Thermodynamics of a BTZ black hole solution with an Horndeski source

Understand

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.