Path integral over black hole fluctuations

Evaluating a functional integral exactly over a subset of metrics that represent the quantum fluctuations of the horizon of a black hole, we obtain a Schrödinger equation in null coordinate time for the key component of the metric. The equation yields a current that preserves probability if we use the most natural choice of functional measure. This establishes the existence of blurred horizons and a thermal atmosphere. It has been argued previously that the existence of a thermal atmosphere is a direct concomitant of the thermal radiation of black holes when the temperature of the hole is greater than that of its larger environment, which we take as zero.