Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 37C (General Relativity)

The upper-left block has eigenvalues $1$ and $-1$, while the remaining two eigenvalues are both $-1$. The metric signature is therefore $(1,3)$ up to sign convention. It is Lorentzian.

Let $F=1-2M/r$, define the tortoise coordinate by $d{r}_{\ast }/dr=F^{-1}$, and set the advanced time $v=t+r_{\ast }$. Since $dt=dv-F^{-1}dr$, substitution givesThese are Ingoing Eddington-Finkelstein coordinates; all coefficients are regular at the event horizon $r=2M$.

Outside the ingoing null shell the enclosed mass is $E$, so use the ingoing Ingoing Eddington-Finkelstein coordinatesThis is the Schwarzschild metric of mass $E$, regular across $r=2E$ and valid for $r>0$ on the exterior side of the shell.

Inside the spherical shell the enclosed mass vanishes by the spherical vacuum solution, so spacetime is flat. In the same ingoing coordinates,valid for $r\ge 0$ on the interior side before the shell reaches the centre.

Because the shell lies well inside its Schwarzschild radius once it reaches $r=2E$, an event horizon forms and the final exterior is a black hole of mass $E$. In the ideal instantaneous-shell model the collapse ends at a spacetime singularity.