Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 38D (General Relativity)

Forthe nonzero Christoffel symbols needed here areOn a line of constant longitude, $\dot{\varphi }=0$, and the geodesic equation reduces to $\ddot{\theta }=0$ after choosing an affine parameter. Thus every meridian is a geodesic.

For a nonconstant line of constant latitude, $\dot{\theta }=0$ and the $\theta$ equation requiresAway from the coordinate-degenerate poles, this holds only whenso only the equator is a constant-latitude geodesic.

Along the constant latitude, use $\varphi$ as parameter. The parallel transport equations areWith $W=\sin \theta V^{\varphi }$, these becomeThe initial components $(1,0)$ therefore evolve asAfter one circuit, the parallel transport around a latitude of the unit sphere gives

Contracting the stated Einstein field equations with $g^{\alpha \beta }$ in $n$ dimensions givesThus $R=-2T/(n-2)$, and substitution back yields the Trace-reversed Einstein field equations in D dimensions

At any point choose orthonormal coordinates. The antisymmetries and pair symmetry of the Riemann curvature tensor leave only $R_{1212}$ independent. In these coordinates,while $R=2R_{1212}$. Since this is a tensor identity, it follows in every coordinate system that the Ricci tensor in two dimensions satisfiesThe Einstein tensor consequently vanishes identically. Hence the two-dimensional vacuum field equation contains no local gravitational dynamics, while coupling the unmodified equation to matter would require $T_{\alpha \beta }=0$. This is the basic degeneracy of two-dimensional general relativity.