On geodesics in Schwarzschild spacetime 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?
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.
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.