Extended phase space thermodynamics for charged and rotating black holes and Born-Infeld vacuum polarization

TL;DR

The paper extends black hole thermodynamics by treating the cosmological constant as pressure and exploring charged and rotating AdS BHs across dimensions, including Born-Infeld nonlinearity. It demonstrates Van der Waals–like phase transitions with mean-field exponents for d>3 RN-AdS and slowly rotating BHs, while showing no critical behavior in d=3 BTZ cases. A novel Born-Infeld vacuum polarization B is introduced to preserve the first law and Smarr relation, unveiling a rich BI–AdS phase structure with RN-type and BI-type branches and zeroth-order transitions. Across these analyses, critical exponents remain universal and match the Van der Waals values, highlighting the robustness of mean-field behavior in extended black hole thermodynamics.

Abstract

We investigate the critical behaviour of charged and rotating AdS black holes in d spacetime dimensions, including effects from non-linear electrodynamics via the Born-Infeld action, in an extended phase space in which the cosmological constant is interpreted as thermodynamic pressure. For Reissner-Nordstrom black holes we find that the analogy with the Van der Walls liquid-gas system holds in any dimension greater than three, and that the critical exponents coincide with those of the Van der Waals system. We find that neutral slowly rotating black holes in four space-time dimensions also have the same qualitative behaviour. However charged and rotating black holes in three spacetime dimensions do not exhibit critical phenomena. For Born-Infeld black holes we define a new thermodynamic quantity B conjugate to the Born-Infeld parameter b that we call Born-Infeld vacuum polarization. We demonstrate that this quantity is required for consistency of both the first law of thermodynamics and the corresponding Smarr relation.

Figures (32)

• Figure 1: Gibbs free energy of charged AdS black hole in $d=4$. Gibbs free energy $G$ of 4D RN-AdS black hole is depicted as function of temperature $T$ for fixed $q=1$ and various pressures $P/P_c=1.6, 1, 0.6$ and $0.2$. The behaviour for $d>4$ is qualitatively similar, see next figure.
• Figure 2: Gibbs free energy in $d=10$. Gibbs free energy $G$ of 10D RN-AdS black hole is depicted as a function of temperature $T$ for fixed $q=1$ and pressures $P/P_c=1.6, 1, 0.6, 0.2$.
• Figure 3: $P-v$ diagram of charged AdS black hole in $d=4$. The temperature of isotherms decreases from top to bottom. The two upper dashed lines correspond to the "ideal gas" one-phase behaviour for $T>T_c$, the critical isotherm $T=T_c$ is denoted by the thick solid line, lower (red) solid lines correspond to two-phase state occurring for $T<T_c$. We have set $q=1$. The behaviour for $d>4$ is qualitatively similar -- see figure \ref{['fig:PV10D']}.
• Figure 4: $P-v$ diagram in $d=10$. The description coincides with that of the previous figure.
• Figure 5: Coexistence line of charged AdS black hole in $d=4$. This figure displays the coexistence line of the small-large black hole phase transition of a charged AdS black hole for $d=4$ system in the $(P, T)$-plane. The critical point is highlighted by a small circle at the end of the coexistence line. The behaviour for $d>4$ is qualitatively similar, as shown in figure \ref{['fig:PT6D']}.