On scalar charges and black hole thermodynamics

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.

Hairy black holes, scalar charges and extended thermodynamics

“…2. In static black-hole spacetimes the volume integral that gives the globally conserved charge usually vanishes when integrated over a spacelike hypersurface [6].…”

“…In [6,8] a covariant definition of scalar charge of a stationary black hole as the integral of an on-shell closed 2-form was proposed 6 . In the cases considered so far, this definition gives the same value as the conventional definition based on the asymptotic expansion, but with the new definition one can go farther: the closedness of the 2-form charge implies that this scalar charge satisfies a Gauss law 7 and the covariant definition can be used to recover the scalar term in the first law of black-hole mechanics found in [7].…”

“…The covariant definition of scalar charge of [6,8] can be used to determine the particular value of the scalar charge allowed in presence of a bifurcate black-hole horizon, which is, as a matter of fact, equivalent to a 'no-hair theorem'.…”

“…In this paper we want to study the extensions of the results obtained in [6,8] to the case in which a real scalar is coupled to itself via a scalar potential instead of being coupled to vector fields. This is a very simple case which has been very much studied in the past and several 'no-hair theorems' have been proven for more or less general classes of scalar potentials in [20][21][22][23][24][25].…”

“…One of the main assumptions in the proofs of these theorems is the positivity of the scalar potential, related to the energy conditions and, not surprisingly, asymptotically-flat black-hole solutions with scalar hair have been found in theories whose scalar potential violates that condition [26][27][28][29][30] 9 . These solutions and their thermodynamics have not been studied from the point of view of their scalar charges 10 and our main goal is to do so, which, first of all requires a generalization of the definition of scalar charge of [6,8]: the theories considered in [6,8] have global symmetries and the scalar charges are related to them. As we are going to see, the covariant definition can be extended to the theories that we are going to consider, which have no global symmetries and it can be used to determine which values are allowed in the presence of a bifurcate black-hole horizon.…”