Regular Article – Theoretical Physics
Quasilocal Smarr relation for an asymptotically flat spacetime
Department of Physics and Research Institute of Basic Science, Kyung Hee University, 02447, Seoul, Republic of Korea
2 Department of Physics and Astronomy, McMaster University, 1280 Main St. W., L8S 4M1, Hamilton, ON, Canada
3 Department of Physics and Astronomy, Faculty of Science, University of Calgary, 2500 University Drive NW, T2N 1N4, Calgary, AB, Canada
4 School of Physics, Korea Institute for Advanced Study, 85 Hoegiro, Dongdaemun-gu, 02455, Seoul, Republic of Korea
Accepted: 31 March 2021
Published online: 15 April 2021
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 Euler’s theorem. Then we calculate the quasilocal energy and surface pressure by employing a Brown–York quasilocal method along with Mann–Marolf counterterm and find entropy from the quasilocal thermodynamic potential. These quasilocal variables are consistent with the 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 be present in a quasilocal Smarr relation and the quasilocal first law of black hole thermodynamics. For dyonic black hole solutions having dynamic dilaton field, a non-trivial dilaton contribution should occur in the quasilocal first law but not in the quasilocal Smarr relation.
© The Author(s) 2021
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