On the Stability of MOTS

TL;DR

This paper demonstrates that the MOTS stability operator can be reformulated as a self-adjoint operator in certain geometric models of general relativity, facilitating analysis of black hole horizons.

Contribution

It introduces a method to transform the MOTS stability operator into a self-adjoint form using null rotations, applicable to specific classes of spacetimes like Kerr-Newman.

Findings

MOTS stability operator can be made self-adjoint in certain models

The method applies to Kerr-Newman spacetimes

Requires existence of a non-expanding null horizon

Abstract

In this paper, it is shown (using the NP-spin coefficient formalism) that the MOTS eigenvalue problem can be formulated - for certain classes of geometric models in GR - such that the MOTS stability operator takes a self-adjoint form, despite being a manifestly non-self-adjoint differential operator. This form is obtained by performing a suitable null rotation, which is chosen so that the MOTS under consideration remain such, i.e., transition into new MOTS. Next to the requirement that certain components of the Einstein tensor - resp. the stress-energy tensor - be zero, the main prerequisite for bringing the MOTS operator into the required form is the existence of a non-expanding null horizon together with a set of eigenfunctions of the MOTS operator that does not change along the Lie flow of the generator of said horizon. For illustration, the developed method is applied to the Kerr-Newman family of spacetimes.