On Charged Lifshitz Black Holes

TL;DR

This work constructs exact charged Lifshitz black holes in general $(d+2)$ dimensions by introducing a second Maxwell field, showing that the mass cannot be defined by standard Hamiltonian or non-relativistic boundary prescriptions and must be obtained from the first-law relation $M=rac{z V_d L^d}{48\, ext{pi}\,G_{d+2}} r_0^{3d}$. It then analyzes Gauss-Bonnet corrections in five dimensions, deriving a perturbed solution with $z=z_0+2\lambda_{GB}(z_0-1)$ (where $z_0=6$) and computing transport properties: $rac{ ext{eta}}{s}=rac{1}{4\, ext{pi}}[1-6\lambda_{GB}]$ and a Gauss-Bonnet-corrected DC conductivity that scales as $ ext{T}^{1/z_0}$ with $z_0=6$. The results demonstrate non-relativistic holography with higher-derivative corrections, including a violation of the classic $rac{1}{4\, ext{pi}}$ bound and nontrivial temperature dependence for conductivity in Lifshitz backgrounds, while highlighting the role of stability and locality in constraining such theories.

Abstract

We obtain exact solutions of charged asymptotically Lifshitz black holes in arbitrary (d+2) dimensions, generalizing the four dimensional solution investigated in 0908.2611[hep-th]. We find that both the conventional Hamiltonian approach and the recently proposed method for defining mass in non-relativistic backgrounds do not work for this specific example. Thus the mass of the black hole can only be determined by the first law of thermodynamics. We also obtain perturbative solutions in five-dimensional Gauss-Bonnet gravity. The ratio of shear viscosity over entropy density and the DC conductivity are calculated in the presence of Gauss-Bonnet corrections.