007): 5

49

N rr



N zz



Nr r

0

(21)

and

Nr2



N

2 z

 8 G 

r2

c2

(22)

Equation (21) can be represented as a function of the gravitational potential  for
rotating galaxies: 2  0 (23)
where the zero value is due to the absence of pressure and friction in a system of particles in rotational motion. If there were no rotational motion, the system would need pressure (a non-zero value) to be stable, as in the Poisson equation of Newtonian gravity for weak fields:
2  4 G  (24)

In a way analogous to the derivation of the Newtonian gravitational field and potential,51 Cooperstock and Tieu obtain the gravitational potential of a system of

particles in rotational motion not subject to pressure, neither to friction:







N r

dr



 r



 r



N r

dr



N r

(25)

From equations (20) and (25), we obtain the rotational or tangential velocity: V  c N  c  (26) r r

In polar cylindrical coordinates, the solution to equation (23) is: 52   Cek z J 0 (kr) (27)
where J 0 is the Bessel function J m (kr) of cero order ( m  0 ) and C is an arbitrary
constant. Given the fact that equation (18) is linear, we can rewrite equation (27) as a linear summary:
   Cnekn z J 0 (knr) (28)
n

From equations (26) and (27), we obtain:

 V



 c



c

r

Cnknekn z J1(knr)
n

and from (26) and (29), we obtain:

 N  V r =  c

n

Cnknr ekn z J1(knr)

(29) (30)

51 For the derivation of the Newtonian gravitational field and potential, see Appendixes II & VI B of John Auping, Origen y Evolución del Universo (2009). The Poisson equation is number (239) on page 668 52 Fred Cooperstock & Steven Tieu, “Galactic Dynamics via General Relativity”, in: International Journal of Modern Physics vol. 22 (2007): 9

50
 By solving Cnkn from n  1 to n  10 , we obtain the theoretical rotational
n
velocity curves, which are perfectly corroborated by observations, with the Bessel function of order one:
10
 V (r, z)  c Cnknekn z J1(knr) (31) n 1 10
 and N (r, z)   Cnknr ekn z J1 (knr) (32) n1
and from (31) and (32), and taking into account that c  3*108 m / s , we obtain:53 V (r, z)  3*108 N (r, z) (33) r
Each galaxy is different, and has its proper coefficients Cn and kn . Cooperstock and Tieu attached the respective values of these coefficients to their article for four galaxies, among them the Milky Way, for n  1 to n  10 .
What many astrophysicists and cosmologists attribute to a halo of cold dark matter in the context of Newtonian gravitational dynamics is explained by Cooperstock and Tieu with ordinary baryonic matter in the context of relativistic gravitational dynamics. This method yields the following results: “Most significantly, our correlation of the flat velocity curve is achieved with disk mass of an order of magnitude smaller than the envisaged halo mass of exotic dark matter.”54
Their hypothesis is corroborated: “We have seen that the non-linearity for the computation of density inherent in the Einstein field equations for a stationary axially-symmetric pressure-free mass distribution, even in the case of weak fields, leads to correct galactic velocity curves as opposed to the incorrect curves that had been derived on the basis of Newtonian gravitational theory.”55 As a mater of fact, the observations corroborate the predictions of the theoretical model, as can be appreciated in the following graph by Cooperstock y Tieu: 56
53 Ibidem, Appendix, p. 30 54 Fred Cooperstock & Steven Tieu, “General Relativity Resolves Galactic Rotation Without Exotic Dark Matter”, arXiv:astro-ph/0507619 (2005): 11 55 Fred Cooperstock & Steven Tieu, “Galactic Dynamics via General Relativity”, in: International Journal of Modern Physics A vol. 22 (2007): 29 56 Fred Cooperstock & Steven Tieu, “General Relativity Resolves Galactic Rotation Without Exotic Dark Matter”, arXiv:astro-ph/0507619 (2005) and “Galactic Dynamics via General Relativity”, in: International Journal of Modern Physics A vol. 22 (2007): 7.

51 Graph.- The rotational velocity curve for the Milky Way as predicted by the relativistic theory of
Cooperstock y Tieu (the curve) is corroborated by the observations (the points)
It is essential in science that the experiments set up by the authors can be replicated by colleagues. Some cosmologists expressed doubts to me as to the replicability of the proofs offered by Cooperstock and Tieu. With Wolfram’s program Mathematica, version 6, Alfredo Sandoval and I were able to reproduce exactly the same flat rotational velocity curves. The values and the equations of Mathematical Box 4 allowed us to reproduce with Wolfram’s program Mathematica, the same rotational velocity curves as published by Cooperstock and Tieu We discovered, however, that variations in the fourth or fifth or sixth decimal figure of the value of the coefficients Cn and kn may affect the results in a non-trivial way. This means we can not use figures that are rounded up to the third or fourth decimal. The following graph replicates the results by Cooperstock and Tieu, for the Milky Way, in the context of relativistic gravitational dynamics. The only difference with the previous graph by Cooperstock and Tieu is that these authors give the results up to a distance of 30 kilo parsecs, and Sandoval and I, up to a distance of 50 kilo parsecs.
Cooperstock and Tieu comment their findings: “The scientific method has been most successful when directed by ‘Ockham’s razor’, that new elements should not be introduced into a theory unless absolutely necessary. If it should turn out to be the case that the observations of astronomy can ultimately be explained without the addition of new exotic dark matter, this would be of considerable significance.”57
57 Fred Cooperstock & Steven Tieu, “Galactic Dynamics via General Relativity”, in International Journal of Modern Physics A vol. 22 (2007) y arXiv:astro-ph/0610370, p.30

52
Graph.- The rotational velocity curve for the Milky Way by Cooperstock y Tieu is replicated with Wolfram’s program Mathematica by Sandoval and the author
The work of Cooperstock and Tieu has generated much interest and also some public criticism, from Korzynski,58 Vogt and Letelier,59 and Garfinkle.60 Cooperstock and Tieu responded adequately to their critics.61
Other astropysicists have expressed the criticism that even with relativistic gravitational dynamics, the baryonic mass in the spiral galaxies is not enough to explain their rotational velocity. One of them wrote to me in an e-mail of December 2010, that “the masses of the galaxies Cooperstock and Tieu find are greatly in excess of any reasonable estimate of the baryonic mass.” What these astrophysicists sustain is that even in the case of relativistic gravitational dynamics only the sum of baryonic and dark matter in spiral galaxies explains the rotational velocity curves of stars belonging to the galaxy. They refer to Stephen Kent’s estimates of spiral galaxy mass, because Cooperstock & Tieu themselves make that comparison. Kent has three articles on this topic, published in The Astronomical Journal in 1986, 1987 and 1988, titled Dark matter in spiral galaxies I; II; and III, respectively.62
I do not think that Kent’s estimates validate the missing mass hypothesis, as I will now show. First, there is no indication in Kent’s figures of a 1/6 baryon/total mass ratio as Cooperstock himself and his critics assert. Kent does not use the term ‘baryonic mass’, but refers to ‘stellar mass’, being the sum of the stellar ‘bulge’ mass M B and stellar ‘disk’ mass M D that he obtains by means of estimates of the mass/luminosity ratio M / L (the luminosity being the surface brightness of stars) for bulges and disks. He derives the total mass estimate M tot by means of Newtonian equations that establish a causal relationship between mass and rotational velocity at
58 Nikolaj Korzynski, “Singular disk of mater in the Cooperstock-Tieu galaxy model,” arXiv:astro-ph/0508377 59 Daniel Vogt & Patricio Letelier, “Presence of exotic matter in the Cooperstock and Tieu galaxy model,” arXiv:astro-ph/0510750 60 David Garfinkle, “The need for dark matter in galaxies”, arXiv:gr-qc/051182 61 Fred Cooperstock & Steven Tieu, “Perspectives on Galactic Dynamics via General Relativity,” arXiv:astro-ph/ 0512048 y “Galactic Dynamics via General Relativity”, in: International Journal of Modern Physics A vol. 22 (2007): ps. 17-28 62 Stephen M. Kent, “Dark Matter in Spiral Galaxies. I. Galaxias with Optical Rotation Curves”, The Asytrophysical Journal, vol. 9 (June 1986): 1301-1327; “Dark Matter in Spiral Galaxies. II. Galaxies with H1 Rotation Curves”, The Asytrophysical Journal, vol. 9 (April 1987): 816-832; “Dark Matter in Spiral Galaxies. III. The Sa Galaxies, The Asytrophysical Journal, vol. 9 (August 1988): 514-527

53

any chosen radius, where velocity is the observed part (derived from blue and redshifts), radius
the chosen part, and mass the inferred part. The stellar/total mass ratio, for 7 of the 16 galaxies with complete data sets, is M BD / M tot  0.32 , that is a ratio of 1/3, not 1/6.63

Of the three galaxies chosen by Cooperstock & Tieu for analysis in their first article (NGC 3031; NGC 3198; and NGC 7331), Kent has complete data sets only for two of them, NGC 3031 and NGC 7331, since he has no bulge mass estimate fort NGC 3198. In the case of NGC 3031 and NGC 7331, the stellar/total mass ratios are 1/2 and 1/4, respectively. These ratios, however, have to be corrected, because Kent neither includes estimates of interstellar gas, nor of black holes, but “the contribution … from the stellar component alone”.64. In galaxy clusters the gas mass is 85% of the baryonic mass, and the stellar mass, 15%. In galaxies, it is 15% and 85%. The interstellar gas mass, for example in the Via Lactea, is 15% of the total baryonic mass.65. The additional gas mass changes the Kent estimate of baryon mass for NGC 3031 and NGC 7331 to 8.47 *1010 and 14.24 *1010 solar masses, implying 1/1½ and 1/3 baryon/total mass ratios, respectively.

Now we have to add the black hole masses M BH , which is originally baryonic dark matter. The black hole mass is not included in Kent’s M/L ratios because in both galaxies, he has identical M/L ratios for bulges and disks -3.76 and 4.04, respectively—, as he explains: “for NGC 3031 … and NGC 7331, the bulge M/L ratio in the full solution was very poorly constrained and so it was kept fixed equal to the disk M/L ratio”.66 The M/L ratios would have been different for bulges and disks if the black hole had been included in the bulge M/L ratio. In NG 3031, the black hole at the center of the galaxy has a negligible mass of M BH  108 solar masses67, but the authors estimate the total stellar mass to be M BD  7.7 *1010 solar masses, which increases the baryon mass estimate to 8.97*10^10. The NGC 7331 black hole may be as much as 109 solar masses68 which increases the baryon mass from 14.24 to 14.34*10^10.

The following table gives the corrected Kent baryon mass estimates, compared to those found by Cooperstock, and shows the baryon masses in both cases are in fact almost identical.

Galaxy NGC 3031 NGC 3198 NGC 7331

Cooperstock 10.9 *10^10 10.1*10^10 26.0*10^10

Kent corrected 8.97*10^10 / 14.34*10^10

Kent/Cooperstock 0.82 / 0.55

63 Stephen Kent, “Dark Matter in Spiral Galaxies. II. Galaxies with H1 Rotation Curves”, The Asytrophysical Journal, vol. 9 (April 1987): 827 64 Stephen Kent, ““Dark Matter in Spiral Galaxies. I. Galaxias with Optical Rotation Curves”, The Asytrophysical Journal, vol. 9 (June 1986): 1301 65 Katia Ferrière, “The interstellar environment of our galaxy”, arXiv:astro-ph/0106359 (June 2001): 1-56 66 Stephen Kent, “Dark Matter in Spiral Galaxies. II. Galaxies with H1 Rotation Curves”, The Asytrophysical Journal, vol. 9 (April 1987): 826 67 Rohlfs & Kreitschmann, “A Two component mass model for M81/NGC3031”Astronomy & Astrophysicis, (1980):175-182 68 Silchenko, “Chemically decoupled nuclei in the spiral galaxies NGC 4216 and 4501”, Astronomical Journal (1999):186-196

54
The table reveals that in the case of NGC 7331 there is still is some apparent missing mass, but not much. The ‘missing mass’ may be due to many uncertainties, among other things: 1) the uncertain estimates of the M/L ratio; 2) “relationship between luminous and dark matter shows significant variation among galaxies”; 3) “the relative amounts of dark and luminous matter in a galaxy are still not well known”69; and 4) the “optical rotation curves usually do not place strong constraints on the amount of dark matter in these galaxies. Indeed, in agreement with Kalnajs, 1983, some rotation curves are fit well without the need to assume the existence of any dark halo”.70 5) The mass of supermassive black holes at the galaxy center may have been underestimated.71 There are other ways in which black hole dynamics falsify the dark matter hypothesis. Though I donot share the MOND theory, their adherents have convincingly falsified the dark matter hypothesis in galaxies,72 adding this proof to the one offered by Cooperstock ansd his colaborators.
In the case of NGC 2841, NGC 2903 and NGC 5033, Cooperstock again employed the solution of the Einstein field equations of general relativity, as in his earlier studies, and the fits to the data appeared again to be very precise. The known data from galactic rotation curves can be accommodated with “at most relatively little extra [baryonic] matter” when the analysis is performed with Einstein’s as opposed to Newton’s gravity. This ‘little extra baryonic matter’ may be “due to dead stars, planets, neutron stars [black holes] and other normal non-luminous baryonic matter debris.”73
69 Stephen Kent, “Dark Matter in Spiral Galaxies. II. Galaxies with H1 Rotation Curves”, The Asytrophysical Journal, vol. 9 (April 1987): 816 70 Stephen Kent, ““Dark Matter in Spiral Galaxies. I. Galaxias with Optical Rotation Curves”, The Asytrophysical Journal, vol. 9 (June 1986): 1301, 1326 71 If the black hole at the center of NGC 3031 would have a mass equivalent to the heaviest one known today
(  1.8 *1010 solar masses), then Cooperstock’s and Kent’s NGC 3031 baryon estimates would be exactly equal.
72 Karl Gebhardt et al., “A relationship between nuclear black hole mass and galaxy velocity dispersion”, Astrophysical Journal Letters 539 (2000): 75 ss.. See also several recent contributions by Pavel Kroupa. 73 J. D. Carrick and F. I. Cooperstock, General relativistic dynamics applied to the rotation curves of galaxias, arXiv:1101.3224, December 2010, ps. 3,10

55

Part 3.- Brownstein & Moffat’s relativistic approach to gravitational galaxy dynamics

Independently of Cooperstock and Tieu, another Canadian team, consisting of Brownstein and Moffat, designed a relativistic gravitational model, called Modified Gravity (MOG), in which they modify the Newtonian laws of acceleration on the basis of the theory of general relativity.74 They corroborated their theory with the data of more than 160 galaxies.75 The following mathematical box synthesizes the essence of this new relativistic theory.

MATHEMATICAL BOX 5. THE MASS OF SPIRAL GALAXIES IN RELATIVISTIC GRAVITATIONAL DYNAMICS

By adding the relativistic acceleration to the Newtonian one, one finds tha relativistoc equation of gravitational force, as I explain in my book on cosmology:76

FEINSTEIN

 m (a NEWTON

 aEINSTEIN )  m{d(d2sx)2

 

dx  ds

dx } ds

(34)

Brownstein and Moffat start with an analogous, relativistic modification of the Newtonian acceleration law:

a(r)



 G M r2

 G0

M0M

e r / r0

 

r2

(1 

r

 )

r0 

(35)

and given the following effective gravitational constant:

G  G0 1  

M0 M



(36)

and by combining (35) y 36), they obtain:

a(r)





G0 M r2

[1 

M0

 1



e

r

/

r0

(1



r

 )]

M

r0 

(37)

As Brownstein explained to me in an e-mail, a year ago, the terms M 0 and r0 do NOT represent a variable mass and a variable radius, but are parameters the values of which are constant. In the case of high surface brightness galaxies (HSB), the values of these constants are:
M 0  9.60 *1011M SOL (38) and r0  13.92 kpc  4.30 *1020 m (39)
In the case of very low surface brightness (LSB) or dwarf galaxies, the mass and radius have the following values:
M 0  2.40 *1011M SOL (40)

74 Joel Brownstein & John Moffat, “Galaxy Rotation Curves Without Non-Baryonic Dark Matter”, arXiv:astroph/0506370 (2005) 75 Joel Brownstein & John Moffat, “Galaxy Rotation Curves Without Non-Baryonic Dark Matter”, arXiv:astroph/0506370 (2005): 18-28 76 Equation (170 B) of Appendix VI B, in: John Auping, The Origen and Evolution of the Universe (2009): 660

56

r0  6.96 kpc  2.15*1020 m (41)

Given the fact that according to Newton:77

a(r)  v2 (42) r
we obtain, combining equations (37) and (42), the law of modified velocity:

v(r) 

G0M [1 

M0

 1



e

r

/

r0

(1



r

)]1/ 2

(43)

r

M

r0 

In the case of a symmetric galaxy, the mass density of which contains an interior

core at a distance r  rc , the acceleration of equation (37) is transformed in equation

(44) for HSB galaxies and in (45) for LSB and dwarf galaxies LSB, respectively:

HSB:

a(r)





G0 M



r

r 

r2

rc

3  

[1 

M0

 1



e

r

/

r0

(1



r

 )]

M

r0 

(44)

LSB:

a(r)





G0 M



r

r 

r2

rc

6 

[1 

M0

 1



e

r

/

r0

(1



r

 )]

M

r0 

(45)

In these cases, the rotational velocity for HSB galaxies is transformed in equation

(46) and for LSB and dwarf galaxies in (47), respectively:

HSB: v(r) 

G0 M r



r

r 

rc

3/ 2  [1  

M0

 1



e

r

/

r0

(1



r )]1/ 2

M

r0 

(46)

LSB: v(r) 

G0 M r



r

r  rc

3  [1

M0

 1



e

r

/

r0

(1

r

)]1/ 2

M

r0 

(47)

These equations of the acceleration and rotational velocity differ from the classical,

Newton rotational ones:

a NEWTON





G0



r

r 

rc

r2

3  

(48)

with   1 for HSB and   2 for LSB and dwarf galaxies.

vNEWTON 

G M (where r   ) r

(49)

Even though Cooperstock & Tieu and Brownstein & Moffat, use different relativistic models, the results are identical, as can be appreciated comparing, for example, the rotational velocity

77 See mathematical box 1

57 curves of the Milky Way, produced by both teams. In the following graph, I reproduce the rotational velocity curve of the Milky Way, produced by Brownstein and Moffat,78 that can be compared with the one produced by Cooperstock y Tieu, reproduced above..
Graph.- The rotational velocity curve of the Milky Way according to the relativistic theory MOG
In synthesis, the gravitational dynamics of spiral galaxies are well explained by Einstein’s theory of general relativity, without any necessity to introduce speculations about a halo of non baryonic dark matter
Image.- The dynamics of spiral galaxies is explained by Einstein’s general relativity 79
78 Joel Brownstein & John Moffat, “Galaxy Rotation Curves Without Non-Baryonic Dark Matter”, arXiv:astroph/0506370, p. 29 79 The spiral galaxy NGC 6946. Photo by John Duncan, Astronomía (2007): 223

58

Part 4.- Relativistic dynamics of galaxy clusters without cold dark matter

4.1.- Cooperstock & Tieu ¿Can we extend this analysis of galaxy rotational velocity to galaxy clusters? Cooperstock &
Tieu have shown that it is indeed possible80: “For the dynamics of clusters of galaxies, the virial theorem is used. This is based on Newtonian gravity theory. It would be of interest to introduce a general relativistic virial theorem for comparison. It is only after possible effects of general relativity are explored that we can be confident about the viability or non-viability of exotic dark matter in nature. ”81

As a matter of fact, in 2008, Cooperstock and Tieu applied general relativity to the gravitational dynamics of galaxy clusters, and corroborated their hypothesis,82 especially their hypothesis about total cluster mass and the rotational velocity of galaxies.83 The following mathematical box synthesizes their main argument, based on a relativistic model of a weak gravitational field constituted by many bodies that suffer mutual gravitational attraction but no friction or pressure.

MATHEMATICAL BOX 6. TOTAL MASS ESTIMATE OF GALAXY CLUSTERS IN A RELATIVISTIC GRAVITATIONAL MODEL

Cooperstock and Tieu start with Schwarzschild’s solution to Einstein’s equations, that uses a metric of spherical coordinates for a spherical mass M ,84 the same that one uses to derive the perihelion rotation of Mercury in a plane:85

ds 2

 
1 

1 2 GM c2r

 dr 2   

 r 2 (d 2



sen 2

d 2 )  1  

2GM c2r

c 2dt 2 

(50)

The four terms between parentheses constitute the metric coefficients, that together determine Schwarzschild’s metric tensor in fourdimensional space-time, where the

80 Fred Cooperstock & Steven Tieu, “Perspectives on Galactic Dynamics via General Relativity,” arXiv:astro-ph/ 0512048 (2005) and Fred Cooperstock, “Clusters of Galaxies”, in: General Relativistic Dynamics (2009): 135-159 81 Fred Cooperstock & Steven Tieu, “Perspectives on Galactic Dynamics via General Relativity,” arXiv:astro-ph/ 0512048, p. 3. For the virial theorem, see Appendix VIII, Section C 1, in: John Auping, El Origen y la Evolución del Universo (2009), p. 736 82 Fred Cooperstock & Steven Tieu, “General relativistic velocity”, in: Modern Physics Letters A vol. 23 (2008): 1745-1755 and Fred Cooperstock, “Clusters of Galaxies”, in: General Relativistic Dynamics (2009): cap. 10 83 John Moffat, “Scalar-Tensor-Vector Gravity Theory”, arXiv:gr-qc/0506021; “A Modified Gravity and Its Consequences for the Solar System, Astrophysics and Cosmology,” arXiv:gr-qc/0608074; y Joel Brownstein & John Moffat, “Galaxy Cluster Masses Without Non-Baryonic Dark Matter,” arXiv:astro-ph/0507222; and Monthly Notices of the Royal Astronomical Society (2005): 1-16 84 See equations (381) and (382) of appendix VI B of John Auping, Origen y Evolución del Universo (2009): 692.
Cooperstock and Tieu invert the signs and simplify the equation, omitting the constants G and c , see Fred
Cooperstock & Steven Tieu, “General relativistic velocity: the alternative to dark matter”, in: Modern Physics Letters A vol. 23 (2008): 1746, equations (1) and (2) 85 See equation (382) of Appendix VI B and equation (4) of Appendix VI C of John Auping, Origen y Evolución del Universo (2009): 692, 698

59

mass M is not small. In the normalized version of Cooperstock and Tieu, the units are selected so as to make c  G  1 and the signs of the metric are inverted:86

ds2  1  2 m 1 dr 2  (r 2 )d 2  (r 2sen2 ) d 2  1  2 m  dt 2 (51)

 r

 r

A big difference between this procedure and the perihelion analysis of Mercury is that we do not make the simplifying assumption, justifiable in the solar system, that the proper time d of the observed mass and the time dt of the observer are one and the same. Normally, in the case of strong gravitational fields, this difference is considered to be important. But, Cooperstock and Tieu show that also in the case of a weak gravitational field, the difference between the proper time d of the observed mass and the time dt of the observer is crucial. The transformation of the coordinates of the observer ( r and t ) into the ‘co-moving’ coordinates of the observed object with its proper time ( R and  ) is the following:

2m





t




1

r 2m

dr

(52)

r

R t

1 2m (1  2m)

(53)

r

r

and r   3 (R  r)2 / 3 (2m)1/ 3 (54)

2



which gives us the following transformed Schwarzschild metric, that Cooperstock took from Landau & Lifshitz’s The Classical Theory of Fields, and that depends on the proper time of the massive object:87

ds 2



d

2





dR 2

3 2(2m)

(

R





)



2

/

3

  

3 2

(R

 )4 / 3 (2m)2 / 3 (d 

2



sen 2

d 2 )

(55)

In the case that the value of  comes close to the value of R , we are in a strong gravitational field and the singularity of a black hole arises where R   . But Cooperstock and Tieu are interested in the case of a weak gravitational field, where R  for all R , implying that r  2m for all r and the coordinates ( r,t ). The

86 Fred Cooperstock & Steven Tieu, “General relativistic velocity: the alternative to dark matter”, in: Modern
Physics Letters A vol. 23 (2008): 1746, equations (1) and (2) and note 6 ( c  G  1). This is equation (100.2) of L. Landau & E. Lifshitz, The Classical Theory of Fields, 4ª ed. revisad (2002):321, if one takes into account that rg (the ‘gravitational radius’) in Landau y Lifshitz is the mass m in Cooperstock and Tieu.
87 This is equation (102.3) of L. Landau & E. Lifshitz, The Classical Theory of Fields, 4ª revised ed. (2002):332, if
one takes into account that rg (the gravitational radius) in Landau and Lifshitz is the mass m in Cooperstock and
Tieu, and that Landau and Lifshitz normalize only half way ( G  1, but c  1).

60

radial velocity, measured by the external observer is:

vrad



dr dt



1  

2m r



2m r



(56)

The radial velocity in the proper time of the observed moving object is:

vrad



dR d





 g11 dr   g00 dt

(1



1 2m
r

)

2

1 



2m r

 

2m   r

2m (nota 88) r

(57)

In a weak gravitational field, the radial velocity measured in the proper time of the co-moving object is equal to the radial velocity measured in the time of the terrestrial observer, because the mass m of the field is so reduced that the factor
1 2m   1 0  1, as Cooperstock and Tieu explain: “the local measures, both  r
proper and external, of the radial velocity are approximately equal in the value of  2m / r .”89 However, this is only true in the case that almost all the mass of the
systrem is concentrated in the centre of mass, as for example in the solar system. So far, the weak gravitational field is originated by one massive object. But things get complicated, when we focus on the collapse of a cloud of particles, where each particle contributes to the total mass and field. It is in this case, that the radial velocity as measured in the proper time of the co-moving object, even in the case of non-relativistic velocities, starts differing considerably from the time of the external, terrestrial observer. Parting from the geodesic equation in general relativity, for a cloud of dust particles, taken from the classic work of Landau and Lifshitz,90 Cooperstock and Tieu obtain the following geodesic equation for dust particles or objects, as measured by an external (terrestrial) observer:
ds2  d 2  e( ,R)dR2  r 2 ( , R)(d 2  sen2 d 2 ) (58)

In this case, “a freely falling dust particle maintains constant space coordinates for all time,” and the exact solution of the four non-trivial Einstein field equations that apply in this case, “assumes a surprisingly simple form” in the form of the following two equations:91

  88 The Schwarzschild metric in Cooperstock and Tieu is g00  (1  2m) / r and g11   1/(1 2m / r) , the
difference with the Schwarzschild metric in John Auping, Origen y Evolución del Universo (2009), Appendix VI B,
  equation 382, p. 692, is that g00 in Cooperstock and Tieu is my  g44 ( g44   1  2GM / c2r ), and g11 in   Cooperstock and Tieu is my  g11 ( g11  1/ 1  2GM / c2r ), with C. & T. normalizing with G  1 and c  1
89 Fred Cooperstock and Steven Tieu “General relativistic velocity: the alternative to dark matter”, in: Modern Physics Letters A vol. 23 (2008): 1748 90 The equation (103.1) in L. Landau & E. Lifshitz, The Classical Theory of Fields, 4ª revised ed. (2002):339 is the equation (9) in Fred Cooperstock and Steven Tieu, “General relativistic velocity: the alternative to dark matter”, in: Modern Physics Letters A vol. 23 (2008): 1748 91 Fred Cooperstock, General Relativistic Dynamics (2009): 142

61

e  (r)2

(59)

1 E(R)

and r2  E(R)  F(R) (60) r
where E(R) and F(R) are functions of integration. This leads to the following

average radial velocity equation:

dr dt



 (

  )(1   2 ) 

8 r 2  2

 

F







(

F  F )

2



1 2F

 1 

 t

(61)

which “stands in sharp contrast to the very simple Newtonian-like expression”92

vlocal    

F r

(62)

and where   rF  (63) 3F

and

  F 8 rr 2

(64)

The factor F is the “accumulated mass function”93 conceived as a function of the radius R of the galaxy cluster (wherein the average radial velocity is supposed to be known)
F(R)  k1Rk2 and M (R)  F(R) / 2 (note 94) (65)
For example, in the case of the Coma cluster, Cooperstock has the following values of the mass function: F  6.641*1016 R1.453  F  9.649 *1016 R0.453  F   4.371*1016 R0.547

These equations permit us to reconstruct the relation between radial velocity, galaxy cluster mass and galaxy mass density, in a relativistic model, without necessity of non-baryonic dark matter. For example, in the case of the Coma cluster, the radial velocity expressed as the ratio 2M (R0 ) / r0 is of the order of 104 “if we assume, as would a Newtonian, that there exists dark
matter present to account for the observed velocities” and of the order of 105 “if we accept only the existence of the matter that we see.”95 Now the problem we face is whether we can reconcile the ‘observed velocities’ and the ‘baryonic matter that we see’, without resorting to dark matter. Cooperstock argues that we can, with the help of the relativistic radial velocity equation (61) and the accumulative mass function of BOX 6. In stead of boosting the mass of the galaxy by adding dark matter, we boost the velocity based on visible baryonic matter using relativistic gravitational dynamics. Assuming the baryonic, visible mass is 20% or 30% or 40% of the supposed total mass within a sphere of 3 Mpc of 1.3*1015 solar masses, we obtain a ‘boost factor’ n of the supposed Newtonian radial velocity associated with only observed baryonic mass ( dr / dt  n ) of n  2.23, n  1.82 and n  1.58 , respectively, to obtain the relativistic

92 Fred Cooperstock, General Relativistic Dynamics (2009): 146 93 Fred Cooperstock, GeneralRrelativistic Dynamics (2009): 149 94 Equations (10.26) and (10.19), respectively in Fred Cooperstock, General Relativistic Dynamics (2009): 149,143 95 Fred Cooperstock, General Relativistic Dynamics (2009): 148

62
radial velocity. Since the observed average radial velocity and all the terms at the right hand side of the relativistic radial velocity equation (61) are known, we can obtain the value of the change of mass density over time (   /  t ), that is 2.13*1041 kg / m3 / s , 2.62 *1041 kg / m3 / s and 3.02 *1041 kg / m3 / s , respectively. “Rates of density changes of the order of magnitude 1041kg / m3 / s are quite reasonable as over a period of one billion years”,96 which is the time of the evolution of the Como galaxy cluster. Cooperstock concludes that, while this is only one example, “and a very rough one at that, ... we have been able to account for the observed velocities of galaxies within a cluster .... solely within the framework of general relativity and without any extraneous dark matter.”97
Cooperstock comments his findings:
“When the gravity was deduced to be weak within these clusters, astronomers naturally turned to Newtonian gravity to correlate the seemingly anomalously large galactic velocities that they measured with the masses that they believed to be present. In this manner they initially deduced that there must be unseen “dark matter” in the order of 100 times as much as the visible matter to make the mass totals accord with the veloicities. However, with the later discovery of very large quantities of gaseous matter, this figure was reduced dramatically but there still remained a large quantity of matter yet to be accounted for. This apparent need is still promoted vigorously by researchers throughout the world. It has spawned a plethora of paspers advocating new particles that would conceivable play the role of this exotic missing material. However, we have seen that, insofar as high rotational velocities of stars in galaxies as the basis for the need for dark matter is concerned, the replacement of Newtonian gravity by general relativity removes this requirement. An essential point is that the nonlinearities of general relativity play an important role in systems of freely falling gravitating masses, leading to expressly non-Newtomian behavior, even when the gravitational field is weak. (...) Had Zwicky done this calculation 70 years ago with general relativity in mind, he might have come to very different conclusions regarding the requirement for vast stores of exotic dark matter. ”98
4.2.- Brownstein & Moffat Brownstein and Moffat too presented a relativistic model of galaxy clusters explaining their
radial velocity and total mass without the necessity of non-baryonic dark matter.99 They do not start with the geodesic equation, as Cooperstock and Tieu do, but with Newton’s laws of acceleration and gravitation, transformed by Einstein’s general relativity. In Mathematical Box (8), I present their physical-mathematical argument and thereafter, by way of illustration of the results, I reproduce the graph of the galaxy cluster Coma, which permits us to compare the total mass estimates in the Newtonian and relativistic gravitational models.
96 Fred Cooperstock, General Relativistic Dynamics (2009): 152 97 Fred Cooperstock, General Relativistic Dynamics (2009): 152 98 Fred Cooperstock, General Relativistic Dynamics (2009): 148, 153. 99 Joel Brownstein and John Moffat, “Galaxy Cluster Masses Without Non-Baryonic Dark Matter”, in: Monthly Notices of the Royal Astronomical Society (2005): 1-16

63

MATHEMATICAL BOX 7. RELATIVISTIC ESTIMATE OF THE MASS OF GALAXY CLUSTERS ACCORDING TO BROWNSTEIN AND MOFFAT

Brownstein and Moffat apply general relativity to the gravitational dynamics of 106

galaxy clusters that emit X-ray radiation that had been previously analyzed by

Reiprich and Böhringer with Newtonian gravitational parameters. Brownstein and

Moffat part from a Riemannian pseudo metric tensor and a third rank skewed

symmetric tensorial field, called metric-skew-tensor-gravity. The cluster mass derived

from their relativistic model is M MSTG . On the other hand, the same cluster mass

derived from Newtonian dynamics is M N . The mathematical argument permits

comparing both mass estimates. The Newtonian acceleration is:

aN

(r)





G0

M r2

(r)

(66)

so that the total Newtonian mass estimate is:

M

N

(r)





a(r) r G0

2

(67)

For a spherical, isotropic and isothermal gas cloud, the acceleration, in both Newtonian and relativistic models, is:

a(r)





3 k T  mp



r2

r 

rc 2



(68)

From equations (67) y (68), we obtain the total cluster mass equation in the Newtonian model:

M

N

(r)





3 k T  mpG0

 

r2

r3  rc 2

 

(69)

and since the relativistic acceleration is:

a(r)





G(r)

M r

MSTG 2

(r)

(70)

the total cluster mass in the relativistic model is:

M

MSTG(r)





a(r) r2 G(r)

(71)

From equations (68) and (71), we obtain the total cluster mass in the relativistic

model:

M

MSTG (r)





3 k T  mpG(r)



r

2

r3  rc 2



(72)

Combining the equations (69) and (72), we obtain the relationship between the total mass estimates in the Newtonian and relativistic models:

64

M MSTG (r)



G0 G(r)

M

N

(r)

(73)

In other publications, Moffat and Brownstein obtained the equation for the gravitational constant in a very large galaxy cluster with radius r :100

G



lim G(r)
r r0



G0

1 



M0

 

M gas 

(74)

From equations (73) and (74), we obtain:

M MSTG



1  

M0 M gas

1  

M

N

(75)

The values of r0 and M 0 are constant:

r0  rout /10 for r out 650 kpc (76 A)

r0  139.2 kpc for r out 650 kpc (76 B)

and

M 0



58.8 *1014

M

SOL



M gas 1014 M S

OL

 0.3 9 

(77)

From equation (75), the reader may appreciate that the relativistic total mass estimate is much less than the Newtonian estimate, which allows us to get rid of the speculations on ‘missing mass’ and non-baryonic dark matter’, according to Brownstein and Moffat: “we have used the simplest [relativistic] isotropic  -model
based upon hydrostatic equilibrium to fit the X-ray galaxy cluster data without the need for exotic dark matter.”101

The following graph of Brownstein and Moffat allows us to appreciate the difference between the relativistic and Newtonian total mass estimates of the Coma galaxy cluster. The difference between the relativistic and Newtonian total mass estimates is equivalent to the Newtonian nonbaryonic dark matter estimate, so that, in the relativistic model, the non-baryonic dark matter speculation is not needed. The authors reproduce similar results for another 105, X-ray radiating, galaxy clusters from the Reiprich and Böhringer sample.102

100 Joel Brownstein & John Moffat, “Galaxy Rotation Curves Without Non-Baryonic Dark Matter”, arXiv:astroph/0506370 (2005); and John Moffat, “Gravitational Theory, Galaxy Rotation Curves and Cosmology without Dark Matter”, arXiv:astro-ph/0412195 (2005) and “Scalar-Tensor-Vector Gravity Theory”, arXiv:gr-qc/0506021 (2005) 101 Joel Brownstein and John Moffat, “Galaxy Cluster Masses Without Non-Baryonic Dark Matter”, in: Monthly Notices of the Royal Astronomical Society (2005): 5
102 Joel Brownstein y John Moffat, “Galaxy Cluster Masses Without Non-Baryonic Dark Matter”, in: Monthly Notices of the Royal Astronomical Society (2005): 8-16

65 Graph.- Relativistic and Newtonian total mass estimates of the galaxy cluster COMA 103

We saw above the case of the Bullet Cluster 1E0657-558, the mass of which could not be explained, within Newtonian gravitational dynamics, without the presence of non-baryonic dark
matter, with a non-baryonic / baryonic mass rate of M NB / M B  3.17 , according to Clowe and his team.104 However, Brownstein and Moffat have proven that in relativistic gravitational dynamics, the dark matter hypothesis is not needed. The following table gives the results for the Bullet Cluster in the different models.

TABLE.- NEWTONIAN AND RELATIVISTIC TOTAL MASS ESTIMATES OF THE BULLET CLUSTER 1E0657-558

Type of matter
Baryonic -ICM gas -visible stars (galaxies) Non-baryonic

Newtonian model Clowe et al.105 24 % No estimate given No estimate given 76 %

Relativistic model Brownstein & Moffat106 100 % 83 % 17 % 0 %

103 Joel Brownstein y John Moffat, “Galaxy Cluster Masses Without Non-Baryonic Dark Matter”, in: Monthly Notices of the Royal Astronomical Society (2005): 7 104 See Section 1 105 Douglas Clowe et al., “A direct empirical proof of the existence of dark matter”, arXiv:astro-ph/0608407, reproduced thereafter inn: Astrophysical Journal Letters (2006). For big galaxy clusters, Böhringer estimates that the baryonic mass, on average 15% of total mass, is distributed between the interstellar medium (13%) and stars (2%), and non-baryonic dark matter is 85% (See Section 1). 106 Joel Brownstein & John Moffat, “The Bullet Cluster 1E0657-558 shows Modified Gravity in he Absence of Dark Matter”, arXiv:astro-ph/0702146. Moffat’s and Brownstein’s Modified Gravity model (MOG) is not to be confused with Milgrom’s MOND model, because the former is a relativistic model and the latter, Newtonian (See mathematical BOX 6).

66

Part 5.- Concluding remarks from the point of view of philosophy of science.-

It is time for a conclusion, in the words of Cooperstock: “For the most part, astronomers continue to ignore general relativity in making deductions from observations. Thus an industry has arisen of massive computer simulations with billions of conjectured dark matter particles. The claim has been made that these simulations confirm that the CDM (cold dark matter) model of structure formation is in accord with observed structures in galaxy surveys such as the Sloan Digital Sky Survey. However, the basis for these simulations is Newtonian gravity. The lesson from our work is that the best theory of gravity, general relativity, is capable of providing surprises,” in making the dark matter hypothesis superfluous.107

We can put this conclusiobn in terms of Popper’s philosophy of science. Popper defines refutability as the demarcation between scientific theories and not-scientific theories. Let us see the following example:
a) Universal statement: “all swans are white”. b) Basic statement that refutes the universal statement in a specific space-time region: “right
here and now we observe this black swan”. c) Existential statement: “black swans exist.”

Logically, the verification of the basic statement (b) refutes the universal statement (a) and verifies the existential statement (c). However, the statement “no black swans to be seen here”, which refutes the basic statement (b), does not refute the existential statement (c), nor does it corroborate the universal statement (a), because there may be other space-time regions where black swans do exist. This is why we say that universal statements can be refuted, but cannot be corroborated, and existencial statements can be corroborated, but cannot be refuted.

TABLE.-REFUTABILITY AND NON-REFUTABILITY OF 3 KINDS OF STATEMENTS

UNIVERSAL STATEMENT

BASIC STATEMENT

EXISTENTIAL STATEMENT

Refutable by the Yes

Yes

No

facts

Verifiable by the No

Yes

Yes

facts

Since both universal and basic statements are refutable, they are both scientifc. However, existential, or metaphysical, or theological statements or those of science fiction cannot be refuted by the facts of the physicsal world and are therefore not scientific, which does not mean that they cannot be useful. For example, philosophy of science is metaphysics, its statements cannot be refuted by the facts, but it is very useful. The frontier between scientific and notscientific statements, asccording to Kartl Popper, is the refutability principle.108

I will now add a few points to Popper’s philosophy of science. He did not take into account the possibility that two theories, one scientific and one speculative, can both be corroborated by the facts. What would be the demarcation principle in such a case? Let me first explain the

107 Fred Cooperstock, General Relativistic Dynamics (2009): 159 108 Karl Popper, Conjeturas y Refutaciones (1989):63-64

67
difference between scientific and speculative statements. A scientific hypothesis establishes a physical law that causally relates an observable cause and andIan observable effect. A speculative hypothesis establishes a physical law that explains an observable effect by an unobservable cause. For example, the hypothesis that explains galaxy rotational velocity curves by dark matter in a Newtonian gravitational theory is a speculative hypothesis. And the hypothesis that explains the flat rotational velocity curves of spiral galaxies in a relativistic gravitational theory is a scientific hypothesis.
Speculative hypotheses are necessary in the history of science. After some time, however, with the advance of scientific theories and/or scientific observations, a speculative hypothesis may end up competing with a scientific theory, where the originally unobservable cause can be substituted by an observable cause. It is my view that when two explanations, derived from two different, but orthodox scientific theories, compete with each other, we have to give preference to the orthodox theory that postulates an observable cause over the orthodox theory that postulates an unobservable cause.
The following scheme explains these different possible developments of a speculative hypothesis
Graph.- Scientific and speculative hypothesis

68
Part 6.- The origin of the speculation on dark energy
Modern cosmology supposes that, on a large scale, the Universe is flat, that is to say, tot  1 and k  0 . Furthermore, empirical observations seem to indicate that M  0.3 .109 The difference between tot and M is usually explained by the speculation on dark energy and the dark energy density  . This interpretation is based on the CDM model, that has been developed and refined during the last ten years,110 and that implies the assumptions of Newtonian gravitational dynamics and a homogeneous Universe. I evaluate the dark energy hypothesis in Part 6, and in Part 76, I analyze the alternative Buchert-Wiltshire model, which reached a certain degree of maturity only two years ago.111 The latter model makes the speculation on dark energy superfluous, and is based on the assumptions of relativistic gravitational dynamics and an inhomogeneous Universe. This new model is capable of explaining the same astrophysical observations as the CDM model, that is to say the acceleration of the expansion of the Universe; the evolution of large structures over time; and the anisotropies of the Cosmic Microwave Background Radiation ( CMBR ), though from a radically different theoretical perspective,.
Part 6 has the following sections: 1) the evidence in support of the hypothesis on the acceleration of the expansion of the Universe 2) Bayesian probability calculus 3) the evolution of the gravitational dynamics of galaxy clusters 4) the form of the anisotropies of the cosmic background radiation ( CMBR ) 5) the attempts at theoretical explanation of the nature of dark energy
6.1.- Dark energy and the acceleration of the expansion of the Universe
Kirshner points out that the dark energy in Guth’s speculation about early inflation is not the same dark energy as the one driving the recent acceleration of the expansion of the Universe: ”a large dollop of dark energy whose negative pressure drove the inflation era and another, much longer-lived dark energy that drives cosmic acceleration now.”112 At the end of the 1990s evidence was presented in favor of the conjecture about the recent acceleration of the expansion of the Universe, in a local close by region (on the basis of the model with M  0.3,   0.7 ). I am speaking of the observations of luminosity and redshift of supernovae type 1a, discovered in the Supernova Cosmology Project of Saul Perlmutter and his team113 and the High-z Supernova Search Team, of Robert Kirshner and Adam Riess and their team.114 According to Kirshner, the
109 See Section 13.5, in John Auping, El Origen y la Evolución del Universo (2009) 110 Joshua Frieman, Michael Turner & Dragan Huterer, “Dark Energy and the Accelerating Universe”, arXiv:astro-ph/0803.0982 (2008) 111 David Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages,” in: New Journal of Physics (2007) 112 Robert Kirshner, The Extravagant Universe (2002): 138, mis negrillas 113 Saul Perlmutter, “Medidas de Omega y Lambda de 42 supernovas de gran corrimiento al rojo”, en: Astrophysical Journal vol. 517 (1999): 565-586 114 Adam Riess, “Prueba observacional de las supernovas para un Universo en aceleración y una constante cosmológica,” en: Astronomical Journal , vol. 116 (1998): 1009-1038; y Robert Kirshner, The Extravagant Universe (2002)

69 estimate of the distance of supernovae on the basis of their redshift, supposes a Hubble constant of  70 / s / Mpc , and implies a reduction of the error margin of 40 to 70%.
According to these data, type 1a supernovae that are relatively close by have a redshift that is larger than would be expected in the case of a decelerating expansion of the Universe, indicating that in the last thousands of millions of years the expansion is accelerating. Many cosmologists attribute this recent expansion acceleration to a modern edition of the ancient cosmological constant, first proposed by Einstein. 115 The sum of the mass density of M  0.3 and the dark energy density of   0.7 yields a total density of tot  1 , as can be seen in the next graph.
Graph.- Computer simulation of the Friedmann-Lemaître model with cosmological constant
Graph.- The apparent acceleration of the expansion of the Universe 116
115 Robert Kirshner, The Extravagant Universe (2002): 223 116 Image reproduced in Robert Kirshner, The Extravagant Universe (2002): 223

70
The same data presented by the Kirshner-Riess and Perlmutter teams led Karttunen and his colleagues, in 2003, to a more cautious interpretation: “a choice of model cannot be made on the basis of these observations.”117 In 2001, Robinson had made the same point, that is, that the original supernovae type 1a observations can be consistent with   0.7 and tot  1 , as well as with   0 and tot  0.3.118
However, years later both teams presented again observations of the same phenomenon, but this time more precise ones, made with the Hubble Space Telescope and reduced the error margin of the observed luminosity considerably. In 2003, Knop and Perlmutter and their team presented data of 11 supernovae of high redshift observed by the Hubble Space Telescope119 and in 2004, Riess and Kirshner and their team used the same telescope for more precise observations of 16 recent supernovae and revaluated the past evidence of 170 supernovae type 1a and affirmed having corroborated once again the hypothesis of the recent acceleration of the expansion of the Universe.120 They also affirmed that the historical transition from deceleration to acceleration occurs at a distance that corresponds to a redshift of z  0.46  0.13. In 2005, Astier and his team published their estimates of the cosmological parameters on the basis of observations of 71 high redshift type 1a supernovae discovered during the first year of the Supernova Legacy Survey (SNLS) that will last a total of five years.121
6.2.- Baysesian probability calculus.
Recently, in the cosmological literature, the concept has emerged of “model-independent cosmology” in recognition of the fact that the supposed corroboration of certain interpretations of observational data depend on certain values of the model’s parameters which in themselves are dependent on the truth of certain assumptions, for example, the validity of Newtonian gravitational dynamics, and the homogeneity of the Universe. The interpretations are “model dependent”. First, certain parameter values are programmed in the computer software and then the computer produces results that are compatible with these assumptions and validate the interpretations. From the point of view of logic, these model-dependent interpretations are not really corroborated by the data but rather shown to be compatible with them, and other interpretations, based on other models, might also be compatible with the same data.
In 2005, Moncy John, an astrophysicist from India, was among the first to propose “a modelindependent, cosmographic approach to cosmology,”122 using Bayesian probability calculus.
117 Hannu Karttunen and others, Fundamental Astronomy, Fourth Edition (2003): 374 118 Michael Robinson, Los nueve números del Cosmos (2001): 172
119 Rob Knop, et al., “New constraints on M ,  , and w from an Independent Set of Eleven High-Redshift
Supernovae Observed with the HST ” (2003), arXiv:astro-ph/0309368 120 Adam Riess et al., “Type Ia Supernova Discoveries at z<1 From the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution”, arXiv:astro-ph/0402512 and Astrophysical Journal, vol. 607 (2004): 665-738
121 Pierre Astier et al., “The Supernova Legacy Survey: Measurement of M ,  and w from the First Year Data Set”, arXiv:astro-phy/0510447 (2005)
122 Moncy John, “Cosmography, Deceleration Past, and Cosmological Models: Learning the Bayesian Way”, in: The Astrophysical Journal vol. 630 (2005): 667

71

This method permits calculating the probability that certain data, based on empirical observations are generated in a Universe that corresponds to a certain theoretical model. At the same time, one calculates the probability that the same data correspond to another theoretical model of the Universe. Then one compares these different probabilities, in order to decide which model has a greater probability of not being false. This decision does not impede two important facts, in the first place, the fact that various theoretical models are compatible with the same data, and in the second place, the fact that as a result of the variation of these empirical data in different samples, different samples (for example, of type 1a supernovae), can generate different levels of probability for the same theoretical model.

Using the data of the Knop-Perlmutter and Riess-Kirshner teams, Moncy John discovered that “the Bayesian analysis shows that ... there is no evidence from supernovae data to conclude that a changeover from deceleration to acceleration occurred anywhere in the past 5*1017 s .”123 The past 5*1017 seconds are the last 15 thousand million years, that is, the total age of the Universe. A second important conclusion of Moncy John is “that the present analysis rules out neither the accelerating nor the decelerating models; instead we can safely conclude that the data cannot discriminate between these models.”124
MATHEMATICAL BOX 8, BAYESIAN PROBABILITY CALCULUS125

“Bayesian evidence” E(M ) in favor of some cosmological model M is defined as the probability P that certain empirical data D are observed in a sample, in the case that this model would be the one that corresponds to the physical reality of the Universe:
E(M )  P(D M ) (78)

and the Bayes factor is the rate of the Bayesian evidence for both models M i y M j :

Bij



E(Mi ) E(M j )

(79)

If Bij  1, we prefer model M i over model M j and vice-versa, if 0  Bij  1, we

prefer model M j over M i . Then we draw the natural logarithm of the Bayes Factor, in

order to compare the different models M i and M j with a basic model M 0 :

if

0

B0i



E(M0 ) E(Mi )

1



ln B0i  0

(80)

 so we prefer M i over M 0 ;

if

B0i



E(M0 ) E(Mi )

1



ln B0i  0

(81)

 both models, M i and M 0 are plausible;

123 Moncy John, “Cosmography, Deceleration Past, and Cosmological Models: Learning the Bayesian Way”, in: The Astrophysical Journal vol. 630 (2005): 672 124 Moncy John, “Cosmography, Deceleration Past, and Cosmological Models: Learning the Bayesian Way”, in: The Astrophysical Journal vol. 630 (2005): 672 125 Harold Jeffreys, The Theory of Probability, 3rd edition (1998)

72

if

B0i



E(M0 ) E(Mi )

1



ln B0i  0

(82)

 we prefer M 0 over M i .

Of course, ln B0i values close to zero make the model comparison inconclusive, as can be appreciated in the following table from Roberto Trotta.126 For example if the

Bayes factor is ln B0i  0.139762  (P(M 0 )  1.15P(Mi ) :

ln B0i

B0i  P(M 0 ) / P(Mi ) Probability Strength evidence

<1.0

 3:1

<0.750

Inconclusive

1.0

 3:1

0.750

Weak evidence

2.5

 12 :1

0.923

Moderate evidence

5.0

 150 :1

0.993

Strong evidence

This inductive method is analogous to the  2 , where the observed distribution is
compared to the expected distribution under the null hypothesis, and a decision is made whether this difference is statistically significant.127
Following the road initiated by Moncy John, Elgaroy and Multamäki analyzed two type 1a supernovae samples, that is, the Riess-Kirshner —also called the ‘Gold’ sample by Elgaroy and Multamäky— and the Astier sample (SNLS).128 The base model M 0 with which both samples are compared is a flat Universe ( k  0 ) with a constant slightly negative deceleration factor q and linear expansion,129 which differs from the CDM model that has a transition from deceleration to acceleration. There are various surprising results of this Bayesian analysis: 1.- In the Gold sample, the more probably true model, is a closed Universe ( k  1), with a slight and constantly negative deceleration parameter ( q0  0.04 ).130 The model that comes second is a flat Universe ( k  0 ), also constantly accelerating ( q0  0.29 ).131 2.- In the SNLS sample, the model most probably true is a flat Universe ( k  0 ), also with constant acceleration ( q0  0.42 ).132 In the second place comes a model of a flat Universe

126 R. Trotta, “Bayes in the sky: Bayesian inference and model selection in cosmology,” en: arXiv:0803.4089, p. 14 127 See Philip R. Bevington & D. Keith Robinson, Data reduction and error anslysis for the physical sciences (2003) 128 Øystein Elgaroy & Tuomas Multamäki, “Bayesian analysis of Friedmannless cosmologies,” arXiv:astro-

ph/0603053

 129 q(z) 

qi z i .The authors define

q





1 H2

a
, which, given that
a

H



a
, implies
a

q





a a a 2

,

so

that

if

q  0  a  0 (acceleration of the expansion) and if q  0  a  0 (deceleration).

130 Ln(B0i )  0; q0  0.04;  2  191.1. The authors do not explain how a closed Universe can have a
slightly negative deceleration, which is a slightly positive acceleration.
131 Ln(B00)  0; q0  0.29;  2  182.8

132 Ln(B00)  0; q0  0.42;  2  112.0

73
( k  0 ), with a non-constant and non-linear deceleration parameter and a transition from deceleration q2 to acceleration q0 and q1 : ( q0  0.60; q1  0.60; q2  0.61).133 3.- However, in both samples the most probably true model has a constant linear expansion, and this fact implies there was never a transition from deceleration to acceleration in the expansion of the Universe, according to Elgaroy and Multamäki:
“[T]he best model in both cases has q(z) constant. It therefore seems fair to conclude that there is no significant evidence in the present supernovae data for a transition from deceleration to acceleration, and claims to the contrary are most likely an artifact of the parameterization used in the fit of the data. (...) It is at the moment not possible to say anything about when, or indeed if the Universe went from deceleration to acceleration. ”134
4.- Another important conclusion is “that the two samples do not enable us to draw conclusions about the underlying model, [since] there is no evidence that anything beyond a constant, negative deceleration parameter is required in order to describe the data.”135
Shapiro and Turner follows followed the same line of reasoning as Moncy John and Elgaroy and Multamäki.136 They showed that the interpretation of supernovae type 1a data by the RiessKirshner, Knop-Perlmutter and Astier teams, start from certain assumptions that are not necessarily true. Shapiro and Turner show that other interpretations are possible, for example, “a long epoch of recent deceleration is consistent with the data at the 10% [confidence] level” y “the present SNe 1a data cannot rule out the possibility that the universe has actually been decelerating for the past 3 Gyr [=three thousand million years] (i.e., since z  0.3 ).”137 If we abandon the assumption, that is part of the standard CDM model, 138 that the Universe is flat, another interpretation is possible, that is, “a positively curved universe with constant negative acceleration [=a closed universe, with constant deceleration] fits the gold set surprisingly well, and allowing q [=the deceleration parameter] to vary does not significantly improve the fit.”139
In synthesis, the Bayesian analysis realized by Moncy John, Elgaroy and Multamäki, and Shapiro and Turner, reveals that certain empirical data are compatible with the standard CDM model, but this compatibility does not corroborate this model, because other possible models
133 Ln(B0i )  0.6 ; q0  0.60; q1  0.60; q2  0.61;  2  110.5 . These data do not imply that the M i model (flat Universe, with a transition from deceleration to acceleration) is the more probably true one, but, on the contrary, the M 0 model with constant negative deceleration is almost twice as probably true as the M i model.
134 Øystein Elgaroy & Tuomas Multamäki, “Bayesian analysis of Friedmannless cosmologies,” arXiv:astroph/0603053, p.5, 6 135 Øystein Elgaroy & Tuomas Multamäki, “Bayesian analysis of Friedmannless cosmologies,” arXiv:astroph/0603053, p.7 136 Charles Shapiro & Michael Turner, “What do we really know about cosmic acceleration?”, in: Astrophysical Journal vol. 649 (2006): 563-569 137 Charles Shapiro & Michael Turner, “What do we really know about cosmic acceleration?”, in: Astrophysical Journal vol. 649 (2006): 566, mis negrillas
138 In this model, the parameter for the equation of state has a value of -1 ( w  P / VAC  1).
139 Charles Shapiro & Michael Turner, “What do we really know about cosmic acceleration?”, in: Astrophysical Journal vol. 649 (2006): 568

74
exist, which, with different degrees of probability of not being false, can explain the same observational data.
6.3.- Dark energy and the gravitational dynamics of galaxy clusters
In 2006, Longair pointed out that the observation of large scale structures, as, for example, galaxy clusters in different stages of the evolution of the Universe, do no permit distinguishing between different cosmological models, at the present moment of the history of the Universe. On the basis of some super computer simulations, run by Guinevere Kauffmann and her team,140 he compared the evolution of large scale structures in four types of universes, among them: 141 1.- The same large scale structures generated in a flat Universe with cosmological constant, in the CDM standard model, so that 0    1, with   0.7 . 2.- Large scale structures generated in an open Universe without a cosmological constant, OCDM (=open cold dark matter), with an overall density parameter of approximately 0  0.3 . 3.- CDM Large scale structures generated in an open Universe without a cosmological constant with cold dark matter and decaying neutrinos.
According to Longair, there is no difference in the results of the dynamics of the subjacent model, if we compare the  CDM model (with   0.7 ) with the OCDM and  CDM models (with   0 ). He argues that “this is because the dynamics only differ from   0 (...) in the late stages of evolution of the Universe when the effect of the cosmological constant is to stretch out the timescale of the model, allowing some further development of the perturbations.”142 The structures with a larger redshift are farther away from us in space and time. For that reason, if we go from the right to the left in the following images, from z  0 to z  3 , the size of the same object is diminishing progressively,143 and the angle from which we observe the object is progressively smaller. The important fact is that the  CDM , on the one hand, and the OCDM and  CDM series of images, on the other hand, the first one with and the second and third ones without the cosmological constant, are identical. There do not yet appear differences due to the cosmological constant, in none of the four stages of the evolution of the Universe that are contemplated. According to Longair, the differences would appear in later stages of its evolution that we have not yet reached.
The original authors of these images, Guinevere Kauffmann and her team, say exactly the same as Longair comparing CDM ( M  0.3;   0.7 ) and  CDM (   1): “Although neither model is perfect both come close to reproducing most of the data. Given the uncertainties in
140 Guinevere Kauffmann et al., “Clusters of galaxies in a hierarchical Universe, in: Monthly Notices of the Royal Astronomical Society, vol. 303 (1999): 188-206 141 Malcolm Longair, The Cosmic Century (2006): 414 142 Malcolm Longair, The Cosmic Century (2006): 415, my underlining. 143 If we take the age of the Universe as a function of the matter density and the redshift of light, we can calculate
the different values of z . See Edward Kolb & Michael Turner, The Early Universe (1994): 504, whose equations yield t  2.0571*1017 (0h2 )1/ 2 (1 z)3/ 2 s , ( 0  0.28 ; h  0.7 )  t  17.6 thousand million years for z  0 ; t  11.1 thousand million years for z  1; t  8.5 thousand million years for z  2 ; and t  7.0 thousand million years for z  3 .

75 modeling some of the critical physical processes, we conclude that it is not yet possible to draw firm conclusions about the values of cosmological parameters from studies of this kind.” 144
Graph.- Computer models of the Universe with and without cosmological constant 145
144 Guinevere Kauffmann, Jörg Colberg, Antonaldo Diafero & Simon White, “Clustering of Galaxies in a Hierarchical Universe: I. Methods and Results at z=0”, in: Monthly Notices of the Royal Astronomical Society, vol. 303 (1999): 188-206 (quote on p. 288; also arXiv:astro-ph/9805283, 21 May 1998) & “II. Evolution to High Redshift”, ibidem, vol. 307 (1999): 529-536 (also arXiv:astro-ph/9809168, 18 September 1998). My underlining. 145 Malcolm Longair, The Cosmic Century (2006): 414

76
6.4- Dark energy and the cosmic microwave background radiation CMBR
Modern cosmology analyzes the CMBR from different angles, in order to measure the small variations or anisotropies of that radiation, that were first discovered by George Smoot and his team.146 Supposedly, these anisotropies prove that the Universe has a flat geometry, which would imply that dark energy exists. By way of example, I reproduce a graph to be found in Longair. The curve is generated by the computer, with he help of a Legendre function, while the software has been programmed with the values of the parameters of some predetermined cosmological model. In the case of the graph reproduced below, the assumption is made that the CDM model, with non zero cosmological constant, is the valid one, with a matter density of M  0.3 and a dark energy density of   0.7 . The black points in the graph represent the observations: in the horizontal axis we read the angle from which the observations are made, and in the vertical axis one reads the magnitude of the observed anisotropies. The maximum variation in the temperature is one in hundred thousand.
Graph.- The CDM model is compatible with the anisotropies of the cosmic background radiation CMBR 147
If one looks at this graph, one may think “well, there is no doubt here, the facts corroborate the theory”. Things are not as simple, however, as they appear to be at first sight. In the first place (A), these observations (the dots) are a kind of ‘average’ of many observations that yield wildly varying results among themselves. In the second place (B), the curve (the continuous line) is model-dependent. A.- Uncertainties.- The following graph of Tegmark, Zaldarriaga and Hamilton, published in the year 2000, presents the observations made by 27 different teams of cosmologists. The curve that best fits these discrepant observations is the solid red line, a kind of ‘average’ of the 27 different series of data. It represents a cosmological model with a closed Universe ( tot  1.3),148 which is of course quite different from the standard CDM , where the Universe is flat ( tot  1.0 ). For this
146 George Smoot & Keay Davidson, Wrinkles in Time (1993) 147 Malcolm Longair, The Cosmic Century (2006): 424 148 Max Tegmark, Matías Zaldarriaga y Andrew Hamilton, “Towards a refined concordance model: joint 11parameter constraints from CMB and large scale structure”, arXiv:astro-ph/0008167 (2000): p. 1

77 reason, according to the authors, there is a need for complementing these observations of the CMBR with other observations, to be able to arrive at the standard CDM model .
Graph.- The discrepancy between 27 different observations of the anisotropies in the CMBR
B.- Model dependency.- In this graph of the year 2000, the cosmological model that lies at the basis of the interpretation of the observational data is defined by 11 parameters. However, Uros Seljak and Matías Zaldarriaga of Harvard perfected the model, and their 2009 version has 16 cosmological parameters, assumed to be true, among them the densities of cold dark matter, and dark energy, and another 20 non-cosmological parameters, which makes a total of 36 assumptions. In the table below, I reproduce the 16 cosmological parameters of the CMBFast model of Seljak and Zaldarriaga.149 TABLE.- 16 COSMOLOGICAL PARAMETERS IN THE CMBFAST MODEL
149 Uros Seljak & Matías Zaldarriaga, “List of CMBFAST parameters,” CMBFAST Website

78
If we vary, by way of example, just one of these 16 cosmological parameters, that is, the assumption of homogeneity (the blue line), the maximum anisotropy varies considerably (red and green lines) when we assume a non-homogeneous Universe. 150
Graph.- Variation of  20 % in the observed maximum anisotropy (red and green line)
when the assumed homogeneity of the Universe (blue curve) is abandoned

Explanation: Edward Kolb & Michael Turner, The Early Universe (1990): 384, 386

alm 2



AH

n3 0

16

(3  n)
(4  n) / 22

(2l  n 1) / 2 (2l  5  n) / 2

(83)

   T (xˆ1) T (xˆ2 TT

1 4


(2l 1)
l 2

alm 2

Pl (xˆ1.xˆ2 ) exp  (l  1/ 2)2 2

(84)

Let me explain the graph. The letter “l” in the horizontal axis represents the angle of observation (the order of the polynomial of Legendre) and “C” in the vertical axis represents the correlation between intensities and the anisotropy of the CMBR . The variation of the maximum peak of the curve is due to the variation of the constant “n” that represents the degree of homogeneity, the blue line representing perfect homogeneity. This is just an example of how the selection of the values of the parameters affects the curve and, for that reason, can be used to fit the observational data.

In other words, the assumptions of the model used to interpret the observational data, make these data compatible with the concordant cosmological model but do not corroborate it. Analogously, we may have 20 different economic models to explain a certain rate of inflation. They may all succeed in doing so, though they may be wildly different among themselves. Of course, if we demand that a model explain not one or two but, simultaneously two thousand different types of observational data, at many points in time, the restraints on the model become more severe. If such a model would succeed in doing so, we may say that the data corroborate the theoretical model.

This model dependency, added to the considerable discrepancy of the observations by different teams, makes it impossible to draw definite conclusions about the underlying cosmological

150 Computer simulation run by Alfredo Sandoval and John Auping with Wolfram’s Mathematica, using the equations 9.144 and 9.148 of Edward Kolb & Michael Turner, The Early Universe (1990): 384, 386

79 model. The data do not refute one model and corroborate another, but only prove that different models, among them the standard CDM are compatible wit the data. Given this model dependency and the error margins in each of the 27 observations that vary considerably,151 it comes as no surprise that different models may succeed, with different degrees of probability, in different samples, to predict the same observational data, as I explained in the foregoing section on Bayesian probability.152
From the point of view of the philosophy of science, we can formulate the same problem in yet another way. When the observational data fit a theoretical model with 36 parameters, which of these are really compatible with the data? We might increase the value of one parameter and decrease the value of another one and the same data may fit the ‘new’ model. Of course, not all parameters are that free. Some are not free at all, because they were corroborated independently as part of some other theory. For example, the value of the primordial and the present helium abundances —one of the 36 parameters of the model—, are known with a considerable degree of certainty.153 However, the values of other parameters are completely model-dependent. Among the latter ones we may count the assumed homogeneity of the Universe, the temperature T of the CMBR , the value of the Hubble constant H 0 , the matter density M and the space curvature k . As we shall see below, in relativistic models some of these parameters may obtain very different values and others are completely dropped, among them the supposed dark energy density  .154
151 See above, the graph “The discrepancy between 27 different observations of the anisotropies in the CMBR ”
152 See part 6.2 153 See Section 13.5 of John Auping, El Origen y la Evolución del Universo (2009) 154 See Part 7

80
Part 7.- General relativity refutes the speculation about dark energy
In some textbooks, the expansion of the Universe is compared with an inflating globe, while coins stuck on its surface move away from each other. The expansion of the globe is conceived to be homogeneous and symmetric, so that the same rate of expansion is observed in all directions. Most cosmologists too use the assumption of the homogeneity and isotropy of the Universe to construct their models of its expansion. This assumption does not appear to be true. Even though the Cosmic Microwave Background Radiation ( CMBR ) reveals that the Universe, some 300,000 years after the Big Bang, was almost perfectly homogeneous and isotropic, it is a fact that the small inhomogeneities present at that time —as registered by the small anisotropies of that radiation—, have since been magnified on a very large scale and today the Universe is not homogeneous, but rather an ensemble of enormous voids surrounded by enormous walls of galaxy clusters, like an enormous sponge.
Peebles analyzed the problem of the small scale inhomogeneity of the Universe, but concluded that there is still evidence in favor of the assumption of large scale homogeneity and isotropy of the Universe,155 though he admitted certain biases do occur due to irregular mass distributions:
“The clumpy mass distribution in the real world can play quite different roles in different tests. (...) Irregular mass distribution can produce a systematic error in apparent magnitudes of galaxies, for the mass along the line of sight acts as a lens that determines the rate of change of the convergence of a bundle of light rays, and that fixes the angular size of the image. If the mass distribution is clumpy rather than smooth (...) observations could be biased to favor objects whose images have been magnified, because they appear brighter, or the bias could go the other way, for where there is mass there tends to be dust. Thus, if the line of sight to a distant object passes through a large amount of mass, so that gravitational lensing is magnifying the angular size of the object, the image tends to be obscured.” 156
Peebles thought these biases would cancel each other out, so that the overall “bias may not be large.”157 This may be true for the kind of inhomogeneities Peebles was contemplating, but there appear to be others, not considered by him, that produce important overall systematic biases, especially the so called backreaction, and the different rates at which clocks are running in voids and walls, which do affect the value of the parameters of the model, as we shall see below.
Our galaxy cluster is located in an enormous void of 200 to 300 Mpc that expands between 20 and 30% more rapidly than could be expected according to the global (average) Hubble constant158; there is a superstructure of 400 Mpc known as Sloan’s Great Wall, surrounding part of this void; more locally there are two other minor voids of 35 to 70 Mpc each and Shapely’s super cluster with a diameter of 40 Mpc, at a distance of some 200 Mpc from our galaxy; and
155 James Peebles, “Fractal Universe and Large-Scale Departures from Homogeneity” and “Cosmology in an Inhomogeneous Universe”, in: Principles of Physical Cosmology (1993): 209-224, 343-360 156 James Peebles, Principles of Physical Cosmology (1993): 343 157 Ibidem, p. 343 158 Paul Hunt & Subir Sarkar, “Constraints on large scale inhomogeneities from WMAP-5 and SDSS: confrontation with recent observations”, arXiv:0807.4508, pág.1

81
according to the Hubble Space Telescope Key Project there exists a significant anisotropy in the local expansion, at distances of up to 100 Mpc (note 159).
In general, Wiltshire estimates that “some 40-50% of the volume of the universe at the present epoch is in voids of 30 h1 Mpc [  between 40 a 50 Mpc] in diameter (...) and there is much evidence for voids 3 to 5 times this size, as well as local voids on smaller scales.”160 Obviously, with the passing of time, due to the expansion of the Universe, the participation of the voids in the total volume increases, and that of the gravitationally collapsed regions decreases.
Today, the expansion of the Universe looks more like a river with rapids than a slowly inflating globe. Instead of a homogeneous flow of a slow river, the flow of the rapids makes the total flow inhomogeneous. Some parts flow more rapidly than others and between slow and faster flows friction occurs. Where there are obstacles the current surrounding them is slowed down and vortices are generated in the surface of the water. The expansion of the Universe resembles such a current that meets obstacles in the form of galaxy clusters and black holes that slow down the rate of expansion of some regions, and the friction between faster and slower expanding regions generates shear, while enormous vortices are generated by black holes. Walls of galaxy clusters with the size of hundreds of Megaparsecs surround voids where the expansion is more rapid, than in regions with higher matter density. The Universe is more like a expanding and deforming sponge than a smoothly and homogeneously expanding globe, something resembling the image produced in a recent article in the Scientific American. 161
Image.- The deceleration of the expansion of the Universe is greater in the walls than in the voids
159 Nan Li & Dominic Schwarz, “Scale dependence of cosmological backreaction”, arXiv:astro-ph/0710.5073, p. 1 160 David Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages”, in New Journal of Physics (2007): 4 161 Timothy Clifton & Pedro Ferreira affirm in “Dark Energy: Does it really exist?”, in: Scientific American, vol. 300 (2009): 33, that we are in the centre of an enormous spherical void with the size of the observable Universe,
diameter 850 170 h1 Mpc . See also their “Living in a void”, arXiv:0807.1443 (2008): 1-4.

82
7.1.- How averaging parameters in a non-homogeneous Universe produces the backreaction
The Universe is a collection of regions with high matter density (walls), where local clocks run slowly, and regions with low matter density (voids), where clocks run faster. We can disappear this inhomogeneity through a process of averaging also known as smoothing, but that does not take away the fact that the values of cosmological parameters in walls and voids differ among themselves and from the global-average. This real inhomogeneity invalidates one of the basic assumptions of the standard model and affects the curvature of space. Even though the Universe has a global-average curvature, that does not take away the differences between gravitationally collapsed regions, like galaxies and galaxy clusters, where space does not expand and the curvature is positive ( k  0 ), especially close to the galaxy centers where black holes can be found, and the enormous voids in the Universe, where space expands faster, and the curvature is negative ( k  0). The curvature of space is not homogeneous.
A numerical example helps to understand the process of smoothing. Let us choose 27 random numbers between 01 and 100,162 with a maximum difference between the biggest and the smallest number of 94. We then average every three numbers and obtain nine new numbers163 that differ less among themselves than the original 27, with a maximum difference of 59 between the extreme values. We now repeat the process of averaging groups of three numbers, and obtain three new numbers, 164 with a maximum difference among themselves of 33.0. If we average these three numbers we obtain the global-average of 48.6.165 That last number is analogous to the global-average of the cosmological parameters and the initial differences between the 27 original numbers is analogous to the differences between the local values of the cosmological parameters. The process of getting from the global inhomogeneity to the final, global average is the process of smoothing. In that process, the inhomogeneity of the Universe totally disappears.
How can we obtain the global-average values of the cosmological parameters? Obviously, that could be done in theory by obtaining a weighed average of their values in different regions of the Universe, of its voids and walls. There is a complication, however, since the values of its parameters evolve and change with time, so that their magnitude is not constant, neither on the local scale, nor on the large scale. Consequently, we have two options, the first one of which would be to obtain the average of the original values of the parameter in different regions at the beginning of the Universe, and then see how this average evolves. The second option would be to let the parameter evolve with time in different regions of the Universe and obtain an average of these independent evolving values in the final stages of the evolution of the Universe. Normally, the operation of averaging and the operation of evolving in time are commutative, so that the same result is obtained, independently of the order in which these two operations are executed, as can be appreciated in the following mathematical box.
162 The random numbers are taken from Hubert Blalock, Social Statistics (1960): 437, the numbers are: 10, 09, 73, 25, 33, 76, 52, 01, 35, 86, 34, 67, 35, 48, 76, 80, 95, 90, 91, 17, 39, 29, 27, 49, 45, 37 y 54. 163 The averages are: 30.7, 44.7, 29.3, 62.3, 53, 88.3, 49, 35 y 45.3. 164 The averages are: 34.9, 67.9 y 43.1. 165 The mean is: 48.6.

83

MATHEMATICAL BOX 9. THE OPERATIONS OF AVERAGING AND DERIVING OVER TIME ARE COMMUTATIVE

Normally, the operations of obtaining the average or the derivative are

commutative. In the first equation, we first obtain the average and then the

derivative over time:

t

x3  2x2  3x  6



t

(1 4

x3



1 2

x2



3 4

x

 1.5)



3 4

x2



x



3 4

(85)

and in the second one, we first obtain the derivative and then the average:

t x3  t 2x2  t 3x  t 6



3x2  4x  3  0

 3 x2  x  3

4

4

(86)

but even so, in both cases, we obtain exactly the same result. This is what

we mean when we say that even in the case of non-linear equations the

operations of obtaining the average or the derivative are commutative.

The problem with many cosmological parameters is that they are determined by tensorial equations, where averaging and deriving over time are NOT commutative operations. It is not the same to let an average matter distribution and its corresponding spatial geometry evolve in time, or let the matter distributions of different regions and their corresponding spatial geometries evolve in time and then average the final results. Cosmologists tend to first average matter distributions and its corresponding geometries and then use Einstein’s equations to obtain the homogeneous geometry that results from the evolution in time of this average. Actually, the proper procedure would be to first resolve Einstein’s equations for the different geometries of the different regions of the Universe, then let these results evolve in time, and then average the final results. Since these operations are not commutative, not following the proper order of operations yields erroneous results, according to Wiltshire, referring to previous work of Buchert: “the geometry which arises from the time evolution of an initial average of the matter distribution does not generally coincide, at a later time, with the average geometry of the full inhomogeneous matter distribution evolved via Einstein’s equations.”166

The first one to draw attention to the fact that the operations of averaging and resolving Einstein’s equations are not commutative, was George Ellis, in 1984.167 He showed that the structure of the non-linear equations of general relativity is substantially modified by the process of large scale smoothing. Let us see this point first graphically (see next graph) and then algebraically (see mathematical BOX 10). The following graph taken from Ellis represents three scales in measuring phenomena in the Universe, that is, the scale of stars and solar systems, the scale of galaxies; and the scale of galaxy clusters and walls of galaxy clusters.168

Obviously, the evolution of the Universe in time has taken place in the opposite order, starting with a homogeneous cloud of hydrogen and helium, which existed some 300,000 years after the

166 David Wiltshire, “Exact solution to the averaging problem in cosmology”, arXiv:0709.0732 167 George Ellis, “Relativistic Cosmology: Its Nature, Aims and Problems”, in: B. Bertotti et al., eds., General Relativity and Gravitation, págs. 215-288. 168 George Ellis, “Relativistic Cosmology: Its Nature, Aims and Problems”, in: B. Bertotti, et al., eds., General Relativity and Gravitation (1984): 230

84 Big Bang, up to the voids and walls of galaxies and galaxy clusters, and the stars and solar systems that we observe today, in our inhomogeneous Universe.
Graph.- The operation of averaging cosmic phenomena on an ever bigger scale: scale 1 = details down to stars; scale 3 = galaxies; scale 5 = large scale features
Ellis showed that the operation of averaging and resolving Einstein’s tensorial equations are not commutative: “Thus, a significant problem at the foundation of cosmology is to provide suitable definitions of averaged manifolds169 (..), of metric [ G ] and stress-tensor [ T ] averaging and smoothing procedures, and to show these have appropriate properties”170
We get from the local scale to the global scale by averaging or smoothing. The problem is, as we can see below, in mathematical BOX 10, that in order to get from scale 1 to scale 3, and from scale 3 to scale 5, the tensorial equation that is valid on scale 1, is no longer valid on scale 3, or scale 5. In order to correct the error that occurs when we first average the matter density and its corresponding geometry in different regions of the Universe, and then solve Einstein’s equation, we will have to introduce a term of correction, also known as the backreaction, in the tensorial equations used on scale 3 and on scale 5.
169 “Manifolds” are multiples of different space-time regions of the Universe 170 George Ellis, “Relativistic Cosmology: Its Nature, Aims and Problems”, in: B. Bertotti, et al., eds., General Relativity and Gravitation, p. 231.

85

MATHEMATICAL BOX 10 THE BACKREACTION TERM IN EINSTEIN’S TENSORIAL EQUATION

Einstein’s tensor is:171

G



R



1 2

g  R

 T 





8G c4

T





(87)

and on scale 1, this tensor look as follows:

G1 



R1 

1 2

g1 R1

 T1

(88)

but on scale 3 and 5, the left hand term of the equation, representing the average of

the metric, no longer is equal to the right hand term, representing the mass-energy

average:

G3  T3 (89)

G5  T5 (90)

We therefore need, in equations (89) and (90) a term also known as the backreaction, P3 y P5 , which leaves Einstein’s tensor as follows:172

G3 



R3 



1 2

g3 R3

 T3



P3 

(91)

G5 



R5 



1 2

g5



R5

 T5

 P5

(92)

The theoretical terms of the backreaction, P3 y P5 , are analogous to the term

QD of Buchert, Kolb, Matarrese and Riotto in BOX 11, 15 and 16, but Ellis did not define its empirical value.173 Zalaletdinov, an astrophysicist of Uzbekistan, has given us a precise mathematical definition of Einstein’s ‘average’ tensorial equation, improving, in his view, the previous work by Buchert.174 Einstein’s tensor, applicable on all scales, according to Zalaletdinov, is the following:175

g   M 



1 2



 

g

M



 

T  (micro) 



(Z

 





1 2



 

Q



)

g





(93)

where g   M  is the average curvature tensor and g M   M , the average

curvature scalar.

171 See equation (286) of Appendix VI B of John Aupíng, Origen y Evolución del Universo (2009) 172 George Ellis, “Relativistic Cosmology: Its Nature, Aims and Problems”, in: B. Bertotti et al., eds., General Relativity and Gravitation, págs. 233 173 William Stoeger, Amina Helmi & Diego Torres, in “Averaging Einstein’s Equations: The Linearized Case”, arXiv:gr-qc/9904020, have made an attempt at averaging Einstein’s non-linear equations in linear form. 174 Roustam Zalaletdinov has found the exact way to average Einstein’s non-linear equations in non-linear form, in many publications, of which I mention only two: “Averaging out the Einstein’s Equations”, in: General Relativity and Gravitation, vol. 24 (1992): 1015-1031; and “Averaging problem in general relativity, macroscopic gravity and using Einstein’s equations in cosmology”, in: Bulletin of the Astronomical Society of India (1997): 401-416. 175 Roustam Zalaletdinov, “Averaging out the Einstein’s Equations”, in: General Relativity and Gravitation, vol. 24 (1992): 1025 equation (23)

86

Regrettably, during fifteen years, Ellis’ warnings were not taken into account by cosmologists in the construction of their models. There was a general tendency to estimate the values of the global cosmological parameters at the present time, and then project them back to the origins of the Universe, in simplified models, where Newtonian gravitational dynamics were assumed to be valid at non-relativistic velocities, and the Universe was assumed to be homogeneous, from beginning to end. The assumptions and simplifications of these models did not alarm too many people, and often these assumptions were note even consciously made. However, the problems became more acute at the end of the 90’s, when the apparent acceleration of the expansion of the Universe was discovered by Kirshner, Perlmutter and Riess. Only in the case that the local expansion rates were equal to the global-average expansion rate, as would be the case in a homogeneous and isotropic Universe, the magnitude of this backreaction term would be zero,176 but, as we shall see below, this assumption proves to be invalid.

Thomas Buchert, a German astrophysicist working in France, followed up on Ellis’s

suggestions. I will first define some terms of Buchert’s model, and then present his comparison of homogeneous and inhomogeneous models, both Newtonian and relativistic.177

1) D , a specific spatial-temporal dominion of the Universe;

2) H D , the Hubble constant in this dominion;

3) R , the average curvature of the Universe, represented by Ricci’s scalar; D

4) D , the expansion of the volume of this dominion (of the expansion of elements of the fluid); 5)  , the shear or distortion of elements of the fluid by interaction with surrounding matter;

6) dt  D or

  , the evolution in time of the initial average of the expanding volumes of D

local dominions (first the initial average is calculated, then this average evolves in time);

7)

dt

,
D

the

final

average

of

the

expanding

volumes

of

local

dominions

after

they

have

evolved in time (first different local dominions evolve in time and then an average is obtained;

8) QD , ‘the source’ of non-linear results, also known as the ‘backreaction term’ that measures the discrepancy between perfect homogeneity and the effect of existing inhomogeneities;178

9) aD , the rate of expansion of a dominion of the Universe.

In the case of expanding, spherical, inhomogeneous volumes  , the operations of averaging and evolving in time are NOT commutative, as we saw above, and as a result, the backreaction QD is generated, that represents the difference between the average of the quantities that evolved separately in time, and the final result of the evolution of the average of the original quantities (see mathematical BOX 11).179

176 Because, in that case,

dt

 D

dt D,

so that QD  0 . See Mathematical BOX 11

177 The model of Newtonian (virial) gravitation is not necessary homogeneous. It is possible to construct Newtonian,

inhomogeneous models, see Thomas Buchert, On Average Properties of Inhomogeneous Cosmologies, arXiv:gr-

qc/00010556 (2000): 1-9 178 Thomas Buchert, On Average Properties of Inhomogeneous Cosmologies, arXiv:gr-qc/00010556 (2000): 3 179 Thomas Buchert, On Average Properties of Inhomogeneous Cosmologies, arXiv:gr-qc/00010556 (2000): 4;

Edward Kolb, Sabino Matarrese & Antonio Riotto, “On cosmic acceleration without dark energy”, in: New Journal

of Physics (2006): 6; and idem, “On Cosmic Acceleration from Backreaction,” on-line (2009): 13.

87

MATHEMATICAL BOX 11 HOW THE BACKREACTION IS DERIVED

The difference between dt  D y dt  D produces the quantity also known as the

backreaction QD :

dt

 D

dt D 

( 

 )2 D


D

2  2   D

2 D


D

(94)

 2  2     2  2  2  2   2  2   2

D

D

D

DD

D

D

D

D

D

QD



2 3



d dt

 D

d dt

  2  2 D

D

(95)

Combining equations (94) y (95), we obtain:

  QD



2 3

2


D

2 D

22

D

(96)

The next mathematical box presents a synthesis of Buchert’s model that differs from the Newtonian model only by integrating the term of the backreaction.

MATHEMATICAL BOX 12 SOME FRIEDMANN EQUATIONS IN BOTH HOMOGENEOUS AND INHOMOGENEOUS, RELATIVISTIC MODELS

I present some Friedmann equations, in both homogeneous and inhomogeneous, relativistic models developed by Buchert:

RELATIVISTIC, HOMOGENEOUS RELATIVISTIC, INHOMOGENEOUS

H 2   a 2  1 (8 G ) (97) a 3

H

2





a

2 



8

G

a 3



1 6

R



1 6

QD

=

  3H 2

(99)

8 G

=  

a a

 2 



8 3

G

 eff

(note 180)

(98)

a   4  G  a3

(100)

a   4  G a3





1 3

QD

(101)

t 



3 a a





0

(102)

t



 3 a a



0

(103)

The solution to equation (103) is:   0a0 / a 3 (104)

180 See Sabino Matarrese, Rocky Kolb & Toni Riotto, “On Cosmic Acceleration from Backreaction,” on-line (2009)

This means that,

 eff





 QD  R 16 G 16 G

and

Peff

  QD  R , where the terms with 16 G 16 G

R

indicate the average curvature and the terms with QD , the cinematic backreaction.

88

as can be seen from combining equations (103) y (104): t (0a0a 3 )  3aa 1 (0a0a 3 )  0
 0a0 (3a 4a )  0a0 (3a 4a )  0

(105)

The equation of state of the backreaction, or ’integrability condition’, only exists in

the relativistic model and not in the Newtonian model. Buchert proposes:

 t QD



6

a D aD

QD



t

R

D

 2 aD aD

R D 0

(note 181)

(106)

and Kolb, Matarrese y Riotto, and Wiltshire propose (which is exactly the same):

t (aD6QD )  aD 4t (aD 2 R D )  0 (nota 182) (107)

The following mathematical box compares two inhomogeneous models one Newtonian and one relativistic, according to Buchert. 183

MATHEMATICAL BOX 13, PARAMETERS IN TWO INHOMOGENEOUS MODELS, ONE NEWTONIAN AND ONE RELATIVISTIC

We represent the Hubble constant by H  a / a and the spatial curvature R by the average Ricci scalar. The density parameters, according to Buchert, are:

NEWTONIAN, INHOMOGENEOUS

M    k  Q  1 (108)



D M

8 G  
3H D 2

D

(110)



D 



c 2 3H 02

(note 184)

(114)

RELATIVISTIC, INHOMOGENEOUS

M  k  Q  1 (109



D M

8 G  
3H D 2

(111)

and   0a0 / a 3 (112)

gives



D M

 8 G0a0 3H D 2a 3

(113)

181 Thomas Buchert, “On Average Properties of Inhomogeneous Cosmologies”, arXiv:gr-qc/00010556, p. 12, eq. 45 182 See Edward Kolb, Sabino Matarrese & Antonio Riotto, “On cosmic acceleration without dark energy” in: New Journal of Physics, vol. 8 (2006): 7, eq. (25) & “On Cosmic Acceleration from Backreaction,” online, p.16 & David Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages”, in: New Journal of Physics, vol. 9 (2007): 5 183 Thomas Buchert, “On Average Properties of Inhomogeneous Cosmologies”, arXiv:gr-qc/0001056 (2000): 4, 12. See also a synthesis of Buchert’s model in David Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages”, in: New Journal of Physics, vol. 9 (2007): 9
184 Some authors write   c2 / 3H 02 and Thomas Buchert, in “On Average Properties of Inhomogeneous
Cosmologies”, arXiv:gr-qc/0001056 (2000):4, writes    / 3H 02 . The two versions are compatible if one
takes into account that some authors normalize the equations with c  1 . The same applies to equation (117).

89

 QD



1

3aD2

H

2 D

t
QD
t1

da

2 D

dt1

dt1

(115)



D k





kDc2

H

2 D

a

2 D

(117)



D Q



 QD 6H D2



D k



 RD 6H D2

(116) (118)

Buchert showed that in the Newtonian, inhomogeneous model, the sum of the first three of the four terms found in the definition-equation M    k  Q  1, is 0.99 (note 185) so the fourth term must be very small, that is Q  0.01 (nota 186): “This term, which brought the higher
voltage of having mastered a generic inhomogeneous Newtonian cosmology, shows no global relevance and it seems that we are drawn back to the previous state of low visibility of the standard cosmological models”.187

Roberto Sussman run a series of simulations with the relativistic dust model of LemaîtreTolman-Bondi (LTB), in which he demonstrated that Buchert’s backreaction term can have positive values, in regions with hyperbolic curvature, as well in elliptic dominions —either in isolation or surrounded by a hyperbolic exterior—, that suffer gravitational collapse, and are capable of producing an acceleration of the expansion of a universe without the need for dark energy.188 He stresses, however, that we are dealing with a qualitative evaluation of the model, and that it is necessary, “as a complement to this work, (…) to test numerically how large the effective acceleration, that we have shown here to exist, can be.”189 Besides, according to Wiltshire, “approaches based on the exact LTB models or the exact Szekeres models”, though “immensely useful, both as exact models for isolated systems in an expanding universe or as toy models (...) could only be applied to the universe as a whole if one abandoned the Copernican Principle”, which is more than Wiltshire is willing to do.190

Some cosmologists, notably Edward Kolb, Sabino Matarrese and Antonio Riotto felt that in an

inhomogeneous, relativistic model, the effects of the inhomogeneities appear to be sufficiently

large

to

let the

terms

Q D

and



D Q

may be

to

replace



and



D 

.

In 2006,

they elaborated

Buchert’s model, attributing the apparent acceleration of the expansion of the Universe to the

backreactions of its gravitational perturbations, making the dark energy hypothesis superfluous:191 “Another possibility [different from the standard CDM model] is that the

Universe is matter-dominated and described by general relativity, and the departure of the

expansion rate of the Einstein-De Sitter model is the result of backreactions of cosmological

185 Thomas Buchert, On Average Properties of Inhomogeneous Cosmologies, arXiv:gr-qc/00010556 (2000): 1-9 186 However, even being so small, it has a strong influence on the evolution of the cosmological parameters in time. See Thomas Buchert, Martin Kerscher & Christian Sicka, “Backreaction of inhomogeneities on the expansion: the evolution of cosmological parameters”, arXiv:astro-ph/9912347, p. 17. 187 Thomas Buchert, On Average Properties of Inhomogeneous Cosmologies, arXiv:gr-qc/00010556 (2000): 13 188 Roberto Sussman, “Conditions for back reaction and ‘effective’ acceleration in Lemaître-Tolan-Bondi dust models”, arXiv:0807.1145 (2009) 189 Ibidem, p. 33 190 David Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages”, in: New Journal of Physics (2007): 6 191 Edward Kolb, Sabino Matarrese & Antonio Riotto, “On cosmic acceleration without dark energy” in: New Journal of Physics vol. 8 (2008): 322

90
perturbations. This explanation is the most conservative, since it assumes neither a cosmological constant nor a modification of general relativity.”192
Like Sussman, the authors admit that they have not yet been able to measure the quantitative influence of the backreactions generated by these gravitational perturbations: “The actual quantitative evaluation of their effect on the expansion rate of the Universe would, however, require a truly non-perturbative approach, which is clearly beyond the aim of this paper.”193
One year later, in 2007, David Wiltshire, an astrophysicist from New Zealand, commenting on this essay of Kolb, Matarrese and Riotto, pointed out exactly that: “While perturbative approaches have naturally led to realization of the significance of backreaction, to account for ’74% [of the matter-energy density of the universe by ] dark energy’, the effect of backreaction on the background of the universe would [have to] be so great that a viable quantitative model is beyond the domain of applicability of perturbation theory.”194 The probable magnitude of the backreaction is not large enough to explain the recent, apparent expansion of the Universe and, as a consequence, the backreaction does not serve as a possible substitute of dark energy.
Aseem Paranjape, an astrophysicist from India, applied the mathematical structure developed by Roustam Zalaletdinov, an astrophysicist from Uzbekistan capable of averaging Einstein’s tensorial equations,195 to the problem of the expansion of the Universe196 and discussed his findings with Buchert and Wiltshire. He reached the same conclusion as Wiltshire in his criticism of Kolb, Matarrese and Riotto, that is to say, that the effects of the backreaction are real, but insufficient to explain the recent acceleration of the expansion of the Universe, which is normally attributed to the negative pressure of dark energy: “  Although technically possible, in the real world backreaction does not significantly affect the expansion history of the universe.
 Cosmological perturbation theory is stable against backreaction effects, well into the nonlinear regime.
 Dark energy cannot therefore be an effect of the backreaction of inhomogeneities.”197
192 Ibidem, p. 2 193 Ibidem, p. 15 194 David Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages”, in: New Journal of Physics (2007): 5-6. See also the same article, p. 22 195 Roustam Zalaletdinov has found the exact way to average Einstein’s non-linear equations in non-linear form, in many publications, of which I mention only two: “Averaging out the Einstein’s Equations”, in: General Relativity and Gravitation, vol. 24 (1992): 1015-1031; and “Averaging problem in general relativity, macroscopic gravity and using Einstein’s equations in cosmology”, in: Bulletin of the Astronomical Society of India (1997): 401-416 196Aseem Paranjape, “A Covariant Road to Spatial Averaging in Cosmology: Scalar Corrections to the Cosmological Equations”, arXiv:0705.2380 (2007) and his thesis, The Averaging Problem in Cosmology (2009) 197 Aseem Paranjape, The Averaging Problem in Cosmology (2009): 6

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7.2.- Clocks run at different rates in voids and walls
In 2005, David Wiltshire presented an alternative theory,198 which he called fractal bubble model = FBM ). The name is not that important. Actually, this fractal structure of the Universe does not exist at small scales, as Peebles pointed out in 1993.199 I prefer to think of the Universe as having the structure of a large, expanding sponge: large voids surrounded by large walls of filaments of galaxy clusters. We may drop the theory of the fractal structure of the Universe, without loosing the essence of Wiltshire’s contribution, which we may call the theory of the differential running of watches in voids and walls or, briefly, ‘the time-scape model’. Wiltshire pointed out that after the moment of recombination, some 300,000 years after the Big Bang, the imaginary watches, located in different regions of the Universe, started to differ increasingly, because in regions with high matter density, gravity makes watches run slower, and in voids, faster. Wiltshire revived one of the implications of general relativity, already explained by Einstein himself, who said: ”[L]et us examine the rate of a unit clock, which is arranged to be at rest in a static gravitational field. (..) [T]he clock goes more slowly if set up in the neighborhood of ponderable masses..”200 This lentification of clocks by gravity has three consequences:
1) The wavelength of light coming from objects with high mass density will be redshifted; 2) The velocity of objects that move away from observers located in a ponderable
gravitational field, and the redshift of those objects’ light will be higher when measured by the observer’s wall clock than by its own co-moving clock or clocks in voids. 3) There is a third consequence not mentioned by Einstein, but not less important. Light from a supernova thast passes through the large void that surrounds us is more redshifted than the global-average, because the void expands faster than ghe global-average
This theory of the watches might explain the apparent acceleration of the Universe in the large void surrounding our galaxy cluster, as compared with the global-average deceleration. When the velocity of a supernova at the other end of the void is measured with our watch, or with a globalaverage watch, it might seem to move away from us at a faster rate then when it is measured with a watch mounted on the supernova itself, because our watch moves slower than its one.
However, a problem persisted with the solution offered by Wiltshire in 2005, as he himself observed. If we compare, with Bayesian probability, the ability of both models, that is the standard flat CDM , which includes dark energy, and Wiltshire’s FBM , without dark energy, to explain the same supernovae type 1a data, the CDM model is more probably true than the FBM in the range of 0.2  M  0.5 , which is the empirical range of our Universe.201 The FBM was in need of serious revision.
198 David Wiltshire, “Viable inhomogeneous model universe without dark energy from primordial inflation”, arXiv:gr-cq/0503099 (2005) 199 See James Peebles, “Fractal Universe and Large-Scale Departures from Homogeneity” in: Principles of Physical Cosmology (1993): 209-224 200 Albert Einstein, “The Foundation of the General Theory of Relativity”, in: Annalen der Physik vol. 49 (1916), traducido al inglés en The Collected Works of Albert Einstein, vol. 6 (1989): 197-198 (my underlining) 201 Benedict Carter, Ben Leith, Cindy Ng, Alex Nielsen & David Wiltshire et al., “Type IA supernovae tests of fractal bubble universe with no cosmic acceleration”, arXiv:astro-ph/0504192.

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7.3.- The new relativistic Buchert-Wiltshire paradigm

The problem just mentioned was not resolved until 2007, when Wiltshire proposed his timescape model that integrated Buchert’s backreaction and his own theory on differential clock rates in an inhomogeneous universe.202 This new, integrated model is capable of explaining the apparent acceleration of the expansion of the Universes and other phenomena that have motivated many cosmologists to accept the speculative concept of dark energy to explain them. Wiltshire distinguishes three times or imaginary clocks, that is, slow clocks in gravitationally dense and collapsed regions that measure  w ( w for walls), rapid clocks in the voids with time  v ( v for voids) and a global-average clock with time t (note 203). These three times yield three
differential clock rates, that is dt / d w , dt / d v and d v / d w . Only at the beginning of the
Universe, at the moment of recombination, the Universe was an almost perfectly smooth and homogeneous cloud of hydrogen and helium and, as a consequence, at that time, dt / d w  dt / d v  d v / d w  1. In order to define the passed and present-day parameters as a
function of the global-average time t , Buchert’s formalism is used to average the values of parameters measured with clocks in walls,  w and voids,  v . In the next mathematical box, I
synthesize the Buchert-Wiltshire paradigm.

MATHEMATICAL BOX 14, THE BACKREACTION IN THE NEW BUCHERT-WILTSHIRE PARADIGM

The Hubble constant is defined as the ratio of the expansion velocity vex and
the distance r to a particular object of the Universe ( H  vex r ). With Wiltshire, I define the Hubble constant as a function of global-average time t , both for gravitationally collapsed regions H w (t) , for the large voids Hv (t) and
for the entire Universe H (t) . The last one is also called the subjacent or bare Hubble constant and represents a ponderated average of the former two. Obviously, the three constants have different values ( H w  H  Hv ).

There are two ways to obtain the bare constant, taking into account that

H w / aw  Hv / av , where a is distance. The first equation is:

H (t)  aw aw

 1 daw aw d w

 1 dt daw aw d w dt



dt d w

Hw



 wHw

=



a v av



1 av

dav d v



1 av

dt d v

dav dt



dt d v

Hv

 vHv

(119)

where

Hw



1 aw

da w dt

and

(26 B)

Hv



1 av

dav dt

(120)

202 Ibidem, p. 22, equation (32) 203 David Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages”, in: New Journal of Physics (2007)

93

From equation (119) we obtain:

hr (t)  H w / Hv   v /  w  d w / d v  1/ (d v / d w ) (121)

where the d v / d w function is called the lapse function or the differential

clock rate in walls and voids. The second way to obtain the bare Hubble

constant is by way of a ponderated average of H w and H v :

H (t)  1 3



H



fwHw 

f v H v (nota 204)

(122)

Where the factors f w and fv indicate the volumes of walls and voids as a

proportion of the total volume of the Universe, respectively, so that::

fw (t)  fv (t)  1 (123)

Since, with the passing of time, the volume of the walls does not increase, but

the volume of the voids increases because of the expansion of the Universe,

these two factors f w (t) and fv (t) , are not constant in time. Given the Ellis-

Buchert formalism, the terms  2 and 2 have different magnitudes:

H

H



2 H

 9 fw2Hw2

 9 fv2Hv2

18 fw

fvHwHv

(124)

(31) 2 H  9 f w H w 2  9 f v H v 2 (125)

Let us remind now the backreaction term of Buchert, Kolb, Matarrese and

Riotto and let us suppose, for the time being, a zero value for the shear  . The

backreaction is defined by the difference between dt  D and dt  D , that is

between  2 and 2 , so that:

H

H

QD



2 3

(d

t

 D

dt D )2  2

2 D3

2

2 D3



2 D

0

(126)

By combining equations (124), (125) and (126), we obtain: (33) Q  6 f w H w2  6 fv Hv 2  6 f w2 H w2  6 fv 2 Hv 2 12 f w fv H w Hv (127) and since, by (123),
fw  1 fv (128) it follows that equation (127) can be transformed in (129):
Q  6H w2  6 fv H w2  6 fv H v 2  6H w2  12 fv H w2  6 fv 2 H w2  6 fv 2 H v 2 (129) 12 fv H w H v  12 fv 2 H w H v  Q  6 fv H w2  6 fv Hv 2  6 fv 2 H w2  6 fv 2 Hv 2 12 fv H w Hv 12 fv 2 H w Hv (130)
 (37) Q  6 fv (1 fv )(Hv  H w )2 (nota 205) (131)

204 Peter Smale & David Witshire, “Supernova tests of the timescape cosmology,” en: arXiv:1009.5855v1, p.4 205 This is the same result as obtained by David Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages”, in New Journal of Physics (2007): 21, equation (31), first part of the equation

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We now rewrite equation (119), which is convenient because of the use we are

making of it when we derive the global deceleration of the expansion of the

Universe:

H (t)  1 dt daw aw d w dt

(119)

 dt  H (t) / daw  H (t) awdt

d w

aw dt

daw

(132)

From equations (122) and (132), we obtain:

dt  ( f w H w  fv H v ) awdt  f w H wawdt  fv H v awdt

d w

daw

daw

daw

from equations (119) and (133) we obtain:

(133)

dt d w



fw



fv

Hv Hw

(134)

and from equatios (120) and (134):

dt d w

1

fv



fv

Hv Hw

1

f

v



Hv Hw

1   H w / Hv

 fv (1  Hw / Hv

H

w

/

H

v

)



(135)

and from (119) and (133):

Since:

Hw



Hw

/

H (t)(Hw / Hv ) Hv  fv (1  H w

/

Hv)

(136)

Hv  H w /(H w / Hv ) (137)

it follows, from (136) and (137) that:

Hv



Hw

/ Hv



H (t) fv (1 

Hw

/ Hv)

(138)

Combining (131), (136) and (138), we obtain the backreaction as a function

of the Hubble constants that take into account the differential clock rates:

Q



6 fv (1 

 fv ) [H w

H 2 (1  H w / H v )2 / H v  fv (1  H w / H v )]2



(note 206)

(139)

The interesting thing about equation (139) of mathematical box 14, is that there are two moments in the history of the Universe where the backreaction is zero, and the deceleration has a value of q  0.0635. At the moment of recombination, the inhomogeneities were almost zero
( hr  1), because the Universe was basically a homogeneous cloud of hydrogen and helium, so that the Hubble constants of regions with different degrees of mass density had an almost identical value, generating a zero backreaction.207 And vice-versa, when we approach the end of

206 This is the same result as obtained by David Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages”, in New Journal of Physics (2007): 21, equation (31), second part of the equation
207 H w / Hv  1  Q  0

95
the history of the Universe, the volume of the voids will be so much larger than the volume with galaxy clusters, that, once more, fv  1, and once more, the backreaction will be zero.208 At this particular moment of the history of the Universe, the deceleration is real, with q  0.015348 and the apparent acceleration of the expansion velocity measured with clocks in walls, like our galaxy cluster, is q( w )  0.042785.
The new Buchert-Wiltshire paradigm has important implications for the interpretation of empirical phenomena that have defied cosmology during the last decades. Below we will see how the Buchert-Wiltshire paradigm allows us to disregard dark energy in explaining the apparent acceleration of the expansion of the Universe, the evolution of its large scale structures, and the anisotropies of the Cosmic Microwave Background Radiation CMBR .
The variation of the Hubble constant.- One of the most paradoxical implications of the new paradigm is the variation of the Hubble constant in walls, voids and global-average regions, and if measured with wall-clocks, void-clocks or global-average clocks.
At the beginning of 2007, Buchert pointed out that it was difficult to quantify the effects of the backreaction.209 However, that same year, Nan Li and Dominic Schwarz offered some approximate estimates.210 Below, I reproduce five important results of Li and Schwarz’s study, that take into account observations made by the Hubble Space Telescope in the Key Project.
1) The effects of the backreaction on the variation of the Hubble constant are scaledependent, that is, its variation depends on the inverse square distance ( QD  1/ r 2 ).
2) At a scale of less than 200 Mpc, the influence of these inhomogeneities is much bigger in a relativistic model than in a Newtonian model.
3) The values of the variation of the Hubble constant in the relativistic model coincide with the observations of the Key Project Hubble Space Telescope: “We see that the theoretical band matches the experimental data well, without any fit parameter in the panel.”211
4) The Hubble constant has a comparatively larger value in our neighborhood, at a scale of about 100 Mpc, which constitutes a large void.
5) “[C]osmological averaging (backreaction) gives rise to observable effects up to scales of  200 Mpc. However, it is not sufficient to explain the observed accelerated expansion at this point.”212
Other investigations go much further. Hunt and Sarkar observed that we are located in a huge void with a diameter of 200 to 300 Mpc that expands 20 to 30% faster than would be expected according to the global-average Hubble constant. The expansion acceleration observed by Hunt and Sarkar in this infra dense void is extremely improbable in the context of the standard
208 fv  1 y 1 fv  0  Q  0 . See David Wiltshire, “Cosmic clocks, cosmic variance and cosmic
averages”, en New Journal of Physics (2007): 28 209 Thomas Buchert, “Backreaction Issues in Relativistic Cosmology and the Dark Energy Debate”, arXiv:grqc/06112166 (enero 2007).15 210 Nan Li & Dominic Schwarz, “Scale dependence of cosmological backreaction”, arXiv:astro-ph/0710.5073 211 Nan Li & Dominic Schwarz, “Scale dependence of cosmological backreaction”, arXiv:astro-ph/0710.5073, p. 5 212 Nan Li & Dominic Schwarz, “Scale dependence of cosmological backreaction”, arXiv:astro-ph/0710.5073, p. 5

96

CDM model213 and, in general, several observations in “these real voids are in gross conflict with the concordance CDM model.”214 However, according to the authors, these same data are compatible with a model that correlates the positive variation of the expansion velocity with the differential effects of the backreaction of the inhomogeneities of the Universe.215

In addition to taking seriously the backreaction, calculated by Buchert, by Li and Schwarz and by Hunt and Sarkar, Wiltshire also takes into account the effects of the differential running of time as measured by clocks in voids and walls. In 2008, Wiltshire published his own approximate results of the Buchert-Wiltshire model.216 He showed that the deceleration of the expansion of the Universe is less in voids than in walls. The voids enhance the redshift of the light that passes through them, and since our galaxy is at the center of a huge void, we observe a nearby expansion deceleration that is less than the global-average. In general, the expansion deceleration in walls with time  w differs 5.5 cm per s 2 ( 5.5*1010 m s2 ) from the deceleration
in voids with time  v . This seems little, but the accumulated effect through the entire history of
the Universe, since the Big Bang, is large, that is, the lapse function is 1.42  d v / d w  1.46 if
we do not take into account the backreaction, and d v / d w  1.3800..0063 if we do take it into account, which makes up for a difference of 38%.217 This unequal deceleration in voids and walls implies that the Hubble parameter is not equal in different regions of the Universe.

I will first present the mathematical equation of the Hubble constant and its variation, considered by Wiltshire himself to be one of the most important equations of his model,218 and then reproduce some of the estimates of its differential values.

MATHEMATICAL BOX 15. HUBBLE CONSTANTS IN WILTSHIRE’S MODEL

In mathematical box 14, we already came to know the subjacent, global-average Hubble

constant:

H (t)



dt d w

Hw



dt d v

Hv

(140)

where:

Hw



1 aw

da w dt

and (26 B)

Hv



1 av

dav dt

(141)

Wiltshire defines the average and variation of the Hubble constant as a function of the

213 Paul Hunt & Subir Sarkar, “Constraints on large scale inhomogeneities from WMAP-5 and SDSS: confrontation with recent observations”, arXiv:0807.4508, Figure 5, p. 13 214 Paul Hunt & Subir Sarkar, “Constraints on large scale inhomogeneities from WMAP-5 and SDSS: confrontation with recent observations”, arXiv:0807.4508, p. 1 215 Nan Li & Dominic Schwarz, “Scale dependence of cosmological backreaction”, arXiv:astro-ph/0710.5073, p. 5 216 David Wiltshire, “Cosmological equivalence principle and the weak field limit”, in. Physical Review D, vol. 78 (2008) y “Exact Solution to the Averaging Problem in Cosmology”, in: Physical Review Letters, vol.99 (2007) 217 David Wiltshire, “Cosmological equivalence principle and the weak field limit”, in. Physical Review D, vol. 78 (2008) y arXiv:0809.1183 (2008): 9 218 Private communication of David Wiltshire to the author, April 1st, 2009

97

wall time  w and the global-average time t . 219 The first term of the right side of the

equation represents the Hubble constant as a function of  w and the second term, the rate

of change of the constant ( dt / d w ).

H ( w ) 

1 a

da d w



1 a

da d w



d w dt

d



dt d w

d w



(142)

Since, by definition:

1 da  dt 1 da  dt H (t) a d w d w a dt d w we obtain, combining (142)and (143):

(143)

H ( w )



dt d w

H (t)



d 

dt d w dt



(note 220)

(144)

Equation (144) gives us the Hubble constant and its variation over time, as well as the rate of decrease of the lapse function, and in order to establish its value, one must make measurements with two clocks, that is the one with the global-average-time t and the other one, with the proper time of the observer in a galaxy cluster  w . The value of the Hubble constant is different, when
measured with the same global-average clock in different regions, that is, in walls, in voids, or in the Universe at large; the estimates of its values in this case are H w (t)  34.9 , Hv (t)  52.4 y
H (t)  48.2, respectively. The present Hubble constant also varies, when measured in the same
region, with different clocks, that is the wall-clock, that runs slower, and therefore yields a higher expansion velocity, or the global average clock, that runs faster, and therefore yields a lower velocity, resulting in H0 ( w )  61.7 kms1 Mpc1 and H0 (t)  48.2 kms1 Mpc1 , respectively, a difference of 28% (note 221).

The apparent acceleration of the expansion of the Universe.- Wiltshire also redefines the redshift, taking into account the difference between the observer’s clock and time located in a wall,  w , and the global average time t (see next mathematical box).

219 Wiltshire explained to me that he has omitted the suffix w in his article from equation (38) onwards and stressed
the importance of equation (48): “This equation relates the thing we interpret as the average Hubble parameter H to an underlying bare Hubble parameter H . Both of these are “measurable.” The point is that there is not only an
average Hubble parameter, but a variance in the Hubble parameter, if referred to one set of clocks, such as ours. Equation (42) quantifies both the average and their variance.” 220 David Wiltshire, “Exact solution to the averaging problem in cosmology”, arXiv:0709.0732 (2007): 2, equation 8 and “Cosmic clocks, cosmic variance and cosmic averages”, in: New Journal of Physics (2007): 25, equation 42 221 David Wiltshire, “Cosmological equivalence principle and the weak-field limit”, in: Physical Review D, vol. 78 (2008): 12.

98

MATHEMATICAL BOX 16. THE REDSHIFT IN WILTSHIRE’S MODEL

reTdshheifrtedzsdheiftterzmidneetderbmyinoebdsebryveorbsswerhvoesres

in dense regions watch measures

is defined as a function of the global-average time:

the

1 z



a0



dt d w

a0



dt / d w

(1  z)

and

a

dt0 a dt0 / d w0

d w0

1  z  dt0 / d w0 (1  z) dt / d w

(145)

The redshift of supernovae cannot easily be established, because of the dust in the

host galaxy of supernovae and colour variations, but observers in dense regions observe

larger redshifts of the same supernova then observers in voids or global-average ones.

In the Riess sample of 182 supernovae type 1a, the Wiltshire model is a perfect fit of the relation between distance and redshift, as can be appreciated in the following graph.222 The same cannot be said for other supernovae samples. We will return to this point shortly.

Graph.- The relationship between distance and redshift in the Buchert-Wiltshire paradigm

The Buchert-Wiltshire model results in a new age of the universe (remember that tage  r / v  1/ H ), estimated to be a thousand million years older than is usually assumed in the standard CDM model (see next graph).223
222 David Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages”, in New Journal of Physics (2007): 35 223 Ibidem, p. 37

99
Graph.- The relationship between age of the Universe and redshift,
according to CDM and the new Buchert-Wiltshire paradigm
What is said until now prepares the way to resolve the question of the apparent acceleration of the expansion of the Universe, which is attributed by many cosmologists to the negative pressure of dark energy, and which Kolb, Matarrese and Riotto attribute to the sole influence of the backreaction. In Wiltshire’s view, the backreaction by itself is too small to explain this apparent acceleration of the expansion velocity.224
¿Why do I speak of apparent acceleration? ¿Is the acceleration not real? Yes and no. The fundamental relativistic principle that guides us is the following one, formulated by Wiltshire: “Systematically different results will be obtained when averages are referred to different clocks.”225 If we observe a supernova, that moves away from us, and is located at the other end of the void that surrounds us, the velocity and redshift of its light, passing through this large void, and reaching an observer in a strong gravitational field, will be higher when measured by the observer’s clock than by the clock in the void. The clock in the more dense region runs more slowly than the one in the void. For that reason, the deceleration measured with the wall clock will be different from the one measured with the clock in the void. When measured with the clock in the void, the supernova will appear to move away from us at a slower rate, and its light will appear to have a smaller redshift, then when measured with the wall clock, because of the differential clock rates. A terrestrial observer, measuring with his own, slower running wall clock the redshift of the supernova’s light, might observe an acceleration, whereas an observer located in the void surrounding us, might measure, with his faster running clock, a deceleration “it is quite possible to obtain regimes in which the wall observers measure apparent acceleration,
224 Ibidem, p. 22 225 David Wiltshire, “Cosmic clocks, cosmic variance and cosmic averages”, in: New Journal of Physics (2007): 27