Past exam of the relativity course of the University of Cambridge / 2016 / Paper 1 / 1 / a / Solution

Use an affine parameter $\lambda$, with dots denoting its derivatives, and take $\alpha \ne 0$ so the metric is nondegenerate for $r>0$. The geodesic Lagrangian is $\mathcal{L}=({\dot{r}}^2+{\alpha }^2{r}^4{\dot{\varphi }}^2)/2$. The Euler-Lagrange equations giveor $\ddot{\varphi }+4\dot{r}\dot{\varphi }/r=0$. The conserved cyclic angular momentum isThis is the cyclic angular momentum in a rotational metric, associated with the rotational Killing vector ${\partial }_{\varphi }$. Explicitly, $\dot{J}={\alpha }^2(4r^3\dot{r}\dot{\varphi }+r^4\ddot{\varphi })=0$ by the angular equation.

Differentiating the Lagrangian and substituting both equations givesThus the squared tangent norm $2\mathcal{L}$ is constant. If $\lambda$ is arc length, it equals one; its numerical value otherwise depends on affine normalization. An arbitrary nonaffine parametrization would not preserve this constant-speed statement.