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

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.