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

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 $m$.