[Physics] 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?

Best Answer

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 $\textrm{mass}\to0$.

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