Varying Newton constant, entropy and the black hole evaporation law

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.