Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 311 / 3

The Physical-process first law for a rotating black hole states that a small flux of matter through an initially and finally stationary horizon obeysin units $G=1$. Let $k^a$ be an affinely parametrized horizon tangent with affine parameter $\lambda =0$ on a bifurcation surface. Constancy of the surface gravity from the Zeroth law of black-hole mechanics and Gaussian null coordinates give the horizon generatorTo first order about a stationary horizon, the squared expansion and shear in the Null Raychaudhuri equation are second order. The Einstein field equations reduce it toThe final stationary condition $\theta (\infty )=0$ givesSince $d(\delta A)/d\lambda =\int \theta dA$, reversing the order of integration yieldsThe stress-energy current from a Killing vector gives the horizon Killing-energy fluxwhich proves the stated law.

The second law of black-hole mechanics, or Hawking's area theorem, states that the total area of future-event-horizon cross-sections cannot decrease toward the future under the null energy condition and the usual predictability assumptions. If any horizon generator had $\theta <0$, the twist-free Null Raychaudhuri equation and the null focusing theorem would force a conjugate point at finite affine parameter. A generator complete to the future cannot pass through such a point while remaining on the achronal boundary of $J^{-}({\mathcal{I}}^{+})$. Hence $\theta \ge 0$ everywhere on the regular future horizon, and $dA/d\lambda =\theta A\ge 0$ proves the area law.

The two initial Kerr black holes have total areawhereas the final Schwarzschild black hole has $A_f=16\pi M^{\prime 2}$. Hawking's area theorem impliesSince $\eta =1-M^{\prime }/(2M)$, the radiated fraction is bounded byThis upper bound increases with $|J|/M^2$ and reachesfor two initially extremal holes, $|J|=M^2$. For the final hole to be Schwarzschild, their spins must be oppositely directed so the total angular momentum vanishes. The bound assumes an idealized merger saturating the area law.

First, causal futures of black-hole points remain in the black-hole region. Indeed, if $p\in B$ and $q\in J^{+}(p)$ could send a signal to future null infinity, concatenating the causal curves would put $p$ in $J^{-}({\mathcal{I}}^{+})$, a contradiction. HenceGlobal hyperbolicity supplies the following connectedness lemma: if $C$ is connected on one Cauchy hypersurface, then $J^{+}(C)$ intersects any later Cauchy hypersurface in a connected set. To see the relevant mechanism, flow $C$ to the later surface along a continuous future timelike vector field; the image is connected, and every additional causally reachable point is joined to that image by the endpoint deformation of a causal curve inside the globally hyperbolic diamond. Applying the lemma to the connected component $B$ shows that $J^{+}(B)\cap {\Sigma }_2$ is connected. Since it is contained in $B\cap {\Sigma }_2$, it must lie wholly inside one connected component of that set. This is the black-hole non-splitting theorem: later black-hole components may merge, but one earlier connected black hole cannot split into two future components.