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: $d{S}_{\mathrm{BH}}=-AdK+\frac{dM}{T_{\mathrm{BH}}}$, where the gravitational coupling $K=1/4G$, M is the black hole mass, A is the area of horizon, and $T_{\mathrm{BH}}$ 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. 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.

中文翻译：

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牛顿常数和黑洞至白洞量子隧穿的变化

当牛顿常数G不是常数，而是热力学变量时，讨论了黑洞的热力学。这给出了Schwarzschild黑洞热力学的第一定律：$d{\mathrm{小号}}_{\mathrm{BH}}=-\mathrm{一种}dķ+\frac{d\mathrm{中号}}{{Ť}_{\mathrm{BH}}}$，其中重力耦合 $ķ=\mathrm{1个}/4G$，M是黑洞质量，A是视界面积，并且${Ť}_{\mathrm{BH}}$是霍金温度。从这第一定律，可以得出无量纲的量${\mathrm{中号}}^2/ķ$是绝热不变量，原则上可以遵循贝肯斯坦猜想对其进行量化。从黑洞的欧几里得作用得出，K和A充当动态共轭变量。使用Painleve-Gullstrand度量（在凝聚态中称为声学度量），我们计算了从黑洞到白洞的量子隧穿。所获得的隧穿指数表明白洞的温度和熵为负。