Black hole formation and stability: A mathematical investigation

Abstract

Abstract. The dynamics of the Einstein equations feature the formation of black holes. These are related to the presence of trapped surfaces in the spacetime manifold. The mathematical study of these phenomena has gained momentum since D. Christodoulou's breakthrough result proving that, in the regime of pure general relativity, trapped surfaces form through the focusing of gravitational waves. (The latter were observed for the first time in 2015 by Advanced LIGO.) The proof combines new ideas from geometric analysis and nonlinear partial differential equations, and it introduces new methods to solve large data problems. These methods have many applications beyond general relativity. D. Christodoulou's result was generalized by S. Klainerman and I. Rodnianski, and more recently by these authors and J. Luk. Here, we investigate the dynamics of the Einstein equations, focusing on these works. Finally, we address the question of stability of black holes and what has been known so far, involving recent works of many contributors.

Initial data and black holes for matter models

“…In [9] we have shown that this solution is, although singular, still surprisingly well-behaved in a way that it satisfies the second Bianchi identity weakly. The stability of this solution may be studied using metric convergence, e.g., in the sense of Gromov-Hausdorff convergence or Sormani-Wenger intrinsic flat convergence [1,6,22,29,30].…”

“…We briefly examine how this was achieved for certain matter models. We will not discuss the vacuum case here, as it differs significantly from the treatment for matter models (spherically symmetry cannot be employed due to Birkhoff's Theorem) and an excellent review has already been written by Bieri [6].…”

Singularities, Black Holes, and Cosmic Censorship: A Tribute to Roger Penrose

“…Footnote 11 (continued) but "infinitely extended") for Reissner-Nordström and Kerr. The pde approach to gr, on the other hand, is based on two slogans, appealing to the fundamental theorem of Choquet-Bruhat and Geroch [29]: 12 • All valid assumptions are about initial data;…”

“…13 Finally, an Appendix written by Erik Curiel traces the definitional history of the concept of a black hole. 12 A Cauchy surface in a space-time (M, g) is a subset Σ ⊂ M such that each endless timelike curve intersects Σ exactly once. This makes Σ a closed connected 3d submanifold of M which can be chosen space-like and hence Riemannian.…”

“…See also [41] 13 In particular, using the pde approach Christodoulou [31,33,34] finally established the formation of black holes both in spherically symmetric collapse models with (scalar field) matter and in vacuum solutions through focusing of gravitational waves, by proving both causal geodesic incompleteness and the existence of an event horizon. See also follow-ups by Klainerman and Rodnianski [116] and Klainerman et al [114], and reviews by Bieri [12] and Dafermos [52]. For the incorporation of more realistic matter models see e.g.…”