  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.
