Abstract
In this work it is shown that the thermodynamics of regular black holes with a cosmological horizon, which are solutions of Lovelock gravity, determines that they must evolve either into a state where the black hole and cosmological horizons have reached thermal equilibrium or into an extreme black hole geometry where the black hole and cosmological horizons have merged. This differs from the behavior of Schwarzschild de Sitter geometry which evolves into a de Sitter space, the ground state of the space of solutions. This occurs due to a phase transition of the heat capacity of the black hole horizon. To perform that analysis it is shown that at each horizon a local first law of thermodynamics can be obtained from the gravitational equations.
Highlights
The first law of thermodynamics of black holes was discovered by analyzing black hole geometries with a well defined asymptotic region
The first law was re-derived in terms of the analysis of the variation of Noether’s charges in [15]. It seems that spaces with cosmological horizons may challenge the existence of a first law of thermodynamics since they have no asymptotic region and conserved charges cannot be defined in the usual way
In that case the thermodynamics determines that geometry evolves from a twohorizons geometry into the de Sitter space, the ground state of the theory
Summary
The first law of thermodynamics of black holes was discovered by analyzing black hole geometries with a well defined asymptotic region. The Lovelock gravity equations of motion may have as many different constant curvature solutions as the highest power of the Riemann tensor presented in the Lagrangian [20,21] This defines a set of effective cosmological constants [Λ1, . In that case the thermodynamics determines that geometry evolves from a twohorizons geometry into the de Sitter space, the ground state of the theory This happens due to the temperature of the black hole horizon is always larger than the cosmological horizon one, while the heat capacities, defined as are always negative for the black hole horizon and positive for the cosmological horizon. The analysis of the Pure Lovelock vacuum solutions shows that the temperature of the cosmological horizon can be larger than the black hole one (within a region of the space of parameters). The analysis of a a generic Lovelock theory in this respect will be presented in a future work
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