Condensation for non-relativistic matter in Hořava–Lifshitz gravity

We study condensation for non-relativistic matter in a Hořava–Lifshitz black hole without the condition of the detailed balance. We show that, for the fixed non-relativistic parameter α2α2 (or the detailed balance parameter ϵ), it is easier for the scalar hair to form as the parameter ϵ   (or α2α2) becomes larger, but the condensation is not affected by the non-relativistic parameter β2β2. We also find that the ratio of the gap frequency in conductivity to the critical temperature decreases with the increase of ϵ   and α2α2, but increases with the increase of β2β2. The ratio can reduce to the Horowitz–Roberts relation ωg/Tc≈8ωg/Tc≈8 obtained in the Einstein gravity and Cai's result ωg/Tc≈13ωg/Tc≈13 found in a Hořava–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 ωg/Tc≈3.5ωg/Tc≈3.5 by taking proper values of the parameters.