The bending of light
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 ].
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
This gives [ Assignment 7 ]
so the full solution is
Let us now calculate the deflection of a light ray from a star which just grazes the sun [ see Figure 8.3 ].
When , , so
at the asymptotes and , and taking we get:
The total deflection is :
This works out to be about 1.75'' and was confirmed by Eddington in 1919 during a solar eclipse.
The distance of closest approach corresponds to an angle
Now from the diagram [ see Figure 8.4 ] above we have
since both and are small. It follows that the impact parameter can be written as