Twisted Tales from General Relativity: The Deflection of Light

Supposedly one of the greatest predictions of twentieth century physics, was Einstein's prediction of the deflection of light from a massive star like the sun. His pre-relativistic calculation gave half the value that his post-relativistic calculation gave, and this was heralded as a great triumph for general relativity. Yet, there are so many loop holes in the calculation that it's amazing that no one has ever questioned it. 

The correction term that supposedly gives the angle of deflection has the $1/r^3$ dependency of a quadrupole moment although the numerical coefficient is the product of the Schwarzschild radius, $\mathcal{R}=2GM/c^2$, and the angular momentum, $L$. Without the correction term the equation of the trajectory is

$$\phi=\int_0^r\frac{\Delta dr}{r\sqrt{r^2-\Delta}}=\cos^{-1}(\Delta/r),$$

where $\Delta=L/c$ is the so-called impact parameter. However, this is the equation of a straight line with no central mass to speak of. Why should a term of third-order give the correct deflection of light about the sun? 

The mass comes from the Schwarzschild coefficient $1-\mathcal{R}/r$ when it multiplies the centrifugal potential $L^2/2r^2$. Einstein's result,

$$\phi=2\frac{\mathcal{R}}{r_m},$$

cannot be considered as a "modification" of the classical result

$$\phi_0=\frac{2GM}{c^2\Delta},$$

which is the Rutherford expression,

$$\cot(\phi/2)=\frac{\Delta}{\mathcal{R}},$$

for small angle, $\phi$, where $GM$ replaces the atomic number and the charge, and the speed of light is introduced for the speed of the deflecting particle. The deflection occurs for a hyperbolic orbit,

$$r=\frac{\ell}{1+\epsilon\cos\theta},$$

where $\ell$, the semi-latus rectum, is the product of the eccentricity, 

$$\epsilon=\sqrt{1+\frac{\Delta^2}{\mathcal{R}^2}},$$

and the directrix.

In the Einstein expression, $r_m$, which is the radius of the sun, is confused with the distance of closest approach, 

$$r_m=\frac{\mathcal{R}}{2c^2}(\epsilon-1)\approx\frac{\Delta^2}{2\mathcal{R}}.$$

For if we equate the radius of the sun with the impact parameter, we would obtain an angular momentum $L=6.95\times 10^{10}\mbox{cm}\times 3\times 10^{10}\mbox{cm/sec}$ which is of the order of $10^{21}\mbox{cm^2/sec}$, clearly an outrageous magnitude. 

In the Rutherford calculation, the target is a point target so that if we were to use the impact parameter, and not the radius of the sun, the deflection would be

$$\phi=2\left(\frac{\mathcal{R}}{\Delta}\right)^2,$$

which is precisely the second order terms in $\mathcal{R}$ that Einstein neglected. 

Moreover, the angular momentum,

$$\frac{r^2\dot{\varphi}}{1-\mathcal{R}/r}=L,$$

is not a constant of the motion in the Schwarzschild metric. But, in the weak-field approximation which is used in the calculation of the advance of the perihelion of Mercury, it is retreated as a constant. This means neglecting terms of the order $\mathcal{R}/r$, the same order as both the advance of the perihelion of Mercury and the deflection of light about a massive body. 

Moller, in his Theory of Relativity, [p. 350] claims that the left-hand side of $L$ "cannot in general be interpreted as angular momentum, since the notion of a 'radius vector' occurring in the definition of the angular momentum has an unambiguous meaning only in a Euclidean space." Does he mean to throw out the entire calculations of the perihelion shift and the deflection of light because these are of the same order as the term he would neglect in the expression for the angular momentum?

For if we multiply the above expression for $L$ by the angular speed $\dot{\varphi}$, we get

$$\frac{r^2\dot{\varphi}^2}{1-\mathcal{R}/r}=\frac{L^2}{r^2}\left(1+\frac{\mathcal{R}}{r}\right),$$

neglecting, like Einstein, higher powers of $\mathcal{R}/r$, and call, in agreement with Moller, $L=r^2\dot{\varphi}$. Then the correction term is exactly the term implicated in both the deflection of light and the advance of the perihelion of Mercury. You can't throw that term out at one stage of the calculation and keep it at the next!

Therefore, Einstein's expression for the deflection of light cannot be considered as a relativistic generalization of scattering angle of a hyperbolic orbit, like the classical calculation carried out by Rutherford with $GM$ for the charge and $c^2$ for the energy. For the hyperbolic orbit is formed from the gravitational attraction and centrifugal repulsion in the case where the total energy is positive. (A negative total energy would give a closed, elliptical orbit.)