Thermodynamics of black branes in asymptotically Lifshitz spacetimes

Abstract

Recently, a class of gravitational backgrounds in $3+1$ dimensions have been proposed as holographic duals to a Lifshitz theory describing critical phenomena in $2+1$ dimensions with critical exponent $z\ensuremath{\ge}1$. We continue our earlier work [G. Bertoldi, B. A. Burrington, and A. Peet, preceding Article, Phys. Rev. D 80, 126003 (2009).], exploring the thermodynamic properties of the ``black brane'' solutions with horizon topology ${\mathbb{R}}^{2}$. We find that the black branes satisfy the relation $\mathcal{E}=\frac{2}{2+z}Ts$ where $\mathcal{E}$ is the energy density, $T$ is the temperature, and $s$ is the entropy density. This matches the expected behavior for a $2+1$ dimensional theory with a scaling symmetry $({x}_{1},{x}_{2})\ensuremath{\rightarrow}\ensuremath{\lambda}({x}_{1},{x}_{2})$, $t\ensuremath{\rightarrow}{\ensuremath{\lambda}}^{z}t$.