[Paper Review] Counterterm method in dilaton gravity and the critical behavior of dilaton black holes with power-Maxwell field

Z. Dayyani, Ahmad Sheykhi+1 more|arXiv (Cornell University)|Oct 29, 2016

Black Holes and Theoretical Physics3 citations

TL;DR

This paper introduces a novel counterterm method to regularize divergences in dilaton gravity with curved boundaries, enabling finite action and conserved quantities in both canonical and grand-canonical ensembles. It reveals that power-Maxwell-dilaton black holes exhibit P-V criticality and first-order phase transitions in both ensembles—unlike previous black hole models—while critical exponents match mean-field theory values across all dimensions and parameters.

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.

Motivation & Objective

• To develop a counterterm method that removes action divergences in dilaton gravity for spacetimes with curved asymptotic boundaries.
• To extend the thermodynamic phase space by treating the cosmological constant as pressure and its conjugate as thermodynamic volume.
• To investigate the critical behavior of (n+1)-dimensional topological dilaton black holes coupled to a power-Maxwell gauge field in both canonical and grand-canonical ensembles.
• To determine whether nonlinear power-Maxwell electrodynamics induces phase transitions in both ensembles, unlike previous models.
• To compute critical quantities and exponents and assess their dependence on dilaton coupling α, power parameter p, and spacetime dimension n.

Proposed method

• Derive counterterms from curvature invariants of the boundary at infinity to cancel divergences in the action for dilaton gravity with curved asymptotics.
• Construct the total finite action as the sum of bulk, Gibbons-Hawking, and counterterm contributions: $ I = I_{\text{bulk}} + I_{\text{GH}} + I_{\text{ct}} $.
• Use the finite action to compute conserved quantities (mass, electric charge, thermodynamic volume) and the Gibbs free energy in both ensembles.
• Treat the cosmological constant $ \Lambda $ as thermodynamic pressure $ P = -\Lambda / 8\pi $, and its conjugate as thermodynamic volume $ V $.
• Derive the equation of state $ P(v,T) $ from the Smarr relation and thermodynamic identities, enabling P-v isotherm analysis.
• Analyze the Gibbs free energy $ G(T) $ to identify swallowtail behavior, signaling first-order phase transitions in both ensembles.

Experimental results

Research questions

• RQ1 Can a counterterm method be formulated for dilaton gravity with curved boundaries to render the action finite?
• RQ2 Do (n+1)-dimensional dilaton black holes with power-Maxwell fields exhibit P-V criticality in both canonical and grand-canonical ensembles?
• RQ3 How do the critical exponents of these black holes compare to those of the van der Waals system and mean-field theory?
• RQ4 What is the dependence of critical quantities (P_c, V_c, T_c) on the dilaton coupling constant $ \alpha $, power parameter $ p $, and spacetime dimension $ n $?
• RQ5 Does the inclusion of nonlinear power-Maxwell electrodynamics alter the ensemble dependence of phase transitions compared to linear or Born-Infeld cases?

Key findings

• The authors introduce the first counterterm method for dilaton gravity with curved boundaries, ensuring finite action and conserved quantities despite potentially infinite counterterm series.
• Power-Maxwell-dilaton black holes exhibit P-V criticality and first-order phase transitions in both canonical and grand-canonical ensembles, a feature absent in RN-AdS, Einstein-Maxwell-dilaton, and Born-Infeld-dilaton black holes.
• The critical exponents are universal and match mean-field theory values (e.g., $ \alpha = 0 $, $ \beta = \frac{1}{2} $, $ \gamma = 1 $, $ \delta = 3 $), independent of $ \alpha $, $ p $, and $ n $.
• Critical quantities such as $ P_c $, $ V_c $, and $ T_c $ depend explicitly on $ \alpha $, $ p $, and $ n $, with explicit expressions derived for $ G $, $ P $, $ V $, and $ T $ in terms of $ r_+ $, $ b $, $ U $, and $ \alpha $.
• The Gibbs free energy $ G(T) $ exhibits a swallowtail structure in both ensembles, confirming the presence of a first-order small-large black hole phase transition.
• The system shows complete analogy with the van der Waals liquid-gas system in all dimensions when $ \alpha $ and $ p $ are appropriately chosen, with consistent critical exponents across ensembles.