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Gravitational Bending of Light
S. Baranow

In 1919 a total solar eclipse of the sun was predicted and Eddington(1) saw a chance to compare Newton's Law of Gravitation with Einstein's General Theory regarding the bending of starlight as it passed near the sun. He made the necessary calculations for each law (theory) and one team was sent to Brazil and another to Africa to gather astronomical data. It is the purpose of this paper to show that Eddington's calculations were incorrect. Unfortunately he does not take the reader through the steps by which he arrives at his results nor does he explain what the polar symbols represent. At any rate, he determined that according to Newton the bend angle should be 2Gm/rC2and according to Einstein 4Gm/rC2. This is 0.87 arc seconds for the former and 1.75 arc seconds for the latter. The experimental results were 1.61 +/- 0.30 arc seconds for the African team and 1.98 +/- 0.12 arc seconds for the Brazilian team. The Royal Astronomical Society which sponsored the the expeditions concluded that the results were inconclusive because of the experimental difficulties and tabled any future expeditions .

Newton's Law of Gravitation

Newton's Law states that in rectangular coordinates Gm/(x2 + y2) = A. My purpose was to determine the slope of the x-y curve at the earth and double it which gives the total angle of deviation in radians. This proved to be fraught with difficulties. Then I realized that dy/dx is equal to Cy / Cx which in turn is practically equal to Cy / C. Therefore, all that is really required is to find Cyas a function of x. This would give the entire range of angle deviations from the sun to at least the earth.

I set up a standard x-y diagram with the sun's center at (0,0) and the point of the light's tangency at (0,-r). The key point for integrating the equation lay in recognizing that y in the interval from the sun to the earth is constant to within 0.2%. Therefore, I set y = r. Although this means dy/dx = 0, it was found that Cy is dependent only on the absolute value of y which makes for a feasible solution.

If we write Newton's equation as Gm/(x2 + r2) = A, then we have to get A into a form involving dCx/dx. This can be accomplished through the use of the following equations:

C2 = Cx2 + Cy2
0 = CxAx + Cy2
A2 = Ax2 + Ay2
Ax = CxdCx/dx

Gmdx/(x2 + r2) = CxdCx/[ 1 - Cx2/C2]-1/2]

Integrating the left sde from zero to x and the right side from C to Cx we obtain

Gm arctan(x/r)/r = C2[ 1 - Cx2/C2]-1/2 = CCy

Therefore, the total bend angle is 1.57 Gm/rC2 + Gm arctan(x/r)/rC2 where the left-hand term is the bend angle from a star to the sun and the right hand term the bend angle from the sun henceforward.

At the earth, this gives a total bend of 1.37 arc seconds. Following Eddington, this results in a total bend of 2.74 arc seconds for the General Theory. I trust that this paper puts to bed any further claims that the results obtained in 1919 helped support the veracity of Einstein's General Theory.

Nomenclature

C, Cx, Cy ----Speed of light and its components
A, Ax, Ay ----Acceleration of light and its components
G ----Gravitational constant 6.67 x 10-8
C ----Speed of light 3.00 x1010
m ----Mass of sun 1.99 x 1033
r ----Radius of sun 6.96 x1010-

All values are in the cgs system

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1 Newman, The World of Mathematics, 1956, vol 2, pp 1094-1104