Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 357 / 3

Write the Kruskal–Szekeres plane with $\hat{r}$ horizontal and $\hat{t}$ vertical. The relationshows that constant-$r$ curves are hyperbolae.

For $r>2M$, their right and left exterior branches satisfyFor $0<r<2M$, their future and past interior branches satisfyThe event horizons $r=2M$ are the null diagonalsand the curvature singularities $r=0$ are the spacelike hyperbolaein the normalization stated in the question.

Constant Schwarzschild-$t$ curves are straight rays through the origin. In the exteriors their slopes obeywhile in the interiorsThus the qualitative diagram is the usual four-region Kruskal diagram: two exterior wedges separated from black-hole and white-hole interiors by the two null horizons, with spacelike singularities bounding the interior wedges.

In geodesic slicing, $\alpha =1$ and ${\beta }^i=0$. The normal acceleration therefore vanishes:Hence each integral curve of $n^{\mu }$ is an affinely parametrized timelike geodesic.

On the initial surface $\hat{t}=0$, the diagonal Kruskal metric makes the unit normal point purely in the $\hat{t}$ direction. The observer starts at $\hat{r}=0$ with $\dot{\hat{r}}=0$, so its initial unit four-velocity equals that normal. The observer's geodesic and the normal integral curve solve the same geodesic initial-value problem. Uniqueness therefore givesthroughout their common domain.

The point $\hat{t}=\hat{r}=0$ is the bifurcation sphere $r=2M$. The observer starts there with $\dot{r}=0$, so its radial-geodesic equationgives $E=0$. During infall,Put $x=r/(2M)$. The proper time to the singularity isThusThe normals of the geodesic foliation reach the physical singularity after finite coordinate time because unit lapse identifies coordinate and normal proper time. Nearby normals can also focus and form coordinate caustics. Consequently geodesic gauge is unsuitable for long-term black-hole evolution: the numerical slice encounters singular behavior in finite time rather than avoiding it.

In geodesic gauge,The BSSN evolution equation reduces toThe first term is a squared norm with respect to the positive-definite spatial metric, the second is nonnegative, and the stated energy condition makes the final term nonnegative. ThereforeThe mean curvature can only increase along this geodesic slicing.

In vacuum, neglecting ${\tilde{A}}_{mn}{\tilde{A}}^{mn}$ leavesSeparating variables and imposing $K(t_0)=K_0>0$ giveshenceThe denominator vanishes atThis finite-time blow-up of mean curvature in geodesic slicing is another direct expression of the gauge's singularity problem.