Remarks on the thermofield double state in 4D black hole background

Abstract: Recently it was shown that there is an anomalous singularity of propagators in spacetimes with horizons for thermal states with non--canonical temperatures. In this paper we extend these observations to the situation when the background geometry is given by the four--dimensional Schwarzschild and Reissner-Nordstr\"om black holes. Namely, we demonstrate that the two-point function in the free scalar field theory acquires anomalous singularity when the two points are located on the horizon. This singularity is anomalous in the sense that the coefficient before the divergent term in the two-point function differs from the canonical one and depends explicitly on the temperature of the state. As it was previously shown, it leads to the explosive behavior of the regularised stress-energy tensor on the horizon when the temperature of the state does not coincide with the canonical temperature of the horizon.