Abstract
The thermodynamic properties of the Ba\~nados-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.
Highlights
In the pursuit of a better understanding of quantum gravity, in the past two decades, a lot of interest has been put into the so called gauge/gravity correspondence, whose most celebrated example is the AdS=CFT duality [1,2]
One of the first main results was the renowned article from Brown and Henneaux [3], where they showed that the asymptotic symmetries of General Relativity in three dimensions with negative cosmological constant correspond to the conformal algebra in two dimensions with a classical central charge given by c 1⁄4 3l=2G, where l is the AdS radius and G the Newton constant
Following [22], we will adopt an unconventional approach to this holographic realization for field theories possessing anisotropic scaling properties, where the spacetime anisotropy at the boundary emerge from a very special choice of boundary conditions for general relativity on AdS3, instead of Lifshitz asymptotics
Summary
In the pursuit of a better understanding of quantum gravity, in the past two decades, a lot of interest has been put into the so called gauge/gravity correspondence, whose most celebrated example is the AdS=CFT duality [1,2]. Following [22], we will adopt an unconventional approach to this holographic realization for field theories possessing anisotropic scaling properties, where the spacetime anisotropy at the boundary emerge from a very special choice of boundary conditions for general relativity on AdS3, instead of Lifshitz asymptotics. This new set of boundary conditions is labeled by a nonnegative integer n, and is related with the Korteweg–de Vries (KdV) hierarchy of integrable systems..
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