Solving a light ray worldline with the geodesic equation I'm having trouble solving the geodesic equation for a light ray.
$$ {d^2 x^\mu \over d\tau^2} + \Gamma^\mu_{\alpha\beta} {dx^\alpha \over d\tau} {dx^\beta \over d\tau} = 0 $$
I apologise, but I'm a bit new to this, but I have the initial $x^\mu$ and initial $dx^\mu\over d\tau$. I'm just not sure how to use them to solve the equation for $x^\mu$.
I would logically start with
$$ {dx^\mu \over d\tau}_{initial} = v^\mu $$
and suppose that the initial acceleration would be
$$ {d^2 x^\mu \over d\tau^2}_{initial} = - \Gamma^\mu_{\alpha\beta} v^\alpha v^\beta $$
But that doesn't really help me integrate it, since I've only got constants for the initial condition. How would I solve this for $x^\mu(\tau)$?
Furthermore, I feel that this equation may not apply to light rays, as their proper time ought to be $0$, right?
 A: Assume, that the photon path is parametrized by an affine parameter. The affine parameter for the null geodesic is usually denoted with $\lambda$, so I'll use $\lambda$ instead of $\tau$. Then by velocity I'll mean 4-vector $u^{\alpha} = \frac{d x^\alpha}{d \lambda}$.
I'll use Wikipedia notation for the Schwarzschild metric, assuming in addition, that $c=1$. Particularly, signature will be $(+---)$. Given any initial conditions for a photon, one can always find a "plane", containing initial position and velocity. (By a plane here I mean a big circle on a sphere, parametrized by $\varphi$-$\theta$ coordinates.) Then you can use rotational symmetry to move the initial conditions in $\theta=\pi/2$ plane, so that initially photon is in this plane and $\theta$ component of its velocity is 0. Then photon will always have $u^\theta=0$ and $\theta=\pi/2$.
The Schwarzschild metric is invariant with respect to time translation and rotation. In the traditional language of general relativity, $\frac{\partial}{\partial t}$ and $\frac{\partial}{\partial\varphi}$ are killing fields. This implies, that $u_t$ and $u_\varphi$ are constant (but not $u^t$ and $u^\varphi$). Together with $u^\alpha u_\alpha = 0$ this gives 3 equations for 3 non-zero components of the 4-velocity, so you can find all of them (as a functions of $r$). In this case physicists say, that the equation of motion is integrable. Now you may want to find $r(\lambda)$, $t(\lambda)$, $\varphi(\lambda)$, starting from solving $\frac{dr}{d\lambda} = u^r(r(\lambda))$ as an ordinary differential equation on the function $r(\lambda)$.
There are great notes on solving the same problem for a massive particle by Christopher Hirata. You may also want to look at other sections of the lecture notes for the course he gave at Caltech in 2011-2012. You may also think about photon trajectory as a particle trajectory in the limit $\textrm{mass}\to0$.
A: $\tau$ can be seen as proper time scaled by an arbitrary factor.  For the limit of approaching a null geodesic, yes $\tau$ as proper time goes to zero, but use an arbitrary factor approaching infinity, and the combination can be kept finite.  Or ignore all that; $\tau$ can be any arbitrary parameterization along the geodesic.
I notice someone has already asked about this: What is the physical meaning of the affine parameter for null geodesic? 
