**Corrections to Einstein’s equation for the deflection of
light by the sun**

**Abstract**

**One of Albert Einstein’s 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 Einstein’s 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 Einstein’s original
equation, which leads to some surprising new knowledge of the universe.**

** **

**Einstein’s
logic in deducing that the sun’s 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
star”.
*

**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/c ^{2}R**

** **

**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 **

** **

**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 Einstein’s famous equation as follows:**

** **

**Deflection
of light by the sun = GMK/c ^{2}ln(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 Einstein’s 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.**

**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/c ^{2}. It is an
extremely small factor, and clearly important. The question arises—is 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/c^{2} ^{ }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 (v_{g})?**

** **

** **

**References**

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

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

**http://retro.seals.ch/cntmng?pid=hpa-001:1956:29::942-**

**2. ****.The Relativity Deflection of Light
Charles Lane Poor
Journal of the Royal Astronomical Society of Canada, Vol. 21, p.225
1927JRASC..21..225L**

[1]
**Fourth Test of General
Relativity**

** Phys. Rev. Lett. 13, 789 –
Published 28 December 1964**

** Irwin I. Shapiro**