Solution Phase Space and Conserved Charges: A General Formulation for Charges Associated with Exact Symmetries

TL;DR

This paper introduces a comprehensive framework for calculating conserved charges in covariant gravitational theories, applicable across various dimensions and asymptotic conditions, unifying the treatment of symmetries, parameters, and black hole thermodynamics.

Contribution

It develops a general solution phase space approach using covariant phase space methods to define and compute conserved charges for solutions with exact symmetries, including black hole entropy.

Findings

Defines the solution phase space and parametric variations.

Provides a method to compute conserved charges from parametric variations.

Generalizes black hole entropy and first law of thermodynamics to broader contexts.

Abstract

We provide a general formulation for calculating conserved charges for solutions to generally covariant gravitational theories with possibly other internal gauge symmetries, in any dimensions and with generic asymptotic behaviors. These solutions are generically specified by a number of exact (continuous, global) symmetries and some parameters. We define "parametric variations" as field perturbations generated by variations of the solution parameters. Employing the covariant phase space method, we establish that the set of these solutions (up to pure gauge transformations) form a phase space, the \emph{solution phase space}, and that the tangent space of this phase space includes the parametric variations. We then compute conserved charge variations associated with the exact symmetries of the family of solutions, caused by parametric variations. Integrating the charge variations over a path in the solution phase space, we define the conserved charges. In particular, we revisit "black hole entropy as a conserved charge" and the derivation of the first law of black hole thermodynamics. We show that the solution phase space setting enables us to define black hole entropy by an integration over any compact, codminesion-2, smooth spacelike surface encircling the hole, as well as to a natural generalization of Wald and Iyer-Wald analysis to cases involving gauge fields.