Varying Newton constant and black hole to white hole quantum tunneling

Understand

The thermodynamics of black holes is discussed for the case, when the Newton constant $G$ is not a constant, but is the thermodynamic variable.

• This gives for the first law of the Schwarzschild black hole thermodynamics: $dS_\text{BH}= -AdK + \frac{dM}{T_\text{BH}}$, where the gravitational coupling $K=1/4G$, $M$ is the black hole mass, $A$ is the area of horizon, and $T_\text{BH}$ is Hawking temperature.
• From this first law it follows that the dimensionless quantity $M^2/K$ 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.