# Quasilocal Smarr relations for static black holes

## TL;DR

This paper derives generalized quasilocal Smarr relations for static black holes using gravitational path integrals and Hamilton-Jacobi analysis, linking thermodynamic variables with gravitational energies and Einstein equations.

## Contribution

It introduces a novel method to obtain quasilocal Smarr relations for static black holes via path integrals and Hamilton-Jacobi analysis, connecting thermodynamics with gravitational energies.

## Key findings

- Derived quasilocal Smarr relations for Schwarzschild and Reissner-Nordström black holes.
- Established connection between quasilocal energy and Komar/Misner-Sharp energies.
- Showed the relation as a thermodynamical realization of Einstein equations.

## Abstract

Generalized Smarr relations in terms of quasilocal variables are obtained for Schwarzschild and Reissner-Nordstr\"om black holes. The approach is based on gravitational path integrals with finite boundaries on which, following Brown and York, thermodynamic variables are identified through a Hamilton-Jacobi analysis of the action. The resulting expressions allow us to construct the relation between the quasilocal energy obtained in this setting and the Komar and Misner-Sharp energies, which are regarded as thermodynamical internal energy in other approaches. The quasilocal Smarr relation is obtained through scaling arguments, and terms evaluated in the external boundary and the horizon are present. By considering some properties of the metric, it is shown that this quasilocal Smarr relation can be regarded as a thermodynamical realization of Einstein equations. The approach is suitable to be generalized to any spherically symmetric metric.