A Note on Black Holes in Asymptotically Lifshitz Spacetime

TL;DR

This work analyzes exact black hole solutions in asymptotically Lifshitz spacetime to explore holographic properties of non-relativistic fixed points. It shows that tidal forces diverge near the horizon in near-extremal regimes, derives Wilson loops analytically in extremal cases and numerically at finite temperature, and computes hydrodynamic quantities. The results demonstrate a universal shear-viscosity to entropy-density ratio $\eta/s = 1/(4\pi)$ in arbitrary dimensions and that the speed of sound satisfies $c_s^2 = 1/d$, with $d=3$ (five bulk dimensions) yielding $1/3$, saturating known bounds. These findings support the consistency of Lifshitz holography with established holographic bounds and point to future directions including string-theory embeddings and holographic renormalization in Lifshitz backgrounds.

Abstract

We investigate several aspects of exact black hole solutions in asymptotically Lifshitz spacetime, which were proposed in 0812.0530. Firstly, we calculate the tidal forces and find that in the near horizon region of such black hole backgrounds, the tidal forces diverge in the near extremal limit. Secondly, we evaluate the Wilson loops in both extremal and finite temperature cases. Finally, we obtain the corresponding shear viscosity and square of the sound speed and find that the ratio of shear viscosity to entropy density takes the universal value $1/4π$ in arbitrary dimensions while the square of the speed of sound saturates the conjectured bound 1/3 in five dimensions.

Figures (2)

• Figure 4.1: The boundary distance between the endpoints of a string $\ell$ as a function of $a$, with $r_{+}=1$.
• Figure 4.2: The potential energy as a function of the boundary distance $\ell$, with $r_{+}=1$.