On geodesics in Schwarzschild spacetime

Question

I am required to show that a circular lightlike geodesic exists in the Schwarzschild spacetime, and to find its radius. What's the best way to start this?

Answer 1

You can start with the system of geodesic equations together with the Schwarzschild metric expression divided by $d\tau^2$ (formulated in terms of $u^\mu := dx^\mu /d\tau$). Coordinates $x^\mu := (x^0 := ct, r, \theta, \varphi)$. Then you can obtain geodesics by assuming as $u^2 \equiv 0$ ($\theta = \pi/2$) due to the spherical symmetry. Solutions in $u^0$ and $u^3$ are evident thanks to this symmetry.