Regular black holes do not violate the first law of thermodynamics

Abstract

Regular black holes in general relativity coupled to nonlinear electrodynamics have been reported to exhibit an apparent mismatch in their thermodynamics: the temperature obtained by differentiating the mass along the family of regular metrics differs from the Hawking temperature by a factor $Ξ$, and the first law seems to need a modified entropy or a corrected internal energy. We examine this question for an electrically charged family of regular black holes in $D\geq4$ dimensions whose metrics reduce to the Bardeen and Hayward geometries in four dimensions. The source is written as a Hamiltonian density $H(P)$ with fixed couplings and no solution parameters. This theory has a two-parameter family of static solutions, of which the regular metrics form a one-parameter subfamily. We prove analytically that on the whole solution space the standard first law of thermodynamics holds with the Hawking temperature, the area entropy and the horizon potential, in agreement with the general first law for nonlinear electrodynamics. The factor $Ξ$ appears when the mass is differentiated along the regular metric family at fixed length parameter, a variation that moves through the space of theories and also changes the conserved charge. We show that $Ξ^{-1}$ is the Jacobian of this projection, equal to the fraction of the mass inside the horizon, give its general form for an arbitrary regular mass function, and show that $Ξ^{-1}dM_{ADM}$ is not a closed one-form on the two-parameter metric family, so this correction cannot be the differential of a local state function there. We also point out that the Legendre map between the $F$ and $P$ frames degenerates at a radius that lies outside the horizon of near-extremal solutions in $D\geq5$.