Abstract
Until now, the critical behavior of Lifshitz black holes, in an extended $P\ensuremath{-}v$ space, has not been studied, because it is impossible to find an analytical equation of state, $P=P(v,T)$, for an arbitrary Lifshitz exponent $z$. In this paper, we adopt a new approach toward thermodynamic phase space and successfully explore the critical behavior of ($n+1$)-dimensional Lifshitz dilaton black holes. The most important advantage of this approach is that we keep the cosmological constant as a constant without needing to vary it. For this purpose, we write down the equation of state as ${Q}^{s}={Q}^{s}(T,\mathrm{\ensuremath{\Psi}})$, where $\mathrm{\ensuremath{\Psi}}={(\ensuremath{\partial}M/\ensuremath{\partial}{Q}^{s})}_{S,P}$ is the conjugate of ${Q}^{s}$, and construct a Smarr relation based on this new phase space as $M=M(S,{Q}^{s},P)$, where $s=2p/(2p\ensuremath{-}1)$, with $p$ the power of the power-law Maxwell Lagrangian. We justify such a choice mathematically and show that with this new phase space, the system admits the critical behavior and resembles the van der Waals fluid system when the cosmological constant (pressure) is treated as a fixed parameter, while the charge of the system varies. We obtain the Gibbs free energy of the system and find a swallowtail shape in Gibbs diagrams, which represents the first-order phase transition. Finally, we calculate the critical exponents and show that although thermodynamic quantities depend on the metric parameters such as $z$, $p$, and $n$, the critical exponents are the same as the van der Waals fluid-gas system. This alternative viewpoint of the phase space of a Lifshitz dilaton black hole can be understood easily since one can imagine such a change for a given single black hole, i.e., acquiring charge, which induces the phase transition. Our results further support the viewpoint suggested in [A. Dehyadegari, A. Sheykhi, and A. Montakhab, Phys. Lett. B 768, 235 (2017)].
Highlights
The Lifshitz spacetime is not a vacuum solution of Einstein gravity and so needs matter source
We cannot have an analytical equation of state, P = P (v, T ), to investigate the critical behavior or calculate critical quantities of Lifshitz black holes
We find out that in case of Lifshitz dilaton black holes, the system admits a critical behaviour provided we take the electrodynamics in the form of power-Maxwell field and considering Qs as a thermodynamic variable with Ψ = (∂M /∂Qs)S,P as its conjugate, where s = 2p/(2p − 1)
Summary
We are going to review the solutions of charged Lifshitz black holes with power Maxwell field [32], with emphasizing on their thermodynamic properties. The (n+1)dimensional action of Einstein-dilaton gravity in the presence of a power Maxwell electromagnetic and two linear Maxwell fields can be written as. As one can see from expression (2.23) it is nearly impossible to solve this equation for P (or more precisely for l) and write an analytical equation of state, P = P (v, T ) for an arbitrary Lifshitz exponent z. This implies that, for the Lifshitz dialton black holes, one cannot investigate the critical behavior of the system through an extended P − v phase space by treating the cosmological constant (pressure) as a thermodynamic variable. As we shall see it is quite possible to investigate the critical behaviour of this system through a new Qs − Ψ phase space and show its similarity with Van der Waals fluid system
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