A classification theorem for static vacuum black holes. Part I: the study of the lapse
arXiv:1806.00818
Abstract
The celebrated uniqueness's theorem of the Schwarzschild solution by Israel, Robinson et al, and Bunting/Masood-ul-Alam, asserts that the only asymptotically flat static solution of the vacuum Einstein equations with compact but non-necessarily connected horizon is Schwarzschild. Between this article and its sequel we extend this result by proving a classification theorem for all (metrically complete) solutions of the static vacuum Einstein equations with compact but non-necessarily connected horizon without making any further assumption on the topology or the asymptotic. It is shown that any such solution is either: (i) a Boost, (ii) a Schwarzschild black hole, or (iii) is of Myers/Korotkin-Nicolai type, that is, it has the same topology and Kasner asymptotic as the Myers/Korotkin-Nicolai black holes. In a broad sense, the theorem classifies all the static vacuum black holes in 3+1-dimensions. In this Part I we use introduce techniques in conformal geometry and comparison geometry á la Bakry-Émery to prove that vacuum static black holes have only one end, and, furthermore, that the lapse is bounded away from zero at infinity. The techniques have interest in themselves and could be applied in other contexts as well, for instance to study higher-dimensional static black holes.
arXiv comment32 pages, 6 figures
References in corpus (7)
- On the nonexistence of quasi-Einstein metrics
- On the structure of solutions to the static vacuum Einstein equations
- On stationary vacuum solutions to the Einstein equations
- On the Bartnik extension problem for the static vacuum Einstein equations
- The Ernst Equation on a Riemann Surface
- A Periodic Analog of the Schwarzschild Solution
- Global and uniqueness properties of stationary and static spacetimes with outer trapped surfaces