Hidden Killing Fields, Geometric Symmetries and Black Hole Mergers

TL;DR

This paper introduces a framework for constructing local geometric deformations called hidden Killing vector fields, enabling the derivation of local conservation laws in spacetimes lacking global symmetries, demonstrated through a black hole merger model.

Contribution

It develops a novel method to derive local conservation laws using hidden Killing fields, extending symmetry-based techniques to less symmetric dynamical spacetimes.

Findings

Derived local balance laws resembling energy and angular momentum conservation.

Constructed hidden Killing vector fields solving the Killing equation locally.

Applied the framework to a black hole merger toy model.

Abstract

In the present work, using the recently introduced framework of local geometric deformations, special types of vector fields - so-called hidden Killing vector fields - are constructed, which solve the Killing equation not globally, but only locally, i.e. in local subregions of spacetime. Taking advantage of the fact that the vector fields coincide locally with Killing fields and therefore allow the consideration of integral laws that convert into exact physical conservation laws on local scales, balance laws in dynamical systems without global Killing symmetries are derived that mimic as closely as possible the conservation laws for energy and angular momentum of highly symmetric models. The utility of said balance laws is demonstrated by a concrete geometric example, namely a toy model for the binary merger of two extremal Reissner-Nordstr\"om black holes.