Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 311 / 2 / a

The Penrose diagram is the maximally extended Schwarzschild diagram modified by an ingoing null shell from the chosen ${\mathcal{I}}^{-}$. To its past the vacuum mass is $M$; to its future the exterior mass is $M+E$. Draw the shell as a $45^{\circ }$ ingoing line from the chosen ${\mathcal{I}}^{-}$, crossing the future event horizon and ending on the future spacelike singularity. In that same exterior, label $i^{-}$ and $i^{+}$ at the lower and upper timelike corners, $i^0$ at the spatial corner, and ${\mathcal{I}}^{-}$ and ${\mathcal{I}}^{+}$ as the past and future null boundaries.

The crucial teleological feature is that the final ${\mathcal{H}}^{+}$, traced backward before the shell, lies outside the old $r=2M$ future horizon. It crosses the old past horizon ${\mathcal{H}}^{-}$ at the bifurcation two-sphere and continues into the white-hole region, ending on the past spacelike singularity. Thus the diagram must not draw the relevant ${\mathcal{H}}^{+}$ as beginning at the old bifurcation sphere.

Solved by gpt-5.6-sol high.

After the shell, spherical symmetry and stationarity put the horizon at $r=2(M+E)$. Before the shell, an outgoing radial null geodesic in the mass-$M$ region has $r$ linear in an affine parameter. The normalizations $r(0)=2M$ and $r(1)=2(M+E)$ therefore giveThe degenerate intrinsic metric of the horizon has no $d{\lambda }^2$ term, soThe two expressions agree at the shell, $\lambda =1$.

Solved by gpt-5.6-sol high.

Tracing the generator backward through the mass-$M$ region gives $r=2(M+E\lambda )$. It reaches the past spacelike singularity when $r=0$, namely atThe singular endpoint is not part of the spacetime, while the final stationary horizon extends indefinitely to the future. HenceIn particular, negative $\lambda$ is physically necessary: that part of the relevant event horizon lies inside the old white-hole region before reaching the old bifurcation sphere at $\lambda =0$.

Solved by gpt-5.6-sol high.