Thermodynamics of black holes with an infinite effective area of a horizon

TL;DR

This paper investigates black holes with infinite horizon area and zero temperature in dilaton theories, revealing conditions for regularity, issues with standard entropy calculations, and proposing a modified variational approach.

Contribution

It demonstrates that infinite horizon area is compatible with zero temperature only if the geometry is regular, and introduces a horizon boundary term method to restore variational consistency.

Findings

Infinite A is compatible with T_H=0 only with regular geometry.

Standard Euclidean approach yields indefinite entropy for infinite effective area.

Modified boundary term approach can produce zero entropy for certain black holes.

Abstract

In some kinds of classical dilaton theory there exist black holes with (i) infinite horizon area $A$ or infinite $F$ (the coefficient at curvature in Lagrangian) and (ii) zero Hawking temperature $T_{H}$. For a generic static black hole, without an assumption about spherical symmetry, we show that infinite $A$ is compatible with a regularity of geometry in the case $T_{H}=0$ only. We also point out that infinite $T_{H}$ is incompatible with the regularity of a horizon of a generic static black hole, both for finite or infinite $A$. Direct application of the standard Euclidean approach in the case of an infinite ''effective'' area of the horizon $A_{eff}=AF$ leads to inconsistencies in the variational principle and gives for a black hole entropy $S$ an indefinite expression, formally proportional to $T_{H}A_{eff}$. We show that treating a horizon as an additional boundary (that is, adding to the action some terms calculated on the horizon) may restore self-consistency of the variational procedure, if $F$ near the horizon grows not too rapidly. We apply this approach to Brans-Dicke black holes and obtain the same answer S=0 as for ''usual'' (for example, Reissner-Nordstr\"{o}m) extreme classical black holes. We also consider the exact solution for a conformal coupling, when $A$ is finite but $F$ diverges and find that in the latter case both the standard and modified approach give rise to an infinite action. Thus, this solution represents a rare exception of a black hole without nontrivial thermal properties.