Non-extreme black holes near the extreme state and acceleration horizons: thermodynamics and quantum-corrected geometry

Abstract

We consider the class of metrics that can be obtained from those of non-extreme black holes by limiting transitions to the extreme state such that the near-horizon geometry expands into a whole manifold. These metrics include, in particular, the Rindler and Bertotti-Robinson spacetimes. The general formula for the entropy of massless radiation valid either for black hole or for acceleration horizons is derived. It is argued that, as a black hole horizon in the limit under consideration turns into an acceleration one, the thermodynamic entropy Sq of quantum radiation is due to the Unruh effect entirely and Sq = 0 exactly. The contribution to the quasilocal energy from a given curved spacetime is equal to zero and the only non-vanishing term stems from a reference metric. In the variation procedure necessary for the derivation of the general first law, the metric on a horizon surface changes along with the boundary one, and the account for gravitational and matter stresses is an essential ingredient of the first law. This law confirms the property Sq = 0. The quantum-corrected geometry of the Bertotti-Robinson spacetime is found and it is argued that backreaction of quantum fields mimics the effect of the cosmological constant eff and can drastically change the character of spacetime depending on the sign of eff - for instance, turn AdS2 × S2 into dS2 × S2 or Rindler2 × S2. The latter two solutions can be thought of as the quantum versions of the cold and ultracold limits of the Reissner-Nordstrom-de Sitter metric.