Varying Newton constant and black hole to white hole quantum tunneling

The thermodynamics of black holes is discussed for the case, when the Newton constant G is not a constant, but it is the thermodynamic variable. This gives for the first law of the Schwarzschild black hole thermodynamics: dS_BH = −AdK + _T^dM_BH, where the gravitational coupling K = 1/4G, M is the black hole mass, A is the area of horizon, and T_BH is Hawking temperature. From this first law, it follows that the dimensionless quantity M²/K is the adiabatic invariant, which, in principle, can be quantized if to follow the Bekenstein conjecture. From the Euclidean action for the black hole it follows that K and A serve as dynamically conjugate variables. Using the Painleve-Gullstrand metric, which in condensed matter is known as acoustic metric, we calculate the quantum tunneling from the black hole to the white hole. The obtained tunneling exponent suggests that the temperature and entropy of the white hole are negative.