Thermodynamics of a BTZ black hole solution with an 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 $\Lambda$ 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 an 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 found 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 $\Lambda \le -\frac{1}{l^2}$ where the limiting case $\Lambda =-\frac{1}{l^2}$ 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-AdS spacetime with planar horizon.

7 pages

References in corpus (2)

• Black holes with non-minimal derivative coupling
• Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario

Cited by in corpus (10)

• Dilatonic BTZ black holes with power-law field
• Lower-dimensional Gauss-Bonnet Gravity and BTZ Black Holes
• Axionic black branes in the k-essence sector of the Horndeski model
• Topological black hole in the theory with nonminimal derivative coupling with power-law Maxwell field and its thermodynamics
• Thermodynamics of Horndeski black holes with non-minimal derivative coupling
• Magnetically Charged Euler-Heisenberg Black Holes with Scalar Hair
• $(2+1)$-Dimensional Black Holes in $f(R,\varphi )$ Gravity
• Three-dimensional energy-dependent $C$-metric: black hole solutions
• Mimicking the LCDM model with Stealths
• Asymptotically AdS black hole with a conformally-coupled scalar field in the first-order formalism of gravity