The entropy of disorder fields in Euclidean Schwarzschild black hole

Abstract

In the Euclidean Section of the Schwarzschild manifold, we discuss the generalized entropy of a black hole. We consider an Euclidean quantum scalar effective field theory with additive quenched disorder. Using the distributional zeta-function method, a series representation of the average of the Gibbs free energy over the ensemble of possible configurations of the disorder is obtained. In this series representation, the effective actions give rise to generalized Schrödinger operators on Riemannian manifolds. Each term of the series is a ratio between the determinant of the Laplace–Beltrami operator (plus the mass term) and the aforementioned Schrödinger operators. This ratio represents how the geometry and the disorder fields are interacting to modify the thermodynamic properties of the system, and the modification of the fluctuations introduced by the disorder. The obtained structure allows to construct an entropy which explicitly shows how the introduction of an additive disorder field affects the thermodynamic properties of the black hole. The disorder field is interpreted as an effective model for unknown degrees of freedom near the event horizon. Finally, is presented the generalized entropy density with the contributions of the black hole geometric entropy, external matter and disorder fields, and the validity of the generalized second law is demonstrated.