From the Chalkboard

This is topic # 06

posted May 2005

last updated: 18 March 2006

"Gravitational Light Bending or just plain Optical Refraction?"

"Fundamental Flaws Encountered in Modern Lensing Tools"

"Not-yet-seen Gravitational Distortions in the Undistorted Orbital Images of Stars orbiting about Sagittarius A*"

now explained with convincing astrophysical evidence

"Extinction Shift Principle predicts NO direct gravitation - electromagnetism interactions!"

We shall now examine a popular tools used by astrophysicists and cosmologists which is based on and derived from the principle of light bending of General Relativity and the basic assumptions made on gravitational light bending. Conventional Physics teaches us that there is a direct influence of gravitation on the path of rays of light, gamma rays, infrared rays and all other forms of electromagnetic waves. Since light requires no medium in which to propagate and the gravitational influence on the electromagnetic waves is, as implied by General Relativity, direct, then some fundamental principles of optical lenses must certainly be applicable here as well.

Most assuredly, since gravitation as well as electromagnetism obeys basically the same fundamental principles, namely, the laws of conservation of energy and the laws regarding the path of least-time as well as the laws of minimum energy path, it should also be proper to assume that the principle of reciprocity of any light ray under the influence of gravitation should also apply. For clarity do review the previous topic "Light Bending at the Galactic Core?": Topic #05

As stated previously in Topic # 05, an observer would view the image of all sources that happen to lie on an optical path leading back to the observer's telescope at a single image location. Thus, the photons emitted from any source that happens to lie on a light ray connecting the observer to the source of the image will take the very same path back to the observers telescope as the photons coming all the way from the source of the image. Since both sources, the near-field and the far-field star, lie on a single optical path, they will both appear together a the same image location. This is a very important principle in optics. As a consequence of axial symmetry, the photons from the near-field star will be subjected to exactly one-half the gravitational light bending effect as the photons coming all the way from the far-field star, again assuming the validity of General Relativity.

For review, the following illustration from Topic # 05, depicts an application of the principle of reciprocity.

It is clearly seen from the illustration that the equation involving the parameters D_L , D_SL and D_S is a clever attempt to scale the light bending effects for the relative distances of the lens-to-observer, the source-to-lens and the observer-to-source, respectively.

This tool, used by many astrophysicists and cosmologists (See Reference: Blandford, R. & Narayan, R. 1992, "Cosmological applications of gravitational lensing", Annual Review of Astronomy and Astrophysics, 30, 331), seems to work very well, but, however, there are some overlooked subtleties involving the sources of emission that happen to be positioned in the vicinity of the gravitational lens. The important principle of reciprocity was apparently not treated or considered in the design or derivation of the lensing tool (See Figure below).

Ray Geometry of Gravitational Lensing

Researchers using this tool would have to be thoroughly familiar with the basic principle of light bending equation

derived using General Relativity considerations!  The geometry as well as the Physics of conservation of energy, the Physics of the minimum energy path, minimum time and the principle of reciprocity must all be considered for a correct interpretation of these astrophysical events. 

In the above Figure, "Ray Geometry of Gravitational Lensing", the angle

corresponds to the radius of the Einstein ring, a circular image of a source that lies anywhere on an optical axis of the observer and the lens, as is predicted by General Relativity. This angle is expressed terms of the adjustable parameters D_d , D_ds and D_s used by Blandford, R. & Narayan, R. (See Reference) as follows:

The calculated radius of the Einstein ring would be expressed in units of radians for a given lens mass M where the mass in the units of Kg. Using M = 3.7±1.5 million solar masses for Sagittarius A*, the above calculated angle would be 7.73E-006 radians which assumes a placement of the gravitational lens half-way between the observer and the source. At a distance of D_d = 26,000 Light-Years, the impact parameter would have a radius of 1.616 arcsec as viewed from Earth, an effect that would be significantly visible with modern optical instruments.  This impact parameter would correspond to a distance of 53 Light-Days (the nearest distance that the light ray passes by the core mass) coming from a point source that is the distance of D_ds = 26,000 Light-Years on the other side of the gravitational lens.  Keep in mind that the orbit of the star S2 about Sagittarius A* with a semi-major axis of 5.5 Light-Days falls well within a field of view, from Earth based observatories, of only 0.12 arcsec, an order of magnitude far less than the expected impact parameter of the Einstein ring, within a very narrow cone-of-view!

Interestingly enough, the light that grazes an analytical Gaussian surface sphere whose radius is 5.5 Light-Days, an impact parameter of an order of magnitude less than the above, enclosing the same core mass M, will cause a light bending effect, according to General Relativity, of at least 15.4 arcsec.  This would certainly be a very visible effect with the use of modern optical instruments.

The important note and main purpose of this Topic # 06 is to show that this frequently used tool, used by astrophysicists and cosmologists to simplify the lens problem as a function of F(  D_d , D_ds , D_s ), appears to breaks down as the parameter D_s is reduced to diminishing values, corresponding to sources that are in the vicinity of the gravitational lens.  For instance, in the case of the orbiting star S2 about Sagittarius A*, the measured semi major axis for S2 is around 5.5 Light-Days and the distance of Sagittarius A* for the Earth based observer is around 26,000 Light-Years. This yields a relative relationship for the astronomical distances as

[ D_ds : D_s ]  = [  5.5 Light-Days : 26,000 Light-Years ]

It is clear that any use of the tools of astrophysics and cosmology that does not consider or treat the important principle of reciprocity can not be used to solve the problem of gravitational light bending of light coming from sources moving about or in the vicinity of gravitating masses such as Sagittarius A*.

The following illustration depicts an example of two observers at two different distances from the lens (two different D_d parameter values). The background sources are stars at distances beyond the lens (distances greater than D_s).  It is graphically illustrated that if the principle of reciprocity holds, assuming the validity of General Relativity, there is always a corresponding impact parameter for any given observer, whether near or far. 

Observers note a hypothetical Einstein Ring of a hypothetical far-field source

from Near to and Far from the lens and principle of optical 

superposition of hypothetical near-field source

Most importantly we can see that the gravitating mass enclosed inside of the analytical Gaussian surface sphere, corresponding to a particular observer of a selected light ray, would have a given magnification and a given radius of the theoretical Einstein ring. At the distance D_d = 26,000 Light-Years, a calculated radius of 

of an Einstein ring for a source at the same distance of D_ds = 26,000 Light-Years beyond the lens is 1.616 arcsec! Note: this is nearly an order of magnitude greater than the field of view of the entire region of 0.12 arcsec encompassing the observed orbital motion of  S2 with a semi-major axis of 5.5 Light-Days!

The image of a star that happens to pass over or sit on a light ray that arrives at the observers telescope should, according to the principle of reciprocity, be seen lying directly on the corresponding Einstein ring, as illustrated.  In the illustration, the near observer sees the image of the orbiting star passing directly over the Einstein ring, while the actual stars position just crosses the Gaussian surface that corresponds to the near observer. The far observer would see the same event only at the time the star crosses the Gaussian surface that corresponds to the far observer. Let us assume now that this event actually happens to take place and the near-field source, (the orbiting blue star), just happens to lie directly on a path (selected light ray) of the photons coming from the far-field source, namely, the red star. The following image illustrates this, using a very important fundamental principle of optics!

Graphical Demonstration of Hypothetical Einstein Ring of a far-field source

and principle of optical superposition of hypothetical near-field source

VERY IMPORTANT NOTE: Each of the rays of light, assuming a point-like gravitating lens, has a theoretical Axis-of-Symmetry {a perpendicular axis to the line connecting the red star and the observer.} It is easily seen, from inspection above, that the gravitating lens serves to rotate this Axis-of-Symmetry for varying impact parameters. This, along with many other fundamental principles are severely missed when utilizing ray geometry techniques of gravitational lenses and many other overly simplified lens tools.

It is easily seen here that the blue  photons coming from a hypothetical near-field source, (the blue star sitting on the light ray),  must take the very same path as the red photons coming from the far-field source, namely, the hypothetical red star. As a consequence, to any observer, the image of the blue star must lie directly on the hypothetical Einstein-ring of radius 1.616 arcsec as depicted in the illustration.  In any event that the blue star crosses a given light ray of photons that would contribute to an optical image of an Einstein-ring, fundamental principles of optics dictate that any observer would see the image of the blue star lying directly on the hypothetical Einstein-ring as depicted in the above illustration.

MOST IMPORTANT NOTE: The recorded events of the orbiting stars about Sagittarius A* is well within a field-of-view of less than 0.12 arcsec.  The calculated impact parameter for a hypothetical Einstein ring, as depicted above, is 1.616 arcsec, a factor of nearly 13 times the semi-major axis of the orbit of S2! Thus, fundamental principles of optics implies that gravitational lensing effects for the region enclosing the orbit of S2 must severely distort the true geometry of the orbital motion of these stars.

With modern optical instruments of today, many opportunities for this observation has already occurred and is still constantly occurring at the region of  Sagittarius A*. The images collected from this region since 1992 yields strong observational evidence in astrophysics that the lensing effect as predicted by General Relativity does not occur!

Assuming the validity of General Relativity, theoretically, there should be no visible emissions coming from background stars inside of the Einstein ring! There should be a complete VOID of all emissions from sources of light at distances beyond the lens when viewing inside of the ring; a completely dark region. As a consequence of the lensing effect, there should be only occasional emissions from sources that happen to pass directly in front of the lens; near-field sources that happen to pass through this very narrow cone-of-view.

There is no observational evidence for an occurrence of any of these effects to date at the site of Sagittarius A*! 

AN ADDITIONAL IMPORTANT NOTE: The thin atmospheres of planets and the plasma rims of stars can act as light refracting media that may cause occasional lensing effects, nothing at all to do with a direct gravitational interaction with the rays of light! Such an occurence would clearly necessitate a colinear (very straight line) alignment of the earth based observer, the light source and the lens. We have clearly seen from the Physics of enclosing the gravitating mass inside of an analytical Gaussian surface that a colinear alignment geometry for the observation of gravitational light bending, as predict by General Relativity, is totally unnecessary!

CONCLUSION and CONSEQUENCE

1) A misuse of the above cited Ray Geometry Gravitational Lensing Tool (Ref Blandford, R. & Narayan, R. 1992) has apparently led to grossly and erroneously misinterpretations of the astrophysical phenomenon surrounding the rapidly orbiting stars about Sagittarius A*. 

2) Use of grossly neglected fundamental principles of optics reveals that the images of the rapidly moving stars about Sagittarius A*, collected since 1992, is in direct contradiction to the expected gravitational light bending effects predicted by General Relativity.

Also, a very much misunderstood fundamental concept is what the equation

of General Relativity really implies! The calculated bending angle a is directly proportional to

and where x is the impact parameter of the path of the photons and the gravitation mass.  General Relativity implies that the bending effect of the light ray would be directed towards the the gravitating mass (a maximum effect at a minimum x )! The predominate bending effect would actually take place in the vicinity of the lens as is theoretically suggested by this equation. Thus, the masses of ALL THE OTHER STARS should have negligible effect or no effect at all on the light rays enclosed within the narrow cone-of-view. At present, at least since 1992, the light rays within this narrow cone have had sufficiently enough clear passage to the telescopes of our observatories. No apparent lensing by other gravitating masses of significance is evident, as is apparent in the undistorted images of the Kepler motion of S2!

Note also that any light bending event would serve only to alter the trajectory of photons influenced by the gravitational lens. The dust clouds, the star dust, the thin atmospheres of hydrogen and helium or the dark matter should have virtually no effect on the observations.  It is very clear that, assuming the validity of General Relativity, the predominate lensing effect would actually take place in the proximity of the point-like super gravitating mass!

The extinction shift principle, derived from pure classical considerations in Euclidean Space, predicts there can be no direct interaction between gravitation and electromagnetism! The recently discovered astrophysical phenomenon taking place at the site of Sagittarius A* seems to confirm this convincingly. A quick search will reveal there is a clear lack of evidence for gravitational lensing at the site of black holes.  The astrophysical observations made since 1992, with all of the recorded images of the undistorted images of the Kepler motions about Sagittarius A* seems to supports this claim.

Do review [topic #05]

"Light Bending at the Galactic Core?; Gedankenexperiment", posted July 2003 

for additional clarity on this subject.