Anisotropic Schr\"odinger black holes with hyperscaling-violation

TL;DR

This paper presents new exact solutions for anisotropic Schrödinger black holes with hyperscaling violation, analyzing their causal structure, thermodynamics, and stability within Einstein-Maxwell-dilaton theories.

Contribution

It extends previous work by providing three families of solutions with arbitrary dynamical exponent and hyperscaling violation, including black holes, and studies their thermodynamics and causal structure.

Findings

Existence of black hole solutions with hyperscaling violation

Thermodynamic properties depend on hyperscaling and dynamical exponents

Non-trivial dilaton fields are supported by hyperscaling violation

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 arXiv: 2012.13412 and 2110.04445, we report three families of exact configurations over a non-relativistic Schr\"odinger background with both, arbitrary dynamical critical exponent $z$ and hyperscaling violating parameter $\theta$. Concretely, we provide field configurations with hyperscaling violation which are asymptotically Schr\"odinger spaces. Our solutions correspond to three kinds: a zero-temperature background, a naked singularity and, more interestingly, a family of black holes. To the later, 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 $\theta$ 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. We provide a thorough study of the thermodynamics including the quasi-local computation of charges, the verification of the first law and arguments concerning the stability. Lastly, we explore the effects in the thermodynamics from varying the rich parameter space. We pay special attention in comparing the qualitative behavior of the thermodynamics scalar-free solutions and the ones with a nontrivial dialton.