[Paper Review] Extended black hole thermodynamics from extended Iyer-Wald formalism

Yong Xiao, Yu Tian+1 more|arXiv (Cornell University)|Aug 24, 2023

Black Holes and Theoretical PhysicsPhysics and Astronomy3 citations

TL;DR

This paper extends the Iyer-Wald formalism to derive a universal framework for extended black hole thermodynamics in diffeomorphism-invariant gravity theories. By treating the cosmological constant and higher-curvature coupling constants as thermodynamic variables, it systematically constructs the first law and conjugate quantities, resolving ambiguities between thermodynamic and geometric volumes through a novel decomposition of the Noether charge variation.

ABSTRACT

In recent years, there has been significant interest in the field of extended black hole thermodynamics, where the cosmological constant and/or other coupling parameters are treated as thermodynamic variables. Drawing inspiration from the Iyer-Wald formalism, which reveals the intrinsic and universal structure of conventional black hole thermodynamics, we illustrate that a proper extension of this formalism also unveils the underlying theoretical structure of extended black hole thermodynamics. As a remarkable consequence, for any gravitational theory described by a diffeomorphism invariant action, it is always possible to construct a consistent extended thermodynamics using this extended formalism.

Motivation & Objective

• To establish a fundamental theoretical framework underlying extended black hole thermodynamics, where the cosmological constant and other coupling parameters are treated as thermodynamic variables.
• To resolve the long-standing ambiguity between thermodynamic volume and geometric volume in higher-curvature gravity models.
• To provide a systematic, first-principles derivation of the extended first law of thermodynamics for any diffeomorphism-invariant gravitational theory.
• To uncover the intrinsic connection between additional terms in the mass and volume corrections, ΔM and ΔV, arising from nontrivial asymptotic behavior of the theory.
• To generalize the formalism to include multiple coupling constants, such as those in Lovelock gravity, and to validate it using perturbative solutions in Gauss-Bonnet gravity.

Proposed method

• Extends the Iyer-Wald formalism by considering variations of the Lagrangian not only in fields but also in coupling constants like the cosmological constant Λ and higher-curvature coefficients αn.
• Derives the extended first law by computing the variation of the Noether charge associated with the time-translation Killing vector, including contributions from explicit coupling dependence.
• Introduces a regularization procedure to handle divergences in the Noether charge integral at spatial infinity, enabling consistent computation of conserved quantities.
• Identifies the thermodynamic volume as the sum of geometric volume V and correction terms ΔV, derived from the integrability condition of the Noether current variation.
• Uses the relation ΔM = 2∑(n−1)αnΔVn to connect mass and volume corrections, ensuring consistency of the extended first law.
• Applies the formalism to a model with Λ, α2, and α3 couplings, computing explicit expressions for ΔV2 and ΔV3 up to second order in perturbation theory.

Experimental results

Research questions

• RQ1 Can a universal formalism be constructed to derive extended black hole thermodynamics for any diffeomorphism-invariant gravity theory?
• RQ2 Why does the thermodynamic volume Vth differ from the geometric volume V in higher-curvature gravity, and how can this discrepancy be systematically resolved?
• RQ3 What is the origin of the correction terms ΔV and ΔM in the extended first law, and how are they related?
• RQ4 How can conjugate variables like thermodynamic volume be computed independently of thermodynamic relations, rather than via partial derivatives?
• RQ5 Can the formalism be generalized to include multiple coupling constants and applied to exact solutions in Lovelock gravity?

Key findings

• The extended Iyer-Wald formalism provides a first-principles derivation of the extended first law, ensuring consistency for any diffeomorphism-invariant gravity theory with dynamical couplings.
• The thermodynamic volume is shown to be Vth = V + ΔV, where ΔV arises from the variation of the Noether charge under changes in coupling constants, not just from field variations.
• Explicit expressions for ΔV2 and ΔV3 in a model with Λ, α2, and α3 are derived as ΔV2 = 32πMΛ/3 + α3×8192π²MΛ³/27 and ΔV3 = 32πMΛ²/3 + α3×10240π²MΛ⁴/27.
• The correction terms satisfy the relation ΔM = 2α2ΔV2 + 4α3ΔV3, confirming consistency between mass and volume corrections.
• The formalism resolves the ambiguity between thermodynamic and geometric volumes by deriving ΔV independently from thermodynamic relations.
• The method is general and applicable to higher-curvature gravity, including Lovelock theories, and is validated using perturbative solutions in Gauss-Bonnet gravity in D=5 dimensions.