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

Understand

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.