Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 38B (General Relativity)

The Killing vectors ${\partial }_t$ and ${\partial }_{\varphi }$ give conserved specific energy and angular momentum,The normalization $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-1$ then rearranges toso the displayed radial energy is also constant. This is the effective potential for timelike Schwarzschild geodesics.

For a circular orbit, the derivative ofvanishes. ThusA real finite angular momentum therefore requires

For a circular orbit, normalization and the value of $h$ giveConsequentlyThis is the relativistic form of the circular-orbit Kepler third law.

Substitute $r=1/u$ and $\dot{r}=-h{u}^{\prime }$ into the conserved radial energy. Differentiation with respect to $\varphi$ givesAfter division by $h^2{u}^{\prime }$ and continuity through turning points,

Put $q=m/h^2$ and substituteKeeping the leading nontrivial powers of $m/h$ and terms through first order in $A$, the constant and oscillatory terms giveThereforeThe radial oscillation has angular period $2\pi /(1+\beta )$, so successive periapses advance byto leading order. This is the Schwarzschild perihelion precession.