# Pressure and chemical potential of quantum Schwarzschild black hole

Just as Stephen Hawking showed that even Schwarzschild black holes have a temperature, shouldn't it also have a pressure and chemical potential? Are there any analytical formulae of those as well as $$T_{BH}=\dfrac{\hbar c^3}{8\pi GMk_B}?$$

Since the analogy with thermodynamics should be complete (the 4 laws of BH are analogues to those of thermodynamics), if you write: $$\def\d{\mathrm{d}}\d E=T\d S-p\d V+μ\d N$$, and since (quantum) black holes have entropy (area), and temperature (surface gravity), one should ask naturally what is the (quantum) pressure and the chemical potential equivalents, even in Schwarzschild black holes. It has volume so it should have a pressure,... However, the chemical potential part is much more intriguing since, indeed, we do not know what BH constituents truly are. Also, even with uncharged black holes, shouldn't we have a chemical term due to e.g. the cosmological constant? What about the NUT "charge" terms?

I guess for the pressure (energy density) of Schwarzschild black holes this quantity:

$$P_S=\dfrac{E_S}{V_S}=\dfrac{k_BT_{BH}}{\dfrac{4\pi}{3}\left(\dfrac{2GM}{c^2}\right)^3}=\dfrac{3\hbar c^9}{256\pi^2G^4M_\odot^4}\left(\dfrac{M_\odot}{M}\right)^4 \approx 8\cdot 10^{-42}\left(\dfrac{M_\odot}{M}\right)^4\;\; Pa$$

However I am not sure if it is meaningful and how to guess the chemical potential (if any) for (quantum) Schwarzschild black holes... Has it any sense or not?

• Can you explain your reasoning. Are you arguing by analogy to a gas or a chemical. A gas has a temperature, and pressure. A chemical species has a temperature and chemical potential. And so by analogy, a black hole has pressure and chemical potential. Is this your reasoning? Or is there something more? Commented Feb 20, 2022 at 15:05
• Yes. Since the analogy with thermodynamics should be complete (the 4 laws of BH are analogues to those of thermodynamics), if you write: $dE=TdS-pdV+\mu dN$, and since (quantum) black holes have entropy (area), and temperature (surface gravity), one should ask naturally what is the (quantum) pressure and the chemical potentical equivalentes even in Schwarzschild black holes, because it has volume it should has a pressure,...However the chemical potentical part is much more intriguing since, indeed, we do not know what BH constituents trule ARE... Commented Feb 20, 2022 at 15:27
• I imagine that the radiation pressure from Hawking radiation is a non-zero but ridiculously small quantity (excepting those with tiny mass). I don't think chemical potential is relevant to black holes because the only external observables are mass, angular momentum and charge. Black holes are odd ducks thermodynamically, because adding energy causes their temperature to fall, and I imagine analogously applying force to their surface would cause them to expand. Commented Feb 21, 2022 at 15:18

There are papers discussing pressure and volume terms, like Johnson, C. V. (2014). Holographic heat engines. Classical and Quantum Gravity, 31(20), 205002. In this paper the pressure is due to the cosmological constant of spacetime $$p=-\Lambda/8\pi$$ and the volume is the volume of the black hole. This allows talking about enthalpy, and analysing heat engines with black holes as working "fluids". The paper speculates that there might be a chemical potential that is some positive power of the number of degrees of freedom, but does not define it.