Counterterm method in Einstein dilaton gravity and the critical behavior of dilaton black holes with a power-Maxwell field

Abstract

We investigate the critical behavior of an (n+1)-dimensional topological dilaton black holes, in an extended phase space in both canonical and grand-canonical ensembles, when the gauge field is in the form of power-Maxwell field. In order to do this we introduce for the first time the counterterms that remove the divergences of the action in dilaton gravity for the solutions with curved boundary. Using the counterterm method, we calculate the conserved quantities and the action and therefore Gibbs free energy in both the canonical and grand-canonical ensembles. We treat the cosmological constant as a thermodynamic pressure, and its conjugate quantity as a thermodynamic volume. In the presence of power-Maxwell field, we find an analogy between the topological dilaton black holes with van der Walls liquid-gas system in all dimensions provided the dilaton coupling constant α and the power parameter p are chosen properly. Interestingly enough, we observe that the power-Maxwell dilaton black holes admit the phase transition in both canonical and grand-canonical ensembles. This is in contrast to RN-AdS, Einstein-Maxwell-dilaton and Born-Infeld-dilaton black holes, which only admit the phase transition in the canonical ensemble. Besides, we calculate the critical quantities and show that they depend on α, n and p. Finally, we obtain the critical exponents in two ensembles and show that they are independent of the model parameters and have the same values as mean field theory.

Reentrant phase transition of Born-Infeld-AdS black holes

“…In this approach the mass of black hole is treated as the enthalpy [6]. Critical behaviour of black holes in an extended phase space have been explored in various setups [7][8][9][10][11][12][13][14][15]. Recently, this approach has been utilized to present the emergence of superfluid-like phase transition for a class of AdS hairy black hole in Lovelock * Electronic address: asheykhi@shirazu.ac.ir gravity [16].…”

Critical behavior of Lifshitz dilaton black holes

“…Critical behavior of the Einstein-Maxwell-dilaton black holes has been studied in [28]. When the gauge field is in the form of Born-Infeld [29] and power-Maxwell [30] field, critical behavior of (n + 1)-dimensional dilaton black holes in an extended phase space have been investigated. Taking into account the dilaton field in the presence of logarithmic and exponential forms of nonlinear electrodynamics, and considering the cosmological constant and nonlinear parameter as thermodynamic quantities which can vary, it was shown that indeed there is a complete analogy between the nonlinear dilaton black holes with Van der Waals liquid-gas system [31].…”

Interaction between two point-like charges in nonlinear electrostatics

“…We considered the electrostatic problem of interaction between two point-like charges q 1 and q 2 placed in the points r = ±R by solving the nonlinear Maxwell equation (18), which follows from the least action principle for the Lagrangian (4), together with the standard Bianchi identity (8). For a small separation between the charges, R r , we found the electric field (31) in the approximation, linear with respect to the ratio R/r , which serves as the nonlinear extension of the usual dipole field.…”

“…[6][7][8][9][10][11][12] and the references therein) with finite selfa e-mail: breev@mail.tsu.ru b e-mail: shabad@lpi.ru energy of the point charge, allied to their famous prototype, the Born-Infeld model [13], where the finiteness of the field is achieved at the cost of square-root nonanalyticity of the Lagrangian, which leads to an infinity to the Maxwell equation. The most popular application [6][7][8][9][10][11][12][14][15][16][17][18][19][20][21][22]] of these models is to combine them with General Relativity in order to study their effect on the initial singularity and on the evolution of the Universe. In contrast to the Born-Infeld model, the models from the class of Ref.…”