Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 1

A null geodesic congruence is a smooth family of nonintersecting null geodesics filling an open spacetime region. For its affinely parametrized tangent $U^a$,Writing $B_{ab}={\nabla }_b{U}_a$, differentiation of the null norm and the geodesic equation give respectively

Choose at one transverse cross-section a null vector $N^a$ satisfying $N^a{N}_a=0$ and $U^a{N}_a=-1$, then parallel transport it along every generator:Metric compatibility and the affine geodesic equation imply that both $N^2$ and $U\cdot N$ are constant along each generator, so the required normalization persists.

The screen-space projectordefines the optical tensor ${\hat{B}}_{ab}=P_a{}^c{P}_b{}^d{B}_{cd}$. In four spacetime dimensions its irreducible decomposition isThese are the null expansion, null twist, and null shear.

On the null hypersurface, the generator covector is proportional to a normal, locally $U_a=f{\nabla }_{a}u$ for a level-set function $u$. The Frobenius theorem therefore givesContracting once with $N^a$ and projecting the remaining indices with $P^a{}_b$ removes every term containing $U$ or $N$ and leaves $P_b{}^d{P}_c{}^e{\nabla }_{[d}{U}_{e]}=0$. Henceon the hypersurface.

Commuting covariant derivatives and using $U^b{\nabla }_b{U}_a=0$ gives the optical evolution equationTaking its screen trace yieldsThe optical decomposition and the symmetry or antisymmetry of its pieces implyThus the four-dimensional Null Raychaudhuri equation is

The hypersurface generators have zero null twist. Contracting the Einstein field equations twice with the null tangent removes the trace term, and the null energy condition gives $R_{ab}{U}^a{U}^b=8\pi T_{ab}{U}^a{U}^b\ge 0$. Since the squared null shear is nonnegative, the Null Raychaudhuri equation impliesWhile $\theta <0$ this is equivalent toStarting from $\theta (0)={\theta }_0<0$, the right-hand side reaches zero no later than $\lambda =2/|{\theta }_0|$. The reciprocal expansion must therefore vanish andThis is the null focusing theorem.

Suppose the closed trapped surface $T$ were not wholly in the black hole. The stated consequence of strong asymptotic predictability then supplies a point $p\in {\dot{J}}^{+}(T)\cap {\mathcal{I}}^{+}$ lying on a future null generator $\gamma$ orthogonal to $T$, with no point conjugate to $T$ before $p$. Both future null expansions at $T$ are negative. The null focusing theorem therefore forces the expansion along $\gamma$ to diverge after finite affine parameter, producing a conjugate point before the generator reaches future null infinity. A null geodesic with such a point cannot continue to generate the achronal boundary ${\dot{J}}^{+}(T)$, contradicting the property of $p$. Hence no point of $T$ can communicate with future null infinity, and