Here is a Wiki link to some methods for solving the metric. I don't understand half of them and the other half don't make sense to me. For instance, I don't see that it is mathematically rigorous to divide by ds or dτ when either is zero for a photon. Even if we were to consider a particle that travels just under c, we are not actually using the particle as the origin as we would for SR, where the distance of the particle from the particle's own origin would be zero, leaving just ds^2 = c^2 dτ^2 in terms of the proper time of the particle, but rather we are using the distance of the particle from the gravitating body, so even in the particle's own frame, this distance would be non-zero and so should be included in the metric. Anyway, I think I have found a very simple way to solve the metric. If L is the locally measured angular momentum, and if this quantity is conserved, then we have
Since m is also a constant, we can drop that and we have
where the primed values are locally measured and unprimed is measured by a distant observer. Since L is constant, then at the point of closest approach b, where the photon travels perfectly tangent to the gravitating body, we have
and solving for dt,
The metric is
and substituting what we found for dt, we get
Integrating this from r=b to r=∞ gives the amount of bending for the gravitational lensing of a photon. It seems much simpler and somewhat more mathematically rigorous than any of the other methods, don't you think? Now I just need to try to find a way to prove that the angular momentum is conserved as measured locally. Does anybody know a way to show that?