[Paper Review] P-V criticality of higher dimensional black holes with nonlinear source

S. H. Hendi, M. H. Vahidinia|arXiv (Cornell University)|Dec 26, 2012

Black Holes and Theoretical Physics10 citations

TL;DR

This paper investigates the P-V criticality of higher-dimensional black holes coupled to a power-Maxwell nonlinear electrodynamics source in an extended phase space where the cosmological constant acts as pressure. It reveals phase transitions in both canonical and grand canonical ensembles—unlike in standard Einstein-Maxwell theory—and finds critical exponents matching mean field theory, with a universal $P_c v_c / T_c$ ratio independent of spacetime dimensions.

ABSTRACT

In this paper, we consider the solutions of Einstein gravity in the presence of a generalized Maxwell theory, namely power Maxwell invariant. First, we investigate the analogy of nonlinear charged black hole solutions with the Van der Waals liquid--gas system in the extended phase space where the cosmological constant appear as pressure. Then, we plot isotherm $P$--$V$ diagram and study the thermodynamics of AdS black hole in the (grand canonical) canonical ensemble in which (potential) charge is fixed at infinity. Interestingly, we find the phase transition occurs in the both of canonical and grand canonical ensembles in contrast to RN black hole in Maxwell theory which only admits canonical ensemble phase transition. Moreover, we calculate the critical exponents and find their values are the same as those in mean field theory. Besides, considerably, we find in the grand canonical ensembles universal ratio $\frac{P_{c}v_{c}}{T_{c}}$ is independent of spacetime dimensions.

Motivation & Objective

• To explore the thermodynamic behavior of higher-dimensional black holes in the presence of a generalized power-Maxwell electrodynamics theory.
• To investigate whether phase transitions occur in both canonical and grand canonical ensembles under this nonlinear electrodynamics framework.
• To determine the critical exponents and assess their universality in relation to mean field theory.
• To examine the dimensionality dependence of the universal ratio $P_c v_c / T_c$ in the grand canonical ensemble.

Proposed method

• Formulate the Einstein field equations coupled to a power-Maxwell Lagrangian, where the Maxwell invariant is raised to a power $p$.
• Introduce the cosmological constant as thermodynamic pressure in the extended phase space, treating it as a variable conjugate to volume.
• Analyze the thermodynamics of asymptotically AdS black holes in the canonical ensemble with fixed charge at infinity.
• Construct P-V isotherms to visualize phase transitions and identify critical points via the Maxwell equal-area law.
• Calculate critical exponents by analyzing the behavior of thermodynamic quantities near the critical point.
• Evaluate the universal ratio $P_c v_c / T_c$ in the grand canonical ensemble and assess its independence from spacetime dimension.

Experimental results

Research questions

• RQ1 Does the inclusion of nonlinear power-Maxwell electrodynamics induce phase transitions in both canonical and grand canonical ensembles for higher-dimensional AdS black holes?
• RQ2 Are the critical exponents of the black hole system consistent with those of mean field theory?
• RQ3 Does the universal ratio $P_c v_c / T_c$ remain invariant across different spacetime dimensions in the grand canonical ensemble?
• RQ4 How does the P-V diagram of the black hole system compare to the Van der Waals liquid-gas system under the same thermodynamic framework?

Key findings

• Phase transitions occur in both the canonical and grand canonical ensembles for higher-dimensional black holes with power-Maxwell nonlinear electrodynamics, a feature absent in standard Einstein-Maxwell theory.
• The critical exponents of the black hole system are found to be identical to those in mean field theory, indicating universal critical behavior.
• The universal ratio $P_c v_c / T_c$ is independent of spacetime dimensions in the grand canonical ensemble, suggesting a deep underlying universality.
• The P-V isotherms exhibit a characteristic swallowtail shape, confirming the presence of first-order phase transitions and critical points.