Anisotropic Lifshitz holography in Einstein–Proca theory with stable negative mass spectrum

Abstract

In this article, we focus on constructing a new family of spatially anisotropic Lifshitz spacetimes with arbitrary dynamical exponent z and constant negative curvature in d + 1 dimensions within the framework of the Einstein–Proca theory with a single vector field. So far, this kind of anisotropic spaces has been constructed with the aid of a set of vector fields. We also consider the spatially isotropic case as a particular limit. The con structed metric tensor depends on the spacetime dimensionality, the critical exponent and the Lifshitz radius; while, the curvature scalar depends just on the number of dimensions. We also obtain a novel spectrum with negative squared mass; we compute the corresponding Breitenlohner–Freedman bound and observe that the found family of spatially anisotropic Lifshitz spaces respects this bound. Hence, these new solutions are stable and can be use ful within the gravity/condensed matter theory holographic duality, since the spectrum with negative squared mass is complementary to the positive ones already known in the literature. We also examine the null energy condition and show that it is essentially satisfied along all the boundary directions, i.e., along all directions, except the r one, of our Lifshitz spacetime with the corresponding consistency conditions imposed on the scaling exponents.