Anisotropic Schrödinger black holes with hyperscaling-violation

Abstract

Abstract We investigate novel exact solutions to an Einstein–Maxwell theory non-minimally coupled to a self-interacting dilaton-like scalar. Extending the results of Herrera-Aguilar et al. (Phys. Rev. D 103(12):124025, 2021. arXiv:2012.13412 [hep-th]; arXiv:2110.04445 [hep-th]), we report three families of exact configurations over a non-relativistic Schrödinger background with both, arbitrary dynamical critical exponent z and hyperscaling violating parameter $$\theta $$ θ in any dimension d. Concretely, we provide field configurations with hyperscaling violation which are locally Schrödinger spaces. Our solutions correspond to three kinds: a zero-temperature background, a naked singularity and, more interestingly, a family of black holes. To the latter, we construct the corresponding Carter–Penrose diagram with a view to understand their causal structure given the non-standard background. We show that a non-trivial hyperscaling violation parameter $$\theta $$ θ is necessary in order to support a real non-constant dilaton field in the configuration. We explore how the relation between the hyperscaling violation parameter and the critical dynamical exponent determine, in combination with the spacetime dimension, the kinematic aspects of the fields. A further refinement on the physically sensible configurations is obtained from the study of the null energy conditions. We provide a thorough study of the thermodynamics including the quasi-local computation of charges and the verification of the first law. We explore the effects in the thermodynamics from varying the rich parameter space, paying special attention in comparing the qualitative behavior of the thermodynamics of the scalar-free solutions and the ones with a nontrivial dilaton. Lastly, it is found that if the reality condition is loosen up on the scalar, the configuration is prone to acquiring a scalar charge.