Varying Newton constant, entropy and the black hole evaporation law

TL;DR

The paper investigates Einstein gravity with time-dependent $G$ and $\Lambda$, allowing non-conservation of the matter energy-momentum and deriving the resulting Bianchi identities. By interpreting the non-conservation through a thermodynamic entropy framework, it links $\partial_t G$ and $\partial_t \Lambda$ to entropy production. Applying this to Schwarzschild black holes, the authors derive a modified evaporation law that yields a lifetime longer by a factor of $\frac{9}{5}$ and shows the average energy density of the shrinking hole remains constant, with horizon entropy playing a central role via $S_{BH}=\frac{4\pi}{\hbar} M^{2} G$ and $T_{BH}=\frac{\hbar}{8\pi G M}$. The work provides a self-consistent horizon-thermodynamics framework for varying constants and suggests observational implications for primordial black holes and cosmological horizons.

Abstract

In Einstein equations we represent the energy-momentum tensor as the one ($T^{μν}$ ) of an ideal fluid plus the cosmological term. We consider time-dependent Newton ``constant" $G$, the cosmological term $Λ$ and non-conserved $T^{μν}$. The Bianchi identity imposes a relation between the energy-momentum (non)conservation and the variation of $G$ and $Λ$. If the energy-momentum $T^{μν}$ is conserved then both the Newton ``constant" $G$ and the cosmological term $Λ$ either do not change or both must depend on time. If the energy-momentum $T^{μν}$ is not conserved then the Bianchi identity implies a relation between the energy-momentum and a variation in time of either $G$ or $Λ$ (or both). We apply thermodynamics in order to express the non-conservation of the energy-momentum of an ideal fluid by entropy and relate the time variations of $G$ and $Λ$ to a change of entropy. Using the relation between a varying Newton constant G and the black hole entropy we derive a modified formula for the Schwarzschild black hole evaporation (a slower evaporation). Its life time is $\frac{9}{5}$ times larger than the one for a constant $G$. The average black hole's density remains constant when the black hole's radius shrinks to zero .