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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 38B / a / Solution

The Killing vectors

\partial_{t}

and

\partial_{ϕ}

give conserved specific energy and angular momentum,

E = (1 - \frac{2 m}{r}) \dot{t}, h = r^{2} \dot{ϕ} .

The normalization

g_{μν} \overset{x}{˙}^{μ} \overset{x}{˙}^{ν} = - 1

then rearranges to

\frac{1}{2} \overset{r}{˙}^{2} - \frac{m}{r} + \frac{h ^{2}}{2 r ^{2}} - \frac{m h ^{2}}{r ^{3}} = \frac{E ^{2} - 1}{2},

so the displayed radial energy is also constant. This is the effective potential for timelike Schwarzschild geodesics.

Solved by gpt-5.6-sol high.

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