Lifshitz black holes with a time-dependent scalar field in a Horndeski theory

Abstract

In arbitrary dimensions, we consider a particular Horndeski action given by the Einstein-Hilbert Lagrangian with a cosmological constant term, while the source part is described by a real scalar field with its usual kinetic term together with a nonminimal kinetic coupling. In order to evade the no-hair theorem, we look for solutions where the radial component of the conserved current vanishes identically. Under this hypothesis, we prove that this model can not accommodate Lifshitz solutions with a radial scalar field. This problem is finally circumvented by turning on the time dependence of the scalar field, and we obtain a Lifshitz black hole solution with a fixed value of the dynamical exponent z = 1 3. The same metric is also shown to satisfy the field equations arising only from the variation of the matter source.

Thermodynamics and Cardy-like formula for nonminimally dressed, charged Lifshitz black holes in new massive gravity

“…where χ and γ are nonnegative constants and, as before, r h denotes the location of the horizon while Φ 0 is a positive constant. As a result, the combination E t t − E r r = 0, with m 2 given previously in (8), yields…”

“…where values of z giving a vanishing value for m 2 and λ are forbidden. Moreover, since we are interested in looking for charged black hole solutions that asymptote the Lifshitz spacetime (1), we not only impose (8) and (9), but we also opt for the following ansatz…”

“…while the remaining constants are given by eqs. (8) and (9). In order to ensure the realness of the solution, we must consider the range of z in accordance to TABLE 1.…”

“…Another aspect that is vital to the idea of a Gauge/Gravity correspondence is the finite temperature effects, through the introduction of black holes whose asymptotic behaviour reproduce the spacetime (1), denominated as Lifshitz black holes. Unfortunately, we have come to learn that General Relativity with a cosmological constant is not enough to support the Lifshitz spacetime with z = 1 [2] (and consequently, cannot support Lifshitz black holes either), this is why many approaches to find these configurations consider one of the following mechanisms: adding matter fields [3,4,5,6,7,8,9,10,11,12,13,14], enriching the gravitational Lagrangian with corrections in the curvature [15,16,17,18,19,20,21] or both simultaneously [22,23,24,25,26]. In particular, in 2+1 dimensions, there exists a ghost-free, parity even gravity theory called New Massive Gravity (NMG) [27], yielding a Lifshitz black hole solution characterized by a dynamical exponent z = 3 [16].…”

Tailoring cosmologies in cubic shift-symmetric Horndeski gravity

“…In what follows, we will show that it is possible to fit any Hubble evolution into a cubic Horndeski theory with a vanishing scalar current. The vanishing of the scalar current is a common physical constraint imposed when obtaining black hole solutions and cosmological solutions in shift-symmetric Horndeski theory [50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65]. In black holes, this constraint arises from requiring regularity of the current, J µ J µ = (J r ) 2 /f (r), particularly at the event horizon.…”