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

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

2002-06-06

arxiv.org/abs/gr-qc/0206018v1

Class.Quant.Grav. 19 (2002) 3783-3798

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

24 pages. Accepted for publication in Class. Quant. Grav

Scientific paper

10.1088/0264-9381/19/14/317

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.

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