We next consider the case of the trajectory of a light ray in a spherically symmetric static gravitational field. The calculation is essentially the same as that given in the last section, except that light rays travel on null geodesics, so that . The differential equation we need to solve is therefore
In the limit of Special Relativity, the last term vanishes and the equation becomes
The general solution can be written in the form
where b is the closest approach to the origin [ or impact parameter, see Figure 8.2 ].
Figure 8.2: Deflection of light ray
This is the equation of a straight line as goes from to . The straight line motion is the same as predicted by Newtonian theory.
We again solve the General Relativity problem by taking the general solution to be a perturbation of the Newtonian solution:
where we have taken for convenience. It follows that the equation for is:
This equation can be solved by trying a particular integral of the form