Thermodynamics of a BTZ black hole solution with a Horndeski source

Abstract

In three dimensions, we consider a particular truncation of the Horndeski action that reduces to the Einstein-Hilbert Lagrangian with a cosmological constant Λ and a scalar field whose dynamics is governed by its usual kinetic term together with a nonminimal kinetic coupling. Requiring the radial component of the conserved current to vanish, the solution turns out to be the BTZ black hole geometry with a radial scalar field well defined at the horizon. This means in particular that the stress tensor associated to the matter source behaves on shell as an effective cosmological constant term. We construct a Euclidean action whose field equations are consistent with the original ones and such that the constraint on the radial component of the conserved current also appears as a field equation. With the help of this Euclidean action, we derive the mass and the entropy of the solution, and find that they are proportional to the thermodynamical quantities of the BTZ solution by an overall factor that depends on the cosmological constant. The reality condition and the positivity of the mass impose the cosmological constant to be bounded from above as Λ ≤ − 1 l 2 where the limiting case Λ ¼ − 1 l 2 reduces to the BTZ solution with a vanishing scalar field. Exploiting a scaling symmetry of the reduced action, we also obtain the usual three-dimensional Smarr formula. In the last section, we extend all these results in higher dimensions where the metric turns out to be the Schwarzschild-anti-de Sitter spacetime with planar horizon.

Black holes in a cubic Galileon universe

“…Per se this does not guarantee the existence of a solution, but shows self-consistency of the method. Indeed, in a number of works [16][17][18], other black hole solutions with a time-dependent Galileon have been found.…”

Thermodynamics of charged Lifshitz black holes with quadratic corrections

“…[15] and [16] or in the case of a particular truncation of the Horndeski theory [17]. Although, Lifshitz black holes can be found analytically, and their temperature as well as their entropy can be easily computed, their thermodynamics issue remains an hard task.…”

Lovelock Galileons and black holes

“…A different way to bifurcate no hair theorems is to involve scalar tensor interactions involving translational invariant Galileons such as the John term of Fab 4 which reads, G µ ν ∇ µ φ∇ ν φ where G µν is the 4 dimensional Einstein tensor. There when one considers additionally a linear time dependence of the scalar field [27] (see also [28][29][30][31][32][33][34], and [35][36][37][38][39][40][41]) it was shown that the time dependence yielded analytic GR like black holes with additionally well defined scalars on the black hole horizon. For a recent review on black holes and scalar fields see [42] and [43].…”