arXiv subjects
Publications and source records attributed to Catherine Williams.
We construct an example of a spherically symmetric black hole interior in which there is NO (spherically symmetric) marginally trapped tube asymptotic to the event horizon. The construction uses a self-gravitating massive scalar field matter model, and the key condition we impose is that the scalar field $ϕ$ be bounded below by a positive constant along the event horizon.
We show that the maximal future development of asymptotically flat spherically symmetric black hole initial data for a self-gravitating nonlinear scalar field, also called a Higgs field, contains a connected, achronal marginally trapped tube which is asymptotic to the event horizon of the black hole, provided the initial data is sufficiently small and decays like O(r^{-1/2}), and the potential function V is nonnegative with bounded second derivative. This result can be loosely interpreted as a statement about the stability of `nice' asymptotic behavior of marginally trapped tubes under certain small perturbations of Schwarzschild.
We prove some related results concerning blow-up solutions for the Jang equation. First: it has been shown that, given an outermost marginally outer trapped surface (MOTS) Σ, there exists a solution to Jang's equation which blows up at Σ. Here we show that in addition, large classes of spherically symmetric initial data have solutions to the Jang equation which blow up at non-outermost MOTSs, i.e. MOTSs which lie strictly inside of other MOTS, and even inside of strictly outer trapped surfaces. Unlike for outermost MOTSs, however, we show that there do not \emph{always} exist blow-up solutions for inner MOTSs, even in spherical symmetry. Secondly, an unpublished result of R. Schoen, whose proof we include here, says that in the time-symmetric case, any MOTS corresponding to a blow-up solution for Jang's equation must be outer-area-minimizing, i.e. cannot be contained in a surface of strictly smaller area. The statement is false without the assumption of time-symmetry, however; we construct an explicit spherically symmetric data set providing a counterexample for the general case.
We give conditions on a general stress-energy tensor T_{αβ} in a spherically symmetric black hole spacetime which are sufficient to guarantee that the black hole will contain a (spherically symmetric) marginally trapped tube which is eventually achronal, connected, and asymptotic to the event horizon. Price law decay per se is not required for this asymptotic result, and in this general setting, such decay only implies that the marginally trapped tube has finite length with respect to the induced metric. We do, however, impose a smallness condition (B1) which one may obtain in practice by imposing decay on the T_{vv} component of the stress-energy tensor. We give two applications of the theorem to self-gravitating Higgs field spacetimes, one using weak Price law decay, the other certain strong smallness and monotonicity assumptions.