Extended black hole thermodynamics from extended Iyer-Wald formalism

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.

• The paper extends the Iyer-Wald formalism by treating coupling constants, including the cosmological constant, as dynamical variables.
• It introduces modified thermodynamic laws and Smarr relations to address ambiguities in energy and volume definitions for AdS black holes and higher curvature theories.
• The research offers new insights into string theory and holography by refining the thermodynamic framework of black holes.

Extended Black Hole Thermodynamics from Extended Iyer-Wald Formalism

Introduction

The paper "Extended black hole thermodynamics from extended Iyer-Wald formalism" (2308.12630) addresses the advancements in black hole thermodynamics, particularly focusing on the extension of the Iyer-Wald formalism to incorporate variability in coupling constants such as the cosmological constant in thermodynamic analyses. This approach reflects a modern paradigm in theoretical physics where constants traditionally held fixed, like the cosmological constant $\Lambda$, are treated as dynamical variables. By integrating these parameters into the framework, a more comprehensive understanding of black hole thermodynamic behavior is achieved, transcending conventional boundaries. The authors aim to establish a robust foundation for these extended thermodynamic principles, paving the way for more unified theories in gravitational physics.

Conventional Iyer-Wald Formalism

The Iyer-Wald formalism is crucial in deriving the laws of black hole thermodynamics within any diffeomorphism invariant theory of gravity. This formalism facilitates the extraction of the first law of thermodynamics by integrating the Noether charge over a hypersurface extending from a black hole's horizon to infinity. In this context, the paper utilizes this formalism to derive the thermodynamic potentials and relations for black holes, like those observed in Kerr and Schwarzschild-AdS solutions. The formalism's adaptability to stationary black holes with bifurcate Killing horizons allows for the derivation of fundamental laws such as the first law of black hole thermodynamics and Smarr relations, which are essential in understanding how mass, angular momentum, and entropy interrelate in various dimensions.

Extension to Higher Order Theories

The extension involves treating $\Lambda$ and other parameters as thermodynamic variables, leading to the augmented variation $\tilde{\delta}$ that considers changes in both the metric and these variables. This alteration affects the thermodynamic first law, now incorporating additional terms that account for variabilities in $\Lambda$ and coupling parameters $\alpha_m$ associated with higher curvature terms like those in Lovelock or Gauss-Bonnet gravity. The authors introduce a novel strategy to compute associated thermodynamic conjugate quantities independently, which becomes particularly relevant in complex black hole solutions such as those incorporating higher curvature terms. This method promises to resolve ambiguities observed in previous theoretical models regarding the thermodynamic vs. geometric volume distinctions.

Implications for AdS Black Holes

The paper discusses specific black hole solutions, such as AdS-Schwarzschild and AdS-Kerr, illustrating how the extended formalism refines our understanding of their thermodynamics. The AdS-Kerr black hole, in particular, showcases discrepancies between conventional and extended thermodynamics, further complicated by its angular momentum and asymptotic structure. The derived first law encompasses an integral balance incorporating rotational parameters, mass, and pressure, demonstrating the multi-faceted nature of these black holes' thermodynamic behavior. Such refined understanding is significant for string theory applications and the AdS/CFT correspondence, where these thermodynamic principles manifest in dual conformal field theories as variations in central charge and other observables.

Application to Higher Curvature Gravity

For gravity theories with higher curvature corrections, exact solutions such as those in Gauss-Bonnet gravity reflect the impact of additional geometric considerations. The paper extends the formalism to these theories by incorporating the extended first law and Smarr relations within their specific context. The explicit calculation for static, spherically symmetric solutions reveals how additional terms influence thermodynamic volume and energy definitions. They underline a systematic approach to evaluating these variations, which may offer broad utility across gravitational theories with complex interactions.

Conclusion

This research substantiates a rigorous framework for extended black hole thermodynamics by adapting the Iyer-Wald formalism. By promoting certain parameters to thermodynamic variables, the theoretical landscape is more aligned with modern physics, providing new insights into the microscopic structure of space-time. Future developments could involve exploring these principles in other gravitational theories or further integrating them into quantum gravity theories. This framework not only encapsulates the intricacies of black holes within a larger cosmos but also aligns with contemporary explorations of holography and gauge/gravity dualities, which continue to challenge and redefine theoretical boundaries.