Description
Apparent horizons are the best-known examples of marginally outer trapped surfaces (MOTS). However, it is now clear that most MOTS are not apparent horizons. Large, likely infinite, families of MOTS are found in the interior of black hole mergers and these engage in a complex set of interactions and evolutions, including the ultimate dissolution of the original apparent horizons inside the final black hole. The key theoretical tool that brings order to these evolutions is the MOTS stability operator, the modern version of which was introduced by Andersson, Mars and Simon in 2005. It is best understood as the linearization of the outward null expansion for surfaces “near” an existing MOTS. It both identifies whether or not a MOTS can be understood as a (local) apparent horizon (forming a boundary between trapped and untrapped regions) and also determines how a MOTS will evolve in a changing spacetime: if the stability operator is invertible then that evolution is unique.
In this talk, I will review that background and then focus on MOTS with non-invertible stability operators and so non-unique evolutions. The MOTS pair-creations and annihilations observed during black hole mergers are the best-known examples of non-unique evolutions but these are not the only possibilities. Understanding the MOTS as fixed points of the outward null expansion equations, a generalization of standard, dynamical system, bifurcation theory can be used to classify all possible non-unique evolutions. MOTS pair-creation/annihilations are then understood as examples of saddle-node bifurcations. There are other possibilities, including pitchfork and transcritical bifurcations. I apply analytical and numerical tools to identify examples of the various bifurcations in a variety of spacetimes. This theory depends only on the geometry of a MOTS and its surrounding spacetime. Hence the classification results apply not only to possible bifurcations observed in numerical time evolutions but also those that occur for any other deformation of the spacetime. In particular, the constraints on possible bifurcations are not restricted to general relativity but apply to any geometric theory of gravity.