Varying Newton constant, entropy and the black hole evaporation law

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 Newtonconstant" $G$ 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 .