Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 56 / 2 / b / Solution

During gravitational collapse, nonspherical time-dependent multipoles can be lost in gravitational radiation or absorbed by the forming black hole. Matter remaining outside must ultimately escape or fall in if the final object is to be an isolated vacuum black hole. Assume that the classical exterior settles to a regular stationary asymptotically flat vacuum geometry, with a connected horizon and the regularity conditions of the black-hole uniqueness theorem. This settling assumption is physical input, rather than a consequence of charge conservation alone.

For a rotating remnant, the black-hole rigidity theorem supplies axial symmetry; the uniqueness theorem then identifies the exterior as Kerr spacetime. The nonrotating equilibrium case is Schwarzschild spacetime. Gravitational radiation carries energy and angular momentum but no net electric charge. In the intended neutral collapse, with no net charge carried away by expelled matter, charge conservation therefore leaves zero black-hole charge. Initial total neutrality alone would not prove this if oppositely charged matter were expelled; the two-parameter conclusion assumes the uncharged vacuum remnant just described. This leavesas its two intrinsic continuous parameters. The final $M,J$ are what remain after escaping matter and radiation have carried off their shares; they need not equal the star's initial values. A spatial rotation may align the spin axis, and a translation or boost chooses the position and rest frame, so these do not add intrinsic parameters. All higher equilibrium multipoles are fixed by $M,J$; for example Kerr's mass quadrupole is $-J^2/M$. A regular Kerr black hole obeys $|J|\le M^2$ in geometrized units. This is the content of the black-hole no-hair theorem in vacuum general relativity, with its stated hypotheses, rather than a statement about arbitrary matter theories or an unproved guarantee of dynamical settling. Hawking evaporation is ignored in this classical late-time equilibrium description.