Renormalization group improved black hole spacetimes

Abstract

We study the quantum gravitational effects in spherically symmetric black hole spacetimes. The effective quantum spacetime felt by a pointlike test mass is constructed by ``renormalization group improving'' the Schwarzschild metric. The key ingredient is the running Newton constant which is obtained from the exact evolution equation for the effective average action. The conformal structure of the quantum spacetime depends on its ADM mass $M$ and it is similar to that of the classical Reissner-Nordstr\"om black hole. For $M$ larger than, equal to, and smaller than a certain critical mass ${M}_{\mathrm{cr}}$ the spacetime has two, one, and no horizon(s), respectively. Its Hawking temperature, specific heat capacity, and entropy are computed as a function of $M.$ It is argued that the black hole evaporation stops when $M$ approaches ${M}_{\mathrm{cr}}$ which is of the order of the Planck mass. In this manner a ``cold'' soliton-like remnant with the near-horizon geometry of ${\mathrm{AdS}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{2}$ is formed. As a consequence of the quantum effects, the classical singularity at $r=0$ is either removed completely or it is at least much milder than classically; in the first case the quantum spacetime has a smooth de Sitter core which would be in accord with the cosmic censorship hypothesis even if $M<{M}_{\mathrm{cr}}.$