Quantum explosions of black holes and thermal coordinates

Abstract

The Hawking temperature for Schwarzschild black hole $T_H=1/8\pi M$ is singular in the limit of vanishing mass $M\to 0$. However, the Schwarzschild metric itself is regular in this limit, it is reduced to the Minkowski metric and there are no reasons to believe that the temperature becomes infinite. This discrepancy may be due to the singularity of the Kruskal coordinates in this limit. To improve the situation, new coordinates for the Schwarzschild metric are introduced, called thermal coordinates, which depend on the black hole mass $ M $ and the parameter $b$. The thermal coordinates are regular in the limit $M\to 0$ when the Schwarzschild metric reduces to Minkowski metric, written in coordinates dual to the Rindler coordinates. Using the thermal coordinates the Schwarzschild black hole radiation is reconsidered and it is found that the Hawking formula for temperature is valid only for large black holes while for small black holes the temperature is $T=1/2\pi(4M+b)$. The thermal observer in Minkowski space sees radiation with temperature $T=1/2\pi b$, similar to the Unruh effect with non-constant acceleration. During evaporation, in the thermal coordinates the black hole mass is decreasing inverse proportional to time and the black hole lifetime is infinite. More general spherically symmetric metrics are considered and it is found that the property to have a temperature is not restricted to the cases of black holes or constant acceleration, but is valid for any spherically symmetric metric written in thermal coordinates. Implications for primordial black holes and for the information loss problem are mentioned.