Numerical thermodynamic studies of classical gravitational collapse in 3+1 and 4+1 dimensions

TL;DR

This study numerically investigates the thermodynamic properties during classical gravitational collapse in 3+1 and 4+1 dimensions, confirming theoretical predictions about black hole free energy and entropy contributions from the interior region.

Contribution

It provides the first numerical validation of the relation between collapse dynamics and black hole thermodynamics in higher dimensions.

Findings

The free energy approaches E/2 in 4D and E/3 in 5D during late collapse stages.

Nonstationary metric regions are confined behind the event horizon.

Entropy mainly originates from the nonstationary interior region.

Abstract

We study a thermodynamic potential during the classical gravitational collapse of a 4D (3+1) massless scalar field to a Schwarzschild black hole in isotropic coordinates. We track numerically the function $F(t)=-dI/dt=-L$, where $I$ is the total action of matter plus gravitation, $L$ the total Lagrangian and $t$ is the time measured by a stationary clock at infinity. At late stages in the collapse, this function can be identified with the free energy $F=E-TS$ of the black hole where $E$ is the ADM mass, $T$ the Hawking temperature and $S$ the entropy. From standard black hole thermodynamics, the free energy of a 4D Schwarzschild black hole is equal to $E/2$. Our numerical simulations show that at late stages of the collapse the function $-L$ approaches a constant to within 5% of the value of $E/2$. We also present numerical results for the thermodynamics of 5D collapse where the free energy in this case is $E/3$. In both 4D and 5D, our numerical simulations show that at late stages of the collapse, the metric fields are nonstationary in a thin region just behind the event horizon (and are basically static everywhere else). The entropy stems mostly from the nonstationary interior region where there is a significant dip (negative contribution) to the free energy.