Entropy of gravitating systems: scaling laws versus radial profiles

TL;DR

This paper investigates the entropy properties of finite, spherically symmetric gravitating systems with perfect fluids, revealing how entropy distribution shifts from volume-based to surface-based as the system approaches black hole conditions.

Contribution

It generalizes non-extensivity methods to gravitating systems under external pressure, analyzing entropy scaling laws and internal profiles in relation to black hole thermodynamics.

Findings

Entropy scales as 1/r inside the system when far from Schwarzschild limit.

Entropy resides mainly in internal layers away from the surface in certain regimes.

Global area scaling of information does not imply boundary confinement, but volume distribution.

Abstract

Through the consideration of spherically symmetric gravitating systems consisting of perfect fluids with linear equation of state constrained to be in a finite volume, an account is given of the properties of entropy at conditions in which it is no longer an extensive quantity (it does not scale with system's size). To accomplish this, the methods introduced by Oppenheim [1] to characterize non-extensivity are used, suitably generalized to the case of gravitating systems subject to an external pressure. In particular when, far from the system's Schwarzschild limit, both area scaling for conventional entropy and inverse radius law for the temperature set in (i.e. the same properties of the corresponding black hole thermodynamical quantities), the entropy profile is found to behave like 1/r, being r the area radius inside the system. In such circumstances thus entropy heavily resides in internal layers, in opposition to what happens when area scaling is gained while approaching the Schwarzschild mass, in which case conventional entropy lies at the surface of the system. The information content of these systems, even if it globally scales like the area, is then stored in the whole volume, instead of packed on the boundary.