Quasilocal Smarr relation for an asymptotically flat spacetime

TL;DR

This paper develops a quasilocal thermodynamic framework for asymptotically flat Einstein-Maxwell(-Dilaton) spacetimes, deriving a Smarr relation and first law that incorporate surface pressure and dilaton effects, consistent with known entropy results.

Contribution

It introduces a quasilocal approach to black hole thermodynamics in asymptotically flat spacetimes, including a novel Smarr relation and analysis of dilaton contributions.

Findings

Quasilocal entropy matches Bekenstein-Hawking entropy.

Surface pressure and area are part of the Smarr relation but not the thermodynamic potential.

Dilaton fields influence the first law but not the Smarr relation for dyonic black holes.

Abstract

We investigate the thermodynamics of Einstein-Maxwell(-Dilaton) theory for an asymptotically flat spacetime in a quasilocal frame. We firstly define a quasilocal thermodynamic potential via the Euclidean on-shell action and formulate a quasilocal Smarr relation from Eulerian theorem. Then we calculate quasilocal energy and surface pressure by employing Brown-York quasilocal method along with Mann-Marolf counterterm and find entropy from the quasilocal thermodynamic potential. These quasilocal variables are consistent with Tolman temperature and the entropy in a quasilocal frame turns out to be same as the Bekenstein-Hawking entropy. As a result, we found that a surface pressure term and its conjugate variable, a quasilocal area, do not participate in a quasilocal thermodynamic potential, but should present in a quasilocal Smarr relation and the quasilocal first law of black hole thermodynamics. For dyonic black hole solutions having dynamic dilaton field, non-trivial dilaton contribution should take part in the quasilocal first law but not in the quasilocal Smarr relation.