Smarr Formula and an Extended First Law for Lovelock Gravity

Abstract: We study properties of static, asymptotically AdS black holes in Lovelock gravity. Our main result is a Smarr formula that gives the mass in terms of geometrical quantities together with the parameters of the Lovelock theory. As in Einstein gravity, the Smarr formula follows from applying the first law to an infinitesimal change in the overall length scale. However, because the Lovelock couplings are dimensionful, we must first prove an extension of the first law that includes their variations. Key ingredients in this construction are the Killing-Lovelock potentials associated with each of the the higher curvature Lovelock interactions. Geometric expressions are obtained for the new thermodynamic potentials conjugate to variation of the Lovelock couplings.

• The paper constructs a novel Smarr formula that links mass, temperature, entropy, and new thermodynamic potentials through scaling arguments.
• It extends the traditional first law by incorporating variations in Lovelock couplings, offering deeper insights into black hole thermodynamics.
• The study employs Komar integrals and asymptotically AdS conditions to explore implications for phase transitions and the AdS/CFT correspondence.

An Analysis of Smarr Formula and Extended First Law for Lovelock Gravity

The paper under discussion offers an intricate examination of the static, asymptotically AdS black holes within Lovelock gravity, providing a Smarr formula for the ADM mass and an extension of the first law of thermodynamics that incorporates variations in the Lovelock couplings. The work of Kastor, Ray, and Traschen explores the distinctive properties of Lovelock gravity and advances the understanding of black hole thermodynamics in these higher-dimensional theories.

Lovelock gravity represents a subset of higher curvature gravity theories notable for leaving the highest derivatives in its field equations at second order. It maintains many appealing features similar to Einstein gravity, such as the potential for stable and ghost-free constant curvature vacua, making it a promising framework for quantization efforts. The study performs extensions to theories of Lovelock gravity, initially built on insights from prior work on Einstein gravity with non-zero cosmological constants.

This paper's primary contribution comes through constructing a Smarr formula for Lovelock gravity, offering a relationship between mass $M$ and various thermodynamic quantities. Equation (4) elucidates the result:

$(D − 3)M = (D − 2)TS − XkΨ^{(k)}b^{(k)}k/2(16πG),$

where $T$ is the temperature, $S$ the entropy, and Ψ^k represents new thermodynamic potentials arising from variations in the Lovelock couplings. The method used pivots on both Komar integrals and an innovative utilization of scaling arguments to frame the formula's derivation. These geometric constructions and first law adaptations are critical in managing Lovelock's additional dimensionful coupling terms that complicate formulations, unlike Einstein gravity's first law where only the cosmological constant varies.

By exploring the boundaries of Lovelock vacua and addressing the divergences within the first law, the paper substantiates how existing techniques fail when extended beyond the cosmological constant to Lovelock scenarios, necessitating new methodologies. These efforts culminate in defining geometric potentials Ψ^k, connecting the Lovelock couplings to generalized thermodynamic behavior.

The derivation provides physical insights beyond conventional applications by linking the extended first law and Smarr formula with the AdS/CFT correspondence. This relationship gives prospective paths for exploring high-temperature behavior within boundary CFTs influenced by Lovelock terms. These endeavors echo significant discussion in past literature regarding stability and phase transitions, notably the Hawking-Page transition discerned through a free energy analysis for Lovelock black holes.

The authors outline future explorations focusing on the utility of these formulas in analyzing free energies and phase transitions, emphasizing higher Lovelock order theories where explicit solutions are inaccessible. These examinations could provide foundational understanding in deciphering classical quantum gravity correspondences, particularly within rotating and dynamical black hole scenarios, dating back to foundational AdS/CFT insights.

The study represents a methodologically rigorous and conceptually innovative approach towards understanding black hole thermodynamics through the lens of Lovelock gravity, highlighting the potential implications of integrating geometric coupling variations into well-established gravitational theories.