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Light Deflection in Comparison

The light deflection angle alpha measures the angle of a light beam bent by the sun. alpha = 180°-beta, where beta is the angle between the asymptotes of the beam. According to Einstein, the angle is determined as an integral in the plane where the light beam passes the sun at a distance R. In his book "Grundzüge der Relativitätstheorie" (Principles of Relativity), page 92 contains the formula:
alpha = ∫1/c'(r)·dc'(r)/dx·dy from -∞ to +∞ The derivation uses the expression T = M·G/c², half the Schwarzschild radius. The integral sums the infinitesimal refractive index changes compared to the current refractive index. Einstein used a similar formula as early as 1911, without curvature, with c' =c·(1+Φ/c²) = c·(1-T/R), using the negative Newton potential: Φ = -M G/R. Note that Einstein is an electrician. Think positive!
The relative speed of light c ' = c P. P is either the Schwarzschild potential or the snow globe potential.

With Schwarzschild potential With snow globe potential
c' = c·(1-2T/r)c' = c·1/(1+T/r)² = c·r²/(r+T)²
dc'(r)/dx = c·2T/r² ·dr/dx = c·2Tx/r³ dc'(r)/dx = c·2/(1+T/r)³·T/r²·dr/dx
= c·2Tx/r³·1/(1+T/r)³
1/c'(r)·dc'(r)/dx ≈ 2Tx/r³ mit c ≈ c' 1/c'(r)·dc'(r)/dx = 2Tx/r³·1/(1+T/r)
= 2Tx/r³·(1-T/(r+T))

For the distance R in the x direction: x = R and z = 0, => r = SQRT(R²+y²),

alpha = ∫2TR/(R²+y²)3/2·(1-F)·dy von -∞ bis +∞

F is either 0 or the factor T/(r+T), positive and at most T/(R+T), i.e., about 2·10-6. This is below the measurement accuracy. The uncertainty for the Schwarzschild potential, at a maximum of 4·10-6, is also below the measurement accuracy. To calculate the integral, one can use symmetry, i.e., twice the integral from 0 to ∞.
d( y/SQRT(y²+R²))/dy = (1·SQRT(y²+R²)- y·2/2·y/SQRT(y²+R²))/(y²+R²)
= 1/SQRT(y²+R²)-y²/(R²+y²)3/2 = (y²+R²-y²)/(R²+y²)3/2 = R²/(R²+y²)3/2.
The calculation then yields alpha = 4·T/R.

Ludwig Resch