607 lines
31 KiB
Plaintext
607 lines
31 KiB
Plaintext
Long Beach 2010
|
||
|
||
PROCEEDINGS of the NPA
|
||
|
||
1
|
||
|
||
Findings Convincingly Show No Direct Interaction between Gravitation and Electromagnetism in Empty
|
||
Vacuum Space
|
||
Edward H. Dowdye, Jr. Founder, Pure Classical Physics Research 8150 Lakecrest Drive, No. 321, Greenbelt, Maryland 20770 e-mail: gravityfindings@verizon.net, ehdowdye@extinctionshift.com
|
||
|
||
Findings show that important fundamental principles of mathematical Physics are consistently misapplied to concepts of gravitational lensing or just simply ignored. The thin plasma atmosphere of the sun represents a clear example of an indirect interaction involving an interfering plasma medium between the gravitational field of the sun and the rays of light from the stars. There is convincing observational evidence that a direct interaction between light and gravitation in empty vacuum space simply does not occur. Historically, all evidence of light bending has been observed predominantly near the thin plasma rim of the sun; not in the empty vacuum space far above the thin plasma rim. An application of Gauss' law clearly shows that, if the light bending rule of General Relativity were valid, then a light bending effect due to the gravitational field of the sun should be easily detectable with current technical mean in Astrophysics at analytical Gaussian spherical surfaces of solar radii, namely, 2R, 3R, 4R, 5R, ..., respectively, where R is one solar radius. An effect of at least one half, one third, one forth, one fifth, etc., respectively, of the observed light bending noted at the solar rim within currently technical means should be easily observable effects. We note that a gravitational bending effect on the rays of starlight is yet to be observed by modern astronomical means at these distances just slightly above the plasma rim of the sun; a clear violation of the light bending rule of General Relativity. Moreover, the events taking place at the center of our galaxy under intense observations by the astrophysicists since 1992, present convincing evidence that a direct interaction between light and gravitation simply does not take place. This highly studied region, known as Sagittarius A*, is thought to contain a super massive black hole, a most likely candidate for gravitational lensing. The past two decades of intense observation in this region have revealed not a shred of evidence for any gravitational lensing. The evidence is clearly revealed in the time resolved images of the rapidly moving stellar objects orbiting about Sagittarius A*.
|
||
Keywords: black hole, gravitational lensing, galactic core, Gauss law, optical reciprocity.
|
||
|
||
1. Introduction
|
||
We shall examine the evidence of gravitational lensing in our region of space near to us, starting with the nearest star to us, our sun. This research has determined that three astronomical regions that are the best chances for an observation of gravitational lensing as predicted by General Relativity, convincingly reveal unseen evidence of:
|
||
1. gravitational lensing in the vacuum space just slightly above the solar plasma rim, easily revealed by applying Gauss' law to the gravitation of the solar mass
|
||
2. Einstein rings in a sky of countless numbers of stars, where the lens and the source are by good chance colinearly aligned with the earth based observer
|
||
3. gravitational lensing in the intensely observed images of the stars orbiting about the super massive object, assumed to be a black hole at the site of Sagittarius A*
|
||
We shall take a closer look at the vacuum space just above the plasma rim of the nearest star to us, namely, the sun, just 8 lightminutes away. We shall examine the nearby stars in our own region of space, less than a hundred light-years away. Here, there are many cases whereby the lens and the source are colinearly aligned with the earth based observer, presenting vast
|
||
|
||
opportunities for the observation of an Einstein ring. Assuming the validity of General Relativity, the night sky should be filled with images of Einstein rings of nearby stars. We shall examine the collected images and the astrophysical data of the stars orbiting about Sagittarius A*, a region thought to contain a super massive black hole located at the center of our Milky Way, right in our own back yard, just 26,000 light-years away.
|
||
A quick search will reveal that in virtually all textbooks on this subject, in our region of space, the effects of gravitational light bending has been historically noted only at the solar rim, the thin plasma rim of the sun. It is currently being taught and printed in all too many textbooks that an observation of a light bending effect, as predicted by General Relativity, has the greatest sensitivity when observed close to the thin plasma rim of the sun. Research shows that this erroneous assumption, which is clearly inconsistent with profound fundamentals in Mathematical Physics, has led to serious misapplications of important fundamentals.
|
||
2. Misapplied Fundamentals on Gravitational Lensing Concepts
|
||
An application of Gauss's law, applied to gravitation as well as to electromagnetism and the principle of optical reciprocity
|
||
|
||
2
|
||
|
||
Dowdye: Findings Convincingly Show No Direct Interaction
|
||
|
||
Vol. 6, No. 2
|
||
|
||
clearly show that a co-linear alignment of the observer, the lens and the source is totally unnecessary for an observation of a gravitational light bending effect, as predicted by the light bending rule of General Relativity. The gravitational effect at the surface of an analytical Gaussian sphere due to the presence of a pointlike gravitating mass that is enclosed inside of the sphere depends only on the quantity of mass enclosed. The size or density of the enclosed mass is not at all important. [12] The study finds that these important fundamentals supported by well founded principles of mathematical Physics are still yet to be applied to the most important astronomical observations that are good candidates gravitational lensing.
|
||
An essential and most fundamental principle in Mathematical Physics may be considered to be that of the Gauss' surface law [1},[4], a Mathematical Physics tool that encloses a gravitating mass particle inside of an analytical Gaussian surface in application to the gravitational field of the particle. An analogy to this principle encloses an electrically charged particle inside of a Gaussian surface in application to the electric field of the charged particle in the discipline of Electromagnetism [4]. The principle of optical reciprocity [2][3] simply states that any photon or wave must take the very same minimum energy path or least time path, in either direction between the source and the observer. A laser beam of a hypothetical laser gun fired by the observer must take the very same minimum energy path back to the source in order to connect with the source. This fundamental principle is an essential tool for the understanding of and design of complex lensing systems in Astronomy and Astrophysics [6]. As a consequence of these combined principles, it is found that the observation of a light bending effect, as predicted by the light bending rule (Eq. (1)) of General Relativity, does not require an exact colinear alignment of the observer, the lens and the source.
|
||
We shall give some clear examples where the application of well founded and proven fundamentals are seriously lacking in the prevailing concepts on gravitational lensing for selected astronomical objects of interest.
|
||
Fig. 1. Gauss' Law applied to Equal Gravitating Masses of Different Radii Enclosed
|
||
|
||
2.1. Gauss Law Applied to a Point-Like Gravitating Mass
|
||
|
||
Any gravitational effect on a light ray due to the presence of a gravitating mass at the impact parameter R would theoretically depend on the amount of Mass M that is enclosed within the analytical Gaussian sphere of radius R as illustrated in Figure 1. Any gravitational effect that would be noted at the surface of the analytical Gaussian sphere should in principle be totally independent of the radius of the mass particle or the density of the mass that is enclosed within the Gaussian sphere. From Gauss's Law, Eq. (2), equal masses of different radii will theoretically have equal gravitational effects at the surface of the Gaussian sphere. The light bending rule
|
||
|
||
|
||
|
||
|
||
|
||
4GM Rc2
|
||
|
||
(1)
|
||
|
||
of General Relativity is essentially a localized 1 R effect. Since
|
||
|
||
we are dealing with astronomical distances, the impact parame-
|
||
|
||
ter R is practically a localized effect only. The predominant gra-
|
||
|
||
vitational effect would take place in the vicinity of the gravitating
|
||
|
||
mass, namely the lens, maximizing at the point where the light
|
||
|
||
ray is tangent to the Gaussian sphere of radius R .
|
||
|
||
A very essential tool of Mathematical Physics known as
|
||
|
||
Gauss' law [1], [4],
|
||
|
||
|
||
|
||
g dA 4GM S
|
||
|
||
(2)
|
||
|
||
is applied directly to the gravitating masses where the gravitational field g is a function only of the mass M enclosed by the
|
||
|
||
spherical Gaussian surface S. [1], [2], [4] The gravitational flux at
|
||
|
||
the surface of the analytical Gaussian sphere is totally indepen-
|
||
|
||
dent of the radius R of the sphere. The idea here is that the gravi-
|
||
|
||
tational field at this analytical Gaussian surface is only a function
|
||
|
||
of the mass that it encloses. [1], [4] Any mass M, regardless of the
|
||
|
||
radius of the mass particle that is enclosed inside of the Gaussian
|
||
|
||
spherical surface of radius R will contribute exactly the same
|
||
|
||
gravitational potential at the Gaussian surface. In Figure 1, the
|
||
|
||
gravitational field points inward towards the center of the mass.
|
||
|
||
Its
|
||
|
||
magnitude
|
||
|
||
is g
|
||
|
||
GM R2
|
||
|
||
.
|
||
|
||
In
|
||
|
||
order
|
||
|
||
to
|
||
|
||
calculate
|
||
|
||
the
|
||
|
||
flux
|
||
|
||
of
|
||
|
||
the
|
||
|
||
gravitational field out of the sphere of area A 4 R2 , a minus
|
||
|
||
sign is introduced. We then have the flux
|
||
|
||
g
|
||
|
||
|
||
|
||
|
||
|
||
gA
|
||
|
||
|
||
|
||
|
||
|
||
|
||
|
||
GM R2
|
||
|
||
|
||
|
||
4 R2
|
||
|
||
4 GM . Again, we note that the
|
||
|
||
flux does not depend on the size of the sphere. It is straightfor-
|
||
|
||
wardly seen that a direct application of Gauss's law to the light
|
||
|
||
bending rule, Equation 1, coupled with the essential principle of
|
||
|
||
optical reciprocity [3], removes any requirement for a co-linear
|
||
|
||
alignment of the light source, the point-like gravitating mass
|
||
|
||
particle (the lens) and the observer for observation of a gravita-
|
||
|
||
tional lensing effect as suggested by General Relativity. [12, 13]
|
||
|
||
From Equation (2) the flux of the gravitational potential
|
||
|
||
through the surface would be the same for all enclosed mass par-
|
||
|
||
ticles of the same mass M, regardless of the size of the mass par-
|
||
|
||
ticle. As a result, as illustrated in Figure 1, each mass particle will
|
||
|
||
produce the very same gravitational light bending effect
|
||
|
||
1 2 , where 1 and 2 are the bending effects on the
|
||
|
||
Long Beach 2010
|
||
|
||
PROCEEDINGS of the NPA
|
||
|
||
3
|
||
|
||
ray of light on approach and on receding the lens, respectively.
|
||
|
||
This of course assumes the validity of Equation (1). This symme-
|
||
|
||
try requirement suggests that 1 2 , and from Equation (1)
|
||
|
||
that
|
||
|
||
|
||
|
||
|
||
|
||
21
|
||
|
||
|
||
|
||
4GM Rc 2
|
||
|
||
.
|
||
|
||
It
|
||
|
||
follows
|
||
|
||
that
|
||
|
||
1
|
||
|
||
2
|
||
|
||
|
||
|
||
2GM Rc2
|
||
|
||
.
|
||
|
||
This
|
||
|
||
says that the total contribution of the light bending effect due to
|
||
|
||
the gravitating point-like mass particle on any given infinitely
|
||
|
||
long light ray is theoretically divided equally at the impact pa-
|
||
|
||
rameter R separating the approaching segment and the receding
|
||
|
||
segment of the optical path. A confirmation of this will be clearly
|
||
|
||
seen later with application of the principle of reciprocity and a
|
||
|
||
demonstration of a simple derivation of the equation of the Eins-
|
||
|
||
tein ring, illustrating the symmetry requirement of General Relativ-
|
||
|
||
ity. From Gauss's law, all that is needed here is the total mass of
|
||
|
||
the particle that is enclosed inside of the analytical Gaussian
|
||
|
||
sphere. The actual size of the gravitating point-like mass particle
|
||
|
||
is totally unimportant and need not be known.
|
||
|
||
2.2. Principle of Optical Reciprocity applied to the Lensed Light Ray
|
||
|
||
Fig. 2. Fundamental Principle of Optical Reciprocity Illustrated on a Lensed Light Ray
|
||
|
||
In any space, the principle of reciprocity, [2], [3] a very fundamental principle of optics, must holds as illustrated in Figure 2. The principle simply states that any photon or wave of light moving on a preferred optical path, from the source to the observer, must take the very same optical path from a hypothetical laser gun of the observer back to the source. As a consequence of this fundamental principle, any additional sources placed along the same preferred optical path will all appear to the observer to be located at the very same image position of the original source. As a consequence of this principle, all light emitting sources on a single preferred optical path will appear to the observer to be colocated at the very same point, appearing as a single light emitting source. This scarcely mentioned fundamental principle of optics is directly applicable to the Astrophysics at the galactic center. The total gravitational light bending effect acting on the light ray upon approach and upon receding a point-like gravitating mass is given by
|
||
|
||
|
||
|
||
|
||
|
||
1
|
||
|
||
approaching the lens
|
||
|
||
|
||
|
||
2
|
||
|
||
receding the lens
|
||
|
||
|
||
|
||
4GM Rc2
|
||
|
||
(3)
|
||
|
||
In this example the gravitating mass M is chosen to be positioned at the midpoint on the line joining the observer and the light source for the simplified special case DL : DSL : DS = 1 : 1 : 2. [5] This simplified special case is illustrated in Figure 3. In the figure the black dotted line represents analytical Gaussian
|
||
|
||
sphere. The red dotted circle that is perpendicular to the observers’ line of site is the impact parameter of the Einstein ring.
|
||
|
||
Fig. 3. Symmetry Requirement of the Accumulative Lensing Effect
|
||
The astronomical distance DL is the distance from the observer to the lens, DSL is the distance from the lens to the source and DS is the distance from the observer to the source. Also again, it is important to note that this case is a simplified special case, where DL = DSL, presented in most academic textbooks. There is no requirement at all that the lens be positioned exactly at the midpoint for an observation of a theoretical Einstein ring. Of course, this again would assume the validity of the light bending rule of General Relativity. This is addressed in the next section dealing with the axis of symmetry of the lensed light ray for corresponding near and far observers. It is readily seen that the axis of symmetry for a given light ray is perpendicular to the line joining the source and the observer only for the special case where the lens is positioned exactly at the midpoint between the source and the observer. The star filled sky reveals a clear lack of gravitational lensing among the countless numbers of stars. There are many cases whereby the lens and the source are colinearly aligned with the earth based observer, thereby presenting the opportunity for the observation of an Einstein ring. Assuming the validity of the light bending rule of General Relativity, the night sky should be filled with images of Einstein rings.
|
||
|
||
2.3. Symmetry Requirement Demonstrated on derivation of Einstein Ring Equation
|
||
|
||
From symmetry we have
|
||
|
||
1
|
||
|
||
|
||
|
||
2
|
||
|
||
|
||
|
||
2GM Rc 2
|
||
|
||
(4)
|
||
|
||
Again, the astronomical distance DL is the distance from the observer to the lens. Since we are dealing with very small angles, from Figure 3, the deflection of the light ray due to the gravitational effect on approach to the gravitating mass is simply
|
||
|
||
1
|
||
|
||
|
||
|
||
R DL
|
||
|
||
|
||
|
||
2GM Rc 2
|
||
|
||
,
|
||
|
||
wherefrom
|
||
|
||
R2 DL
|
||
|
||
|
||
|
||
2GM c2
|
||
|
||
and
|
||
|
||
R2 DL2
|
||
|
||
|
||
|
||
2GM DL c 2
|
||
|
||
12
|
||
|
||
.
|
||
|
||
Solving this for the radius of the impact parameter of the light
|
||
|
||
ray and thus the radius of the Einstein Ring expressed in units of
|
||
|
||
radians we have
|
||
|
||
1
|
||
|
||
2GM DLc 2
|
||
|
||
(5)
|
||
|
||
which is the radius of the Einstein ring in units of radians for a lens place exactly midway between the source and the observer. Note that the gravitational bending effect on the light ray for the
|
||
|
||
4
|
||
|
||
Dowdye: Findings Convincingly Show No Direct Interaction
|
||
|
||
Vol. 6, No. 2
|
||
|
||
approach segment alone is exactly equal to the radius of the solved Einstein ring expressed in radians and is given as
|
||
|
||
1
|
||
|
||
|
||
|
||
2GM Rc2
|
||
|
||
(6)
|
||
|
||
This effect is exactly one half of the total accumulative gravitational effect acting on the light ray for the approach and receding segments. [12, 13] This principle, an essential Mathematical Physics principle on lensing, is often totally missed by researchers attempting to deal with this topic. From symmetry requirement, the integral gravitational effect on a light ray upon approach to a gravitating mass positioned exactly at the midpoint of a line joining the source and the observer, must equal that of the integral gravitational effect on the light ray upon receding the gravitating mass
|
||
|
||
1
|
||
|
||
approaching the lens
|
||
|
||
|
||
|
||
2
|
||
|
||
receding the lens
|
||
|
||
(7)
|
||
|
||
as suggested by Equation 4 and the laws of conservation of energy and of momentum. [12]
|
||
This is a rarely covered fundamental on gravitational lensing in modern academic textbooks. The accumulative gravitational effect along the light ray must sum the total effects of gravity acting on the light ray for both the approach and receding segments of any ray of light passing by a point-like gravitating mass. [12] The total light bending effect is therefore
|
||
|
||
1
|
||
|
||
|
||
|
||
2
|
||
|
||
|
||
|
||
2GM Rc 2
|
||
|
||
(8)
|
||
|
||
mass or lens should see an Einstein ring. Only the mid-field observer depicted in Figure 4 will derive Equation (6) which gives exactly the same value as that given by Equation (5) for a simplified special case, where DL = DSL, which is presented in most academic textbooks. The near-field and the far-field observers depicted in Figure 4 will also derive Einstein ring equations with coefficients corresponding to their unique geometries. Each observer has distinct sets of lensed light rays, each with their corresponding axis of symmetry. It is very important to note that all of the lensed light rays depicted in Figure 4 belong to the very same family of equations derived from the light bending rule (Equation (1)) of General Relativity. Any light ray that is gravitationally bent by a point-like gravitating mass, as predicted by General Relativity, will always have an axis of symmetry which would be perpendicular to the line joining the source and the observer only when the lens is positioned exactly at the midpoint on the line joining the observer and the source. All observers will see, according to General Relativity, an Einstein ring. Figures 4 depicts the geometry of the lensed rays as a function of the position of the observer relative to the lens and the source.12,13 This essential key point is totally missed in many textbooks and lectures on this subject matter.
|
||
3. The Important Fundamentals Applied
|
||
3.1. The Fundamentals Applied to the Thin Plasma Rim of the Sun
|
||
|
||
r 4R
|
||
|
||
|
||
|
||
1 4
|
||
|
||
|
||
|
||
R
|
||
|
||
Fig. 4. The near-field observer sees the largest, most lensed Einstein ring. The far-field observer sees the smallest, least lensed Einstein ring. The axis of symmetry leans towards the near-field observer and away from the far-field observer. The axis of symmetry is perpendicular to the line joining the mid-field observer and the light source where the lens is exactly midway between the observer and the source.
|
||
Regardless of one’s position on gravitational lensing of light, the fundamental principle of optical reciprocity must hold. This is a given. The principle of optical reciprocity simply states that any light ray or a photon of light must take the very same path, along the same minimum energy path, in either direction between the source and the observer as depicted in Figure 2.
|
||
Using the light bending rule1 of General Relativity, it is herewith theoretically demonstrated and graphically illustrated in Figure 4 that all observers of varying distances from a gravitating
|
||
|
||
light bending NOT observed
|
||
|
||
r 3R
|
||
|
||
|
||
|
||
1 3
|
||
|
||
|
||
|
||
R
|
||
|
||
light bending NOT observed
|
||
|
||
r 2R
|
||
|
||
|
||
|
||
1 2
|
||
|
||
|
||
|
||
R
|
||
|
||
light bending NOT observed
|
||
|
||
rR R
|
||
|
||
light bending observed
|
||
rR R
|
||
|
||
R
|
||
|
||
|
||
|
||
4GM Rc2
|
||
|
||
2R
|
||
R
|
||
M
|
||
|
||
Gaussian Surface of 4 Solar Radiii
|
||
Gaussian Surface of 3 Solar Radii
|
||
Gaussian Surface of 2 Solar
|
||
Radii
|
||
solar plasma rim
|
||
|
||
Fig. 5. Gravitational Light Bending as function of various Gaus-
|
||
sian Surface radii Impact Parameters
|
||
Historically, the effect of light bending has been noted only at the solar rim, the thin plasma of the sun’s atmosphere due to a refraction of the light by the solar plasma only. This is a firm and well founded observational fact. Figure 5 illustrates the theoretical light bending effect of the sun at various radii of analytical Gaussian surfaces concentric to the center of the sun as suggested by General Relativity. The past century of astrophysical observations have revealed light bending only at the solar plasma rim. Figure 5 depicts a theoretical solar lensing effect as suggested by General Relativity which is essentially a 1/R effect.
|
||
|
||
Long Beach 2010
|
||
|
||
PROCEEDINGS of the NPA
|
||
|
||
5
|
||
|
||
We note that the total effect of the sun’s gravity on a ray of
|
||
|
||
light on approach and on receding are virtually equal where the
|
||
|
||
relation
|
||
|
||
1
|
||
|
||
approaching the lens
|
||
|
||
|
||
|
||
2
|
||
|
||
receding the lens
|
||
|
||
still holds even though the
|
||
|
||
Earth based observer is relatively close to the sun. [12] Remarka-
|
||
|
||
bly, as it may seem, however, historically the solar light bending
|
||
|
||
effect has been observed only at the solar rim, namely, the light
|
||
|
||
refracting plasma of the solar atmosphere; a well confirmed ob-
|
||
|
||
servational fact. It is widely taught in many Physics lectures that
|
||
|
||
the solar light bending effect noted at the solar rim is done pri-
|
||
|
||
marily to maximize the effect for detection and that gravitational
|
||
|
||
lensing is most sensitive at the solar rim. This erroneous teaching
|
||
|
||
has also contributed to misapplication of the important funda-
|
||
|
||
mentals as well. We note again, that the thickness of the thin
|
||
|
||
plasma shell of the sun, frequently referred to as the solar rim, is
|
||
|
||
very negligible in comparison to the solar radius R.
|
||
|
||
Assuming the validity of the light bending rule of General
|
||
|
||
Relative, the current technical means of the astronomical tech-
|
||
|
||
niques should have easily allowed observations of solar light
|
||
|
||
bending of stellar light rays at different solar radii of analytical
|
||
|
||
Gaussian surfaces, namely at the radius of 2R, 3R and even at 4R,
|
||
|
||
where R is one solar radius, as illustrated in Figure 5. For in-
|
||
|
||
stance, at the analytical Gaussian surface of radius 2R, the pre-
|
||
|
||
dicted light bending effect of General Relativity would have been
|
||
|
||
an easily detectable effect of one half the effect of 1.75 arcsec
|
||
|
||
noted at the solar rim; at the surface of radius 3R, an effect of one
|
||
|
||
third the effect at the solar rim, etc., etc. The equatorial radius R
|
||
|
||
of the sun is approximately 695,000 km. The thickness of the so-
|
||
|
||
lar rim is been recorded to be less than 20,000 km; less than 3
|
||
|
||
percent of the solar radius R. From this, we can easily see that a
|
||
|
||
gravitational lensing effect in vacuum space several solar radii
|
||
|
||
above the solar plasma rim should be a very noticeable effect.
|
||
|
||
3.2. The Fundamentals applied to the Orbit of S2 about Sagittarius A*
|
||
|
||
The past decades of intense observations using modern astronomical techniques in Astrophysics alone reveal an obvious lack of evidence for lensing effects on collected emissions from stellar sources orbiting about Sagittarius A*, believed to be a super massive black hole located at the galactic center of our Milky Way. This is most obviously revealed in the time resolved images collected since 1992 on the rapidly moving stars orbiting about Sagittarius A*. [7–11] The space in the immediate vicinity of a black hole is by definition an extremely good vacuum. The evidence for this is clearly seen in the highly elliptical orbital paths of the stars orbiting about the galactic core mass. The presence of material media near the galactic core mass would conceivably perturb the motion of the stellar object s16 which has been observed to move with a good fraction of the velocity of light. The presence of any media other than a good vacuum would have caused the fast moving stellar object s16 to rapidly disintegrate. Astrophysical observations reveal that s16 has a velocity approaching 3 percent of the velocity of light when passing to within a periastron distance corresponding to 60 astronomical units from Sagittarius A*, perceived to be a massive black hole. This gives solid evidence that the space in this region has to be, without a doubt, an extremely good vacuum.
|
||
|
||
Fig. 6. A Frame from the Time Resolved Imagery recorded by MaxPlanck-Institut für extraterrestrische Physik of Stars moving along Kepler orbits around Sagittarius A* at Center of the Milky Way.
|
||
Fig. 7. Theoretical Fit to the Observed Orbit of S2 about Sagittarius *A and the Predicted Lensing Application of the light bending rule of Equation 4 together with considerations of Gauss’s Law and the principle of optical reciprocity to the data of the observed orbit as depicted in Figure 6, it is clear that some gravitational lensing effect should be detectable in the time resolved images of the orbit of S2 given the current level of today’s observational means. To demonstrate this, we shall assume a worst case in the two possible choices for inclination of i = ±46 given for of the orbit of S2. It is assumed that the projection of the distances between the S2 star and the galactic center mass, indicated in arcsec, represents an instantaneous impact parameter for the light rays coming from the S2 light source, passing by the galactic center mass, and then arriving at the earth based point of observation. The instantaneous
|
||
|
||
6
|
||
|
||
Dowdye: Findings Convincingly Show No Direct Interaction
|
||
|
||
Vol. 6, No. 2
|
||
|
||
impact parameter is used to calculate the worst case expected lensing based on Equation 6 as a function of the position (r,θ) of S2 in the orbit.
|
||
We assume that the elliptical orbit of S2 has the form
|
||
|
||
a 1 e2
|
||
|
||
r 1 eCos p
|
||
|
||
(9)
|
||
|
||
where a = 0.119 arcsec is the semi-major axis, e = 0.87 is the eccentricity, θ is the angle of polar coordinates and θp is the phase angle to orient the orbit according to the best-fit data. Using the best fit observed r given in the first column of Table 2 and expressing r(θ) in units of meters, the radial lengths would serve as a set of instantaneous impact parameters along the orbit of S2, namely the nearest point of approach of the light ray to the galactic center mass, since the orbit is a projection of the actual elliptical orbit of S2 with an inclination of i = ±46. Plugging this result into Equation (4) by setting R = r(θ) we have
|
||
|
||
1
|
||
|
||
|
||
|
||
2GM Rc2
|
||
|
||
|
||
|
||
2 1 eCos p a 1 e2 c2
|
||
|
||
|
||
|
||
GM
|
||
|
||
(10)
|
||
|
||
where the semi-major axis a is expressed in meters. The predicted lensing 1 is expressed in arcsec. Equation (10) is the
|
||
predicted image of the actual orbital path of S2 given by Equation (9), an entirely different configuration as illustrated in Figure 7 where the theoretical fit to the observed orbit of S2 about Sagittarius *A and the predicted lensing is compared.
|
||
In Table 1, some selected points of the observed orbit of S2 and the corresponding predicted lensing of the orbit, based on the light bending rule of General Relativity, are tabulated. It is clear from these calculations that the predicted magnitude of the lensing effect, which is orders of magnitude greater than the observed radial separation between the S2 source and the position of the galactic center mass, should be a very noticeable effect. To date, clear evidence of a gravitational lensing effect based on the light bending rules of General Relativity is yet to be revealed in the time resolved images of the stellar objects orbiting about Sagittarius A*, a region under intense astrophysical observations since 1992.
|
||
Conclusion
|
||
Historically, the light bending effect has been observed only at the thin plasma rim of the sun. A direct interaction between the sun’s gravity and the rays of star light in vacuum space just above the solar plasma rim is yet to be observed. The stellar sky presents vast opportunities to modern Astronomy and Astrophysics to allow for the detection of lensing events due to the large numbers of stellar objects that just happen to be positioned in a near perfect line-of-site to the earth based observers; again of course, this would assume the validity of the light bending rule of General Relativity. With this in mind the entire celestial sky should be filled with Einstein rings. From these fundamentals, we can see that the event taking place at the galactic center of the Milk Way, starring us right in the face, clearly violates the light bending rule of General Relativity. The evidence is clear in the everyday cosmological appearance.
|
||
|
||
|
||
(degrees)
|
||
0 30 70 90 100 120 130 140 150 160 170 180 190 200 240 270 330
|
||
|
||
Observed r (arcsec)
|
||
0.029 0.051 0.159 0.223 0.203 0.118 0.087 0.066 0.051 0.041 0.034 0.029 0.025 0.022 0.017 0.016 0.020
|
||
|
||
Observed r (light-days)
|
||
1.337 2.366 7.238 10.285 9.336 5.423 4.009 3.033 2.366 1.903 1.575 1.337 1.162 1.030 0.763 0.715 0.932
|
||
|
||
Predicted lensing 1
|
||
(arcsec) 63.186 35.700 11.529
|
||
8.214
|
||
9.049 15.579 21.075 27.851 35.700 44.385 53.641 63.186 72.732 81.988 110.794 118.158 90.672
|
||
|
||
Table 1. Data of the Elliptical Orbit Observed and the Predicted Lensing of the Orbit
|
||
|
||
References
|
||
|
||
[ 1 ] G. Arfken, Hans Weber, Mathematical Methods for Physicist, pp. 77-79 (Academic Press, 1995).
|
||
[ 2 ] M. Born & E. Wolf, Principles of Optics , pp. 71, 100-104 (Pergamon Press, London-New York, 1975).
|
||
[ 3 ] R. J. Potton, ”Reciprocity in Optics”, Institute of Physics Publishing, Rep. Prog. Phys. 67, pp. 717-754 (2004).
|
||
[ 4 ] John David Jackson, Classical Electrodynamics, 3rd. ed., pp. 27-29 (John Wiley & Sons, Inc., 1998).
|
||
[ 5 ] R. Blandford, R. Narayan, ”Cosmological applications of gravitational lensing”, Annual Review of Astronomy and Astrophysics, 30, 331 (1992).
|
||
[ 6 ] B. W. Carroll, D. A. Ostlie, An Introduction to Modern Astrophysics (Addison-Wesley Publishing Co., 1996).
|
||
[ 7 ] R. Genzel, R. Schödel, T. Ott, A. Eckart, T. Alexander, F. Lacombe, D. Rouan, B. Aschenbach, (2003); “Near-infrared flares from accreting gas around the supermassive black hole at the Galactic Centre”, Nature 425, pp. 934-937 (1961).
|
||
[ 8 ] F. Melia, H. Falcke, ”The supermassive black hole at the Galactic center”, Anny. Rev. Astrophys 39:309-52 (2001).
|
||
[ 9 ] F. Melia, The Black Hole at the Center of Our Galaxy (Princeton University Press, Princeton, 2003).
|
||
[ 10 ] R. Narayan, ”Black holes: Sparks of interest”, Nature 425:6961, pp. 908-909 (2003).
|
||
[ 11 ] R. Schödel, et al., ”A star in a 15.2-year orbit around the supermassive black hole at the centre of the Milky Way.” Nature, 419, pp. 694-696 (2002).
|
||
[ 12 ] E. H. Dowdye, ”Time resolved images from the center of the Galaxy appear to counter General Relativity”, Astronomische Nachrichten, 328(2), pp. 186-191 (2007); also published online at: http://www3.interscience.wiley.com/search/allsearch.
|
||
[ 13 ] E. H. Dowdye, ”Astrophysical evidence shows no direct interaction between gravitation and electromagnetism in empty vacuum space”, SPIE Paper number: 7421-12, Presented at the Optics and Photonics SPIE Conference 7421, The Nature of Light: What are Photons? III, 3 August 2009, San Diego, Cal., USA.
|
||
|