Black hole thermodynamics: No inconsistency via the inclusion of the missing $P\text{−}V$ terms

The early literature on black hole thermodynamics ignored the $P\text{−}V$ term associated with the existence of a fundamental physical constant in the black hole solution. The inclusion of this constant in the first law becomes inconsistent with the Smarr relation. Once the missing $P\text{−}V$ term is introduced, it becomes customary to introduce it only in problems where there is a negative cosmological constant. This practice is inherited from cosmological approaches which consider the quantity $-\mathrm{\Lambda }/8\pi$ as the constant pressure due to a cosmological fluid. However, the notions of pressure and thermodynamic volume in black hole thermodynamics are very different from their counterparts in classical thermodynamics. From this point of view, there is a priori no compelling reason to not extend this notion of pressure and associate a partial pressure with each “external” density ${8\pi T_t}^t$. In this work, we associate a partial pressure with a variable mass parameter as well as with each $tt$ component of the effective stress-energy tensor ${T_{\mathrm{eff}\mu }}^{\nu }$ but not with the linear component of the electromagnetic field. Using the field equations ${G_{\mu }}^{\nu }=8\pi {T_{\mathrm{eff}\mu }}^{\nu }$, we derive universal expressions for the enthalpy, internal energy, free energies, thermodynamic volume, equation of state, law of corresponding states, criticality, and critical exponents of static (nonrotating) charged black holes, with possibly a variable mass parameter, whether they are solutions to the Einstein field equations or not. We extend the derivation to the case where the black hole is immersed in the field of a quintessence force and to the multiforce case. Many applications and extensions are considered, including applications to regular black holes derived in previous and present work. No inconsistency has been noticed in their thermodynamics.

Figure 1

A generic $P-v$ diagram showing an isotherm for $T_0<T<T_c=1/(3\sqrt{6}\pi |Q|)$ and an isobar $P=\text{const}>0$ (dashed line). Maxwell’s equal area law applies to the oscillating part of the isotherm $T_0<T<T_c$ (dotted curve). This is expressed by the line integral $\oint v\mathrm{d}P=0$.

Figure 2

A generic $P-v$ diagram showing an isotherm for $0<T<T_0=1/(6\sqrt{3}\pi |Q|)$ and an isobar $P=\text{const}<0$. Maxwell’s equal area law is replaced by the law expressing the equality of the dark grey and light grey areas, with $v_1$ and $v_2$ being the specific volumes where the isotherm intersects the $v$ axis and $v_b$, the largest specific volume where the isobar intersects the isotherm.