BTZ black hole with KdV-type boundary conditions: Thermodynamics revisited

Abstract

The thermodynamic properties of the Bañados-Teitelboim-Zanelli (BTZ) black hole endowed with Korteweg-de Vries (KdV)-type boundary conditions are considered. This familiy of boundary conditions for General Relativity on AdS${}_3$ is labeled by a non-negative integer $n$, and gives rise to a dual theory which possesses anisotropic Lifshitz scaling invariance with dynamical exponent $z=2n+1$. We show that from the scale invariance of the action for stationary and circularly symmetric spacetimes, an anisotropic version of the Smarr relation arises, and we prove that it is totally consistent with the previously reported anisotropic Cardy formula. The set of KdV-type boundary conditions defines an unconventional thermodynamic ensemble, which leads to a generalized description of the thermal stability of the system. Finally, we show that at the self-dual temperature $T_s=\frac{1}{2\pi }(\frac{1}{z}{)}^{\frac{z}{z+1}}$, there is a Hawking-Page phase transition between the BTZ black hole and thermal AdS${}_3$ spacetime.

arXiv comment19 pages, 1 figure. Minor clarifications, affiliation added

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• Integrable Systems and Spacetime Dynamics
• Negative flows of generalized KdV and mKdV hierarchies and their gauge-Miura transformations
• Thermodynamics of Inhomogeneously Mass-deformed ABJM Model and Pressure Anisotropy
• Scaling symmetry, Smarr relation, and the extended first law in lower-dimensional Lovelock gravity
• 1/c deformations of AdS${}_3$ boundary conditions and the Dym hierarchy