On conserved charges and thermodynamics of the AdS$_{4}$ dyonic black hole

TL;DR

This paper analyzes the conserved charges and thermodynamics of an AdS$_{4}$ dyonic black hole in a theory derived from eleven-dimensional supergravity, highlighting the role of boundary conditions and the magnetic charge in the first law.

Contribution

It provides a detailed Hamiltonian and thermodynamic analysis of a dyonic AdS black hole with scalar fields, including the identification of magnetic charge from boundary terms.

Findings

Conserved charges include mass and electric charge, with scalar contributions.

The magnetic charge is identified from boundary terms, ensuring the first law holds.

The Hamiltonian action principle remains well-defined despite non-symmetry charges.

Abstract

Four-dimensional gravity in the presence of a dilatonic scalar field and an Abelian gauge field is considered. This theory corresponds to the bosonic sector of a Kaluza-Klein dimensional reduction of eleven-dimensional supergravity which induces a determined self-interacting potential for the scalar field. We compute the conserved charges and carry out the thermodynamics of an anti-de Sitter (AdS) dyonic black hole solution recently proposed. The charges coming from symmetries of the action are computed by using the Regge-Teitelboim Hamiltonian approach. These correspond to the mass, which acquires contributions from the scalar field, and the electric charge. Integrability conditions are introduced because the scalar field leads to non-integrable terms in the variation of the mass. These conditions are generically solved by introducing boundary conditions that arbitrarily relates the leading and subleading terms of the scalar field fall-off. The Hamiltonian Euclidean action, computed in the grand canonical ensemble, is obtained by demanding the action to attain an extremum. Its value is given by a radial boundary term plus an additional polar angle boundary term due to the presence of a magnetic monopole. Remarkably, the magnetic charge can be identified from the variation of the additional polar angle boundary term, confirming that the first law of black hole thermodynamics is a consequence of having a well-defined and finite Hamiltonian action principle, even if the charge does not come from a symmetry of the action. The temperature and electrostatic potential are determined demanding regularity on the black hole solution, whereas the value of the magnetic potential is already identified in the variation of the additional polar angle boundary term. Consequently, the first law of black hole thermodynamics is identically satisfied by construction.