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.
Highlights
Another motivation for studying the black hole solution presented in [1] has to do with the computation of its Gibbs free energy
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
We have carried out the thermodynamic analysis of a new class of AdS4 dyonic dilatonic black holes recently proposed in [1], which are solutions of the bosonic sector of a KaluzaKlein reduction of eleven-dimensional supergravity
Summary
We consider four-dimensional gravity with negative cosmological constant in the presence of an Abelian gauge field and a dilatonic scalar field with a self-interacting potential. Hereafter the gravitational constant is chosen as κ = 1/2.1 The self-interacting potential of the scalar field is given by. The theory given by (2.1) corresponds to the bosonic sector of two possible dimensional reductions, which depend on the coupling constant g in the following way. In the case of vanishing g the action is obtained after a S1 reduction of five-dimensional pure gravity. The dilatonic scalar field is given by φ (r) =. In the case of β1 = β2, the dilatonic scalar field is decoupled and the solution turns out to be an AdS dyonic Reissner-Nordstrom black hole where the electric and magnetic charges have the same value.
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