Condensation for non-relativistic matter in Ho\v{r}ava-Lifshitz gravity

TL;DR

This paper investigates how non-relativistic matter condenses in a Hořava-Lifshitz black hole, revealing dependencies on parameters that influence scalar hair formation and conductivity ratios, bridging results with Einstein gravity, Cai's work, and BCS theory.

Contribution

It demonstrates how non-relativistic parameters affect scalar hair formation and the conductivity gap ratio in Hořava-Lifshitz gravity without detailed balance, extending previous relativistic and detailed balance studies.

Findings

Scalar hair forms more easily with larger epsilon or alpha_2.

The gap frequency to critical temperature ratio varies with parameters, approaching known theoretical values.

The ratio can match BCS theory predictions with proper parameter choices.

Abstract

We study condensation for non-relativistic matter in a Ho\v{r}ava-Lifshitz black hole without the condition of the detailed balance. We show that, for the fixed non-relativistic parameter $\alpha_2$ (or the detailed balance parameter $\epsilon$), it is easier for the scalar hair to form as the parameter $\epsilon$ (or $\alpha_2$) becomes larger, but the condensation is not affected by the non-relativistic parameter $\beta_2$. We also find that the ratio of the gap frequency in conductivity to the critical temperature decreases with the increase of $\epsilon$ and $\alpha_2$, but increases with the increase of $\beta_2$. The ratio can reduce to the Horowitz-Roberts relation $\omega_g/T_c\approx 8$ obtained in the Einstein gravity and Cai's result $\omega_g/T_c\approx 13$ found in a Ho\v{r}ava-Lifshitz gravity with the condition of the detailed balance for the relativistic matter. Especially, we note that the ratio can arrive at the value of the BCS theory $\omega_g/T_c\approx 3.5$ by taking proper values of $\epsilon$, $\alpha_2$, $\beta_2$ and $m$.