Varying Newton constant, entropy and the black hole evaporation law

TL;DR

This paper explores how a varying Newton constant G affects black hole entropy and evaporation, deriving relations between G and mass, and analyzing implications for black hole temperature and luminosity.

Contribution

It introduces a relation G ~ M^(-γ) linking Newton's constant to black hole mass, considering time-dependent G and entropy definitions, and discusses their impact on evaporation laws.

Findings

For S = dM/T, γ = 1, leading to constant evaporation temperature.

Using Bekenstein-Hawking entropy, γ = 2/3, affecting evaporation dynamics.

Different entropy models alter the temperature and luminosity evolution of black holes.

Abstract

In Einstein equations we represent the energy-momentum tensor as the one ($T^{\mu\nu}$ ) of a fluid plus the cosmological term. We consider time-dependent Newton ``constant" $G$, the cosmological term $\Lambda$ and non-conserved $T^{\mu\nu}$. The Bianchi identity imposes a relation between the energy-momentum (non)conservation and the variation of $G$ and $\Lambda$. The covariant divergence $\nabla_{\mu}T^{\mu\nu}$ can be related to the first law of thermodynamics. For compact systems of mass $M$ from the Bianchi identity we obtain a power-law relation $G\simeq M^{-\gamma}$ with $\gamma$ depending on pressure or entropy. We discuss radiation and a mass loss described by the Stefan-Boltzmann law. In this formula we insert an expression for the black hole area and its temperature $T$. The Bianchi identity together with a formula for temperature and entropy $S$ determines the index $\gamma$ in the relation between the Newton constant $G$ and the mass $M$. If the entropy $S$ is defined by the equation $dS=T^{-1}dM$ then $\gamma=1$ (the same as for zero pressure). If the formula of Bekenstein-Hawking entropy holds true for time-dependent $G$ then $\gamma=\frac{2}{3}$. We discuss consequences for the evaporation law of some modified expressions for the entropy appearing in effective models of gravity resulting from an interaction with matter fields. In particular, $\gamma=1$ leads to a constant evaporation temperature whereas $\gamma>1$ to a decreasing temperature and luminosity.