Modification of a Lie-algebra-based approach and its application to asymptotic symmetries on a Killing horizon

TL;DR

This paper introduces a modified method for identifying asymptotic symmetries in general relativity that simplifies the integrability check, demonstrated on Killing horizons where a new class of symmetries is found.

Contribution

It proposes a sufficient condition for integrability of charges that avoids solving differential equations, enhancing the analysis of asymptotic symmetries.

Findings

Discovered a new class of asymptotic symmetries on Killing horizons.

Established that the charge algebra forms a central extension of vector fields.

Provided a practical criterion for integrability without solving differential equations.

Abstract

We develop a new approach to find asymptotic symmetries in general relativity as a modification of the Lie-algebra-based approach proposed in T. Tomitsuka et al. [Classical Quantum Gravity 38, 225007 (2021)]. Those authors proposed an algorithmic protocol to investigate asymptotic symmetries. In particular, their guiding principle helps us to find a non-vanishing charge that generates an infinitesimal diffeomorphism. However, in order to check the integrability condition for the charges, it is necessary to solve differential equations to identify the integral curve of vector fields, which is usually quite hard. In this paper, we provide a sufficient condition of the integrability condition that can be checked without solving any differential equations, avoiding the difficulties in the approach in the above reference. As a demonstration, we investigate the asymptotic symmetries on a Killing horizon and find a new class of asymptotic symmetries. In 4D spacetimes with a spherical Killing horizon, we show that the algebra of the corresponding charges is a central extension of the algebra of vector fields.