Corrections to Einsteinís equation for the deflection of light by the sun


One of Albert Einsteinís most famous predictions was the deflection of starlight by the gravitational effect of the sun.† His famous equation for the amount of deflection was based on his knowledge that the velocity of light is reduced in a gravitational field.† He assumed that this velocity reduction was a function of distance from the source (1/R), the same as gravitational attraction.† But modern experiments show that the reduction in velocity of light due to gravity is actually logarithmic (1/ln(R)).† In this brief paper we modify Einsteinís equation for the deflection of light by a gravitational source to incorporate this new knowledge.† As a result, the deflection of light by a gravitational field is significantly larger at a distance than provided by Einsteinís original equation, which leads to some surprising new knowledge of the universe.



Einsteinís logic in deducing that the sunís gravitational field would deflect light is easily summarized in his writings:


So, it is absolutely true that the speed of light is not constant in a gravitational field [which, by the equivalence principle, applies as well to accelerating (non-inertial) frames of reference]. If this were not so, there would be no bending of light by the gravitational field of stars. One can do a simple Huyghens reconstruction of a wave front, taking into account the different speed of advance of the wavefront at different distances from the star (variation of speed of light), to derive the deflection of the light by the starĒ.

So Einstein used his knowledge that the speed of light is reduced in a gravitational field to derive his famous equation:


Deflection of light by the sun = 4GM/c2R


where M is the mass of the sun, G is the gravitational constant, c is the speed of light and R is the distance from the surface of the sun, in solar radii.† For light just grazing the sun this equation provides a deflection of 1.75 arcseconds.


Since the attractive force of gravity decreases directly with distance from the gravitational source (1/R), Einstein logically assumed that the reduction of the speed of light in a gravitational field would decrease in the same manner and hence the term 1/R in his equation. BUT THIS HAS BEEN FOUND NOT SO!


In 1964, Dr. Irwin Shapiro published a letter titled Fourth Test of General Relativity.[1] In this letter he described an experiment using radio waves bounced off a distant planet to show that the radio waves would be increasingly delayed as they passed nearer and nearer to the sun.† This experiment, now known as the Shapiro effect, was highly successful and has been repeated many times using various targets to verify the effect.† It demonstrates that the velocity of electromagnetic waves (light) decreases as the wave gets nearer and nearer to the sun.


The importance of the Shapiro effect is to demonstrate that the reduction in velocity of light in a gravitational field is a logarithmic function of the distance from the source (1/ln(R) and not the expected function (1/R).† It is a long-range effect! †We may then rewrite Einsteinís famous equation as follows:


Deflection of light by the sun = GMK/c2ln(R)

where K is a constant.


This equation is a better fit to solar eclipse result, as seen below. †Overall, eclipse results have resulted in an average deflection of nearly 20% greater than predicted by Einstein.† More importantly, it suggests that the deflection of light by distant objects is probably much larger than previously thought. The difference between the two functions is illustrated below:


Figure 1 - Comparison between the accepted version of gravitational deflection (1/R) and the logarithm version (1/ln(R)). The logarithmic version results in much greater deflection at a distance.


Figure 2 - Results from eclipse experiments in 1922 and 1929. The dashed line is Einsteinís prediction, while the dot-dashed line is a least-squared fit of the actual data

Clearly the deflection of light by a gravitational field is much larger at a distance than previously thought. This results in some amazing effects in our observable universe.† This new knowledge gives rise to many optical illusions in the observable heavens.

The effect of photon velocity

I hesitate bringing this subject up since it so controversial, but I think it might be important.† The 900 pound gorilla in the deflection equation is the term 1/c2.† It is an extremely small factor, and clearly important. The question arisesóis c a constant, or is it the velocity of the photon being deflected?† Since, as even Einstein claimed, the velocity of light is reduced in a gravity field, if c is actually the velocity, then c would be lower in the vicinity of a truly massive object, and the term 1/c2 would be decreased and the amount of deflection substantially higher than if c were just a constant.† My math skills are insufficient to answer this question, but my observations of apparent significant deflection in the universe suggests to me that the deflection of light by massive objects is quite large, suggesting that the term c should be replaced by a velocity term (vg)?


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1.     Observational Results on the Light Deflection and on Red-shift in Star Spectra

by Robert J. Trumpler (University of California, Berkeley)

2.     .The Relativity Deflection of Light
Charles Lane Poor
Journal of the Royal Astronomical Society of Canada, Vol. 21, p.225

[1] Fourth Test of General Relativity

†† Phys. Rev. Lett. 13, 789 Ė Published 28 December 1964

†† Irwin I. Shapiro