Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 38B (General Relativity)

For coordinates $x^{\alpha }$, the Christoffel symbols of the Levi-Civita connection arewhere $g^{\alpha \delta }$ is the inverse metric.

For the two-dimensional hyperbolic metric in polar coordinates,Only $g_{\varphi \varphi }$ has a nonzero derivative, namelyIt follows that the nonzero connection coefficients are

For an affine parameter $\lambda$, the geodesic equation and the connection coefficients from part (a) givewhere dots denote derivatives with respect to $\lambda$.

On a line of constant $\varphi$, one has $\dot{\varphi }=0$. The equations reduce to $\ddot{r}=0$, so $r$ itself can be chosen as an affine parameter. Every such radial line is therefore a geodesic.

For a circle $r=r_0>0$, the first equation instead requiresBoth hyperbolic factors are positive when $r_0>0$, so $\dot{\varphi }=0$. This gives only a constant point, not a parametrized circle. Thus no circle of constant positive $r$ is a geodesic, in agreement with the coordinate geodesics of the hyperbolic polar metric.

Using the stated convention for the Riemann curvature tensor, the first independent component isThe other isusing the hyperbolic Pythagorean identity ${\coth }^{2}r-{\mathrm{csch}}^{2}r=1$. Antisymmetry in the final two indices supplies the components with those indices reversed. These are the curvature of the two-dimensional hyperbolic polar metric components.

Contracting the Riemann curvature tensor gives the nonzero components of the Ricci tensor:The Ricci scalar is the contraction with the inverse metric. ThereforeIt is independent of both coordinates, so the hyperbolic plane has constant negative scalar curvature.