Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 66 / 3 / b

Let $p^a$ be the future-directed causal four-momentum of the fragment captured by the Kerr black hole. In signature $(-+++)$ its conserved Killing energy and axial angular momentum areThe future-directed null generator of the Killing horizon is ${\chi }^a=k^a+{\Omega }_H{m}^a$, where ${\Omega }_H$ is the Kerr horizon angular velocity. At a crossing of the future event horizon, the scalar product of a future-directed causal vector with a future-directed null vector is nonpositive. For example, in a local orthonormal frame, $\chi =\alpha (1,\mathbf{n})$ with $|\mathbf{n}|=1$ and $\alpha >0$, while $p^0\ge |\mathbf{p}|$; hence $p\cdot \chi =\alpha (-p^0+\mathbf{p}\cdot \mathbf{n})\le 0$. ThereforeThis is a local causal restriction, so it applies also to an extremal Kerr black hole. A massive fragment crossing a regular future event horizon has a strict inequality; equality is the reversible limiting case, with causal four-momentum approaching the generator direction.

The Penrose process exploits the fact that $k^a$ becomes spacelike in the Kerr ergoregion, allowing future-directed fragments with negative Killing energy. For a positive choice of Kerr horizon angular velocity, a captured negative-energy fragment must have sufficiently negative angular momentum to satisfy the inequality. Conservation of energy in the splitting then gives $E_{\mathrm{escape}}=E_{\mathrm{initial}}-E_{\mathrm{capture}}>E_{\mathrm{initial}}$: the escaping fragment extracts rotational energy and the captured fragment reduces the black hole's mass and angular momentum.

For the independent thermodynamic argument, take a subextremal Kerr black hole and a small capture process covered by the Physical-process first law of black-hole mechanics. To first order, $\delta M=E$, $\delta J=L$, soThe second law of black-hole mechanics, under its classical null energy condition and regularity hypotheses, gives $\delta A\ge 0$; since $\kappa >0$, this reproduces $E\ge {\Omega }_{H}L$. The ideal reversible limit has $\delta A\to 0$. One should not infer a general finite-capture equality for an extremal black hole by simply substituting $\kappa =0$: the linearized nonextremal process proof and its differentiability assumptions must first be controlled. The causal proof already establishes the required inequality there without this issue.