Entropy production and the generalised second law of black hole thermodynamics

Abstract

The generalised second law of black hole thermodynamics states that the sum of a black hole’s entropy and the entropy of all matter outside the black hole cannot decrease with time. The violation of the generalised second law via the process in which a distant observer extracts work by lowering a box arbitrarily close to the event horizon of a black hole has two profound ramifications: (1) that the entropy of the Universe can be decreased arbitrarily via this process; and (2) that it is not appropriate to apply the laws of thermodynamics to systems containing black holes. In this paper, we argue that for the generalised second law to not be violated, entropy must be produced during the lowering process. To demonstrate this, we begin by deriving an equation for the locally measured temperature of the vacuum state of an observer that is a finite distance from the event horizon of a Schwarzschild black hole. Then, using this locally measured temperature and the Unruh effect, we derive an equation for the force required to hold this observer in a stationary position relative to a Schwarzschild black hole. These equations form a framework for calculating the change in black hole entropy as a result of the lowering process both in the case where the process is isentropic and in the case where entropy is produced during the lowering process. In the latter case, two requirements: (1) that the resultant change in black hole entropy is finite; and (2) that the resultant change in common entropy is finite, are used to identify two conditions that the form of an entropy production function must satisfy. These, in turn, are used to identify a set of possible functions describing the production of entropy. Using this set of functions, we demonstrate that the production of entropy limits the amount of work that the distant observer can extract from the lowering process. We find that this allows for the generalised second law to be preserved, provided that a coefficient in this set of functions satisfies a given bound. To conclude, we discuss two natural choices of this coefficient that allow for the generalised second law to be preserved in this lowering process. In addition to providing a resolution to this violation of the generalised second law, the framework presented in this paper can be applied to inform theories of gravity and quantum gravity on the form of their entropy relations, such that they do not violate the generalised second law.