Dain's invariant for black hole initial data

TL;DR

This paper extends Dain's geometric invariants to initial data for black hole spacetimes, providing a method to identify approximate symmetries and their relation to actual Killing vectors.

Contribution

It generalizes Dain's invariants to initial data sets for black holes and proves the existence and uniqueness of approximate Killing vectors in these spacetimes.

Findings

Existence and uniqueness of solutions to the boundary value problem for approximate Killing vectors.

Construction of a geometric invariant on a MOTS that detects local Killing data.

Approximate symmetries coincide with actual symmetries when present.

Abstract

Dynamical black holes in the non-perturbative regime are not mathematically well understood. Studying approximate symmetries of spacetimes describing dynamical black holes gives an insight into their structure. Utilising the property that approximate symmetries coincide with actual symmetries when they are present allows one to construct geometric invariants characterising the symmetry. In this paper, we extend Dain's construction of geometric invariants characterising stationarity to the case of initial data sets for the Einstein equations corresponding to black hole spacetimes. We prove the existence and uniqueness of solutions to a boundary value problem showing that one can always find approximate Killing vectors in black hole spacetimes and these coincide with actual Killing vectors when they are present. In the time-symmetric setting we make use of a 2+1 decomposition to construct a geometric invariant on a MOTS that vanishes if and only if the Killing initial data equations are locally satisfied.