Adjoint operators, gauge invariant perturbations, and covariant symplectic structure for black holes in string theory

TL;DR

This paper derives comprehensive gauge-invariant perturbation expressions for static charged black holes in string theory, applicable to extremal and nonextremal cases, and establishes a covariant symplectic structure using adjoint operators.

Contribution

It introduces a novel method using Wald's adjoint operators to obtain gauge-invariant perturbations and covariant symplectic structures for black holes in string theory.

Findings

Derived complete perturbation expressions valid below Planck scale

Established a local continuity law for field perturbations

Identified conserved quantities and symplectic structure in phase space

Abstract

Expressions for the general and complete perturbations in terms of Debye potentials of static charged black holes in string theory, valid for curvature below the Planck scale, are derived starting from a decoupled set of equations and using Wald's method of adjoint operators. Our results cover both extremal and nonextremal black holes and are valid for arbitrary values of the dilaton coupling parameter. The decoupled set is obtained using the Newman-Penrose formulation of the Einstein-Maxwell-dilaton theory and involves naturally field quantities invariant under both ordinary gauge transformations of the electromagnetic potential perturbations and infinitesimal rotations of the perturbed tetrad. Furthermore, using the recent pointed out relationship between adjoint operators and conserved currents, a local continuity law for the field perturbations in terms of the potentials is also obtained. It is shown that such continuity equation implies the existence of conserved quantities and of a covariant symplectic structure on the phase space. Future extensions of the present results are discussed.