# 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 $x^\mu$.

$${dx^\mu \over d\tau}_{initial} = v^\mu$$

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?

• Do you have expressions for the Christoffel symbols as functions of the coordinates? If this is homework, please add the homework tag.
– user4552
Commented May 26, 2013 at 1:03
• This isn't homework, it's an exploration of the workings of the universe, purely out of curiosity. I'm using the schwarzchild metric, and I've worked out the Christoffel symbols for it (just too lazy to LATEX them all into the post). Commented May 26, 2013 at 1:09
• There is no mechanical process for finding a closed-form expression for the trajectories of test particles in a situation like this, and in fact there will not in general be any such closed-form expressions. Are you trying to find the general solution, or just the deflection of a light ray to first order in a weak field?
– user4552
Commented May 26, 2013 at 1:19
• I'm trying to calculate the path of a light ray, given its initial conditions, the exact solution not a first order approximation. Commented May 26, 2013 at 1:21
• Do yourself a favor and look up the literature on the subject. It's complicated. I don't think the solutions can be expressed in closed form in the general case. An example of a book that discusses this kind of thing is Taylor and Wheeler, Exploring Black Holes: Introduction to General Relativity.
– user4552
Commented May 26, 2013 at 2:00

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 $\textrm{mass}\to0$.

• Could you explain in more detail how to get to $r(\lambda)$? Commented May 29, 2013 at 7:56
• What is the specific energy of a massless object? Wouldn't that be undefined? Commented May 30, 2013 at 3:10

$\tau$ 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?

• I read that question, but it doesn't go into detail into how exactly I am supposed to solve a null geodesic. Commented May 26, 2013 at 7:03