Quasilocal Smarr relation for an asymptotically flat spacetime

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.