BTZ black hole with Korteweg–de Vries-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 nonnegative 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 = 1 2π ( 1 z ) z z+1 , there is a Hawking-Page phase transition between the BTZ black hole and thermal AdS 3 spacetime. arXiv:1907.13026v2 [hep-th]

KdV-charged black holes

“…Going back to (6.1), unless µ 1 is large, free energy is positive, and therefore there is no Hawking-Page transition. Here we disagree with [29,38], where thermal AdS free energy was assigned negative sign, F AdS = − c 12 µ 3 for µ 1 = 0, and regard that as a mistake. It is easy to see that second condition in (7.6), which is u 0 > 0, is not implying first condition, which is a consequence of (4.21).…”

“…The corresponding geometry is the conventional BTZ black hole, albeit understood as a state in a theory with the deformed Hamiltonian. These black holes were initially considered in [29] and more recently revisited in [38]. Since a constant solution is invariant under the action of all Q 2k+1 we readily identify corresponding Lorentzian geometry as a holographic dual of a qKdV eigenstate, or, more accurately, exponentially many eigenstates with an approximately equal energy, where the log of the number of states is given by (4.23).…”

KdV-charged black holes

“…The corresponding geometry is the conventional BTZ black hole, albeit understood as a state in a theory with the deformed Hamiltonian. These black holes were initially considered in [28] and more recently revisited in [37]. Since a constant solution is invariant under the action of all Q 2k+1 we readily identify corresponding Lorentzian geometry as a holographic dual of a qKdV eigenstate, or, more accurately, exponentially many eigenstates with an approximately equal energy, where the log of the number of states is given by (4.23).…”

Thermodynamics of inhomogeneously mass-deformed ABJM model and pressure anisotropy

“…It is not straightforward to obtain the Smarr relation of the black hole either. In this respect, the reduced action formalism accompanied by 'scaling symmetry technique' developed in [25][26][27][35][36][37][38][39] is highly useful. We start with the following reduced action:…”

“…By equating the above two variations in Eqs. (39) and Eqs. (40) for the variation under scaling along with the on-shell condition (E Ψ = 0), one obtains the relation:…”