Adjoint operators and perturbation theory of black holes

TL;DR

This paper introduces a novel method for deriving conservation laws in black hole perturbation theory, especially for non-Hermitian systems, potentially advancing understanding in string theory contexts.

Contribution

It develops a general framework linking solutions of differential equations and their adjoints to conservation laws, extending traditional methods to more complex black hole perturbations.

Findings

Approach recovers known results in Schwarzschild black holes

Framework applies to non-Hermitian perturbation equations

Potential for analyzing string theory black hole perturbations

Abstract

We present a new approach for finding conservation laws in the perturbation theory of black holes which applies for the more general cases of non-Hermitian equations governing the perturbations. The approach is based on a general result which establishes that a covariantly conserved current can be obtained from a solution of any system of homogeneous linear differential equations and a solution of the adjoint system. It is shown that the results obtained from the present approach become essentially the same (with some diferences) to those obtained by means of the traditional methods in the simplest black hole geometry corresponding to the Schwarzschild space-time. The future applications of our approach for studying the perturbations of black hole space-time in string theory is discussed.