Quasilocal Smarr relations for static black holes

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.