Entropy: From Black Holes to Ordinary Systems

TL;DR

This paper explores a spacetime-based definition of entropy, linking black hole thermodynamics to ordinary systems, and suggests a more general understanding of entropy beyond microstate counting, emphasizing a dynamical approach.

Contribution

It introduces an alternative, spacetime-based definition of entropy and establishes a relation between action and entropy applicable to both black holes and ordinary systems.

Findings

A local order-based entropy definition exists and aligns with traditional results under equilibrium.

A relation between time interval and inverse temperature is physically meaningful beyond mathematical tricks.

A connection between action and entropy is established, applicable to black holes and ordinary systems.

Abstract

Several results of black holes thermodynamics can be considered as firmly founded and formulated in a very general manner. From this starting point we analyse in which way these results may give us the opportunity to gain a better understanding in the thermodynamics of ordinary systems for which a pre-relativistic description is sufficient. First, we investigated the possibility to introduce an alternative definition of the entropy basically related to a local definition of the order in a spacetime model rather than a counting of microstates. We show that such an alternative approach exists and leads to the traditional results provided an equilibrium condition is assumed. This condition introduces a relation between a time interval and the reverse of the temperature. We show that such a relation extensively used in the black hole theory, mainly as a mathematical trick, has a very general and physical meaning here; in particular its derivation is not related to the existence of a canonical density matrix. Our dynamical approach of thermodynamic equilibrium allows us to establish a relation between action and entropy and we show that an identical relation exists in the case of black holes. The derivation of such a relation seems impossible in the Gibbs ensemble approach of statistical thermodynamics. From these results we suggest that the definition of entropy in terms of order in spacetime should be more general that the Boltzmann one based on a counting of microstates. Finally we point out that these results are obtained by reversing the traditional route going from the Schr\"{o}dinger equation to statistical thermodynamics.