Abstract
We revisit the first law of black hole thermodynamics in 4-dimensional theories containing scalar and Abelian vector fields coupled to gravity using Wald’s formalism and a new definition of scalar charge as an integral over a 2-surface which satisfies a Gauss law in the background of stationary black-hole spacetimes. We focus on ungauged supergravity-inspired theories with symmetric sigma models whose symmetries generate electric-magnetic dualities leaving invariant their equations of motion. Our manifestly duality-invariant form of the first law is compatible with the one obtained by of Gibbons, Kallosh and Kol. We also obtain the general expression for the scalar charges of a stationary black hole in terms of the other physical parameters of the solution and the position of the horizon, generalizing the expression obtained by Pacilio for dilaton black holes.
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Acknowledgments
This work has been supported in part by the MCI, AEI, FEDER (UE) grants PID2021-125700NB-C21 (“Gravity, Supergravity and Superstrings” (GRASS)) and IFT Centro de Excelencia Severo Ochoa CEX2020-001007-S. The work of RB has also been supported by the National Agency for Research and Development [ANID] Chile, Doctorado Nacional, under grant 2021-21211461 and by PUCV, Beca Pasantía de Investigación. The work of CG-F was supported by the MU grant FPU21/02222. The work of MZ was supported by the fellowship LCF/BQ/DI20/11780035 from “La Caixa” Foundation (ID 100010434). TO wishes to thank M.M. Fernández for her permanent support.
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Ballesteros, R., Gómez-Fayrén, C., Ortín, T. et al. On scalar charges and black hole thermodynamics. J. High Energ. Phys. 2023, 158 (2023). https://doi.org/10.1007/JHEP05(2023)158
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DOI: https://doi.org/10.1007/JHEP05(2023)158