Black hole evolutions: Lessons from bifurcation theory

Abstract: We explore the role of the stability operator in regulating the evolution of marginally outer trapped surfaces (MOTSs). In 2005, Andersson, Mars and Simon showed that if the stability operator is invertible, then the time evolution of a MOTS is unique. Here we focus on moments at which that stability operator is not invertible. Understanding MOTSs as analogous to fixed points, and the stability operator as a linearization of the system of the MOTS-defining equations, bifurcation theory can be used to classify possible non-unique evolutions. MOTS pair creation/annihilation is an example of a saddle-node bifurcation but other possibilities can occur, including pitchfork and transcritical bifurcations. Analytical and numerical tools are used to identify examples of the various bifurcations in a variety of spacetimes. To help analyze those results, we define a generalized MOTS stability operator and discuss the (partial) barrier properties of unstable MOTSs. Spherically symmetric examples are given in Reissner-Nordström-de Sitter spacetime and axisymmetric examples are studied in Reissner-Nordström and Weyl-distorted Schwarzschild. This application of bifurcation theory is very general and so applies to any theory of gravity containing MOTSs (or some generalization thereof). Possible bifurcations of those structures are constrained in any such theory in the same way.