Black holes in asymptotically Lifshitz spacetimes with arbitrary critical exponent

TL;DR

The paper constructs and analyzes black-hole/brane solutions in 3+1 dimensional spacetimes that are asymptotically Lifshitz with arbitrary $z\ge1$, using a minimal Einstein-Maxwell-Proca-type action that supports Lifshitz scaling. Through analytic near-horizon and near-infinity expansions and a finite-energy Hamiltonian analysis, the authors show that for $z>2$ there exists a continuous one-parameter family of finite-energy solutions, while for $1\le z\le 2$ the energy conditions initially suggested a discrete set, a point later revisited by boundary-counterterm arguments that can restore a continuous family for all $z$. Numerical integration confirms these trends and reveals a thermodynamic feature: a negative slope in the temperature- horizon-radius curve $T(r_0)$ for $z$ near and below $1.761$, signaling potential instabilities in part of the parameter space. The work also discusses exact solutions at special values of $z$, the role of boundary terms in controlling energy, and directions for embedding these backgrounds into more complete theories. Overall, the results inform holographic modeling of 2+1D critical systems with Lifshitz scaling and highlight the importance of energy and stability analyses in this nonrelativistic setting.

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 $z\geq 1$. We numerically explore black holes in these backgrounds for a range of values of $z$. We find drastically different behavior for $z>2$ and $z<2$. We find that for $z>2$ ($z<2$) the Lifshitz fixed point is repulsive (attractive) when going to larger radial parameter $r$. For the repulsive $z>2$ backgrounds, we find a continuous family of black holes satisfying a finite energy condition. However, for $z<2$ we find that the finite energy condition is more restrictive, and we expect only a discrete set of black hole solutions, unless some unexpected cancellations occur. For all black holes, we plot temperature $T$ as a function of horizon radius $r_0$. For $z\lessapprox 1.761$ we find that this curve develops a negative slope for certain values of $r_0$ possibly indicating a thermodynamic instability.

Figures (6)

• Figure 1: The various fixed points for a given action parameterized by $Z$ and $\hat{L}$. The (dash-dot) green curve is $z=\frac{4}{Z}$, the (long-dashed) red sloped line is $z=Z$ and the (short-dashed) blue flat line is $z=1$. We choose to plot only $Z>2$ because these specify different actions that have different ratios $\frac{-\Lambda}{c^2}$. We will find that for the critical exponent $z<2$ (the dash-dot green curve) that the Lifshitz fixed point is "attractive" when going to larger $r$ and that for $z>2$ (the red sloped line) it is repulsive. Further, for $Z>4$ the solution $z=4/Z<1$ is imaginary, and so we exclude it.
• Figure 2: The above gives a picture of the cell of spacetime under consideration. $r_{-}$ is the interior surface, $r_{+}$ is the outer surface with $r_+>r_-$. The red (dashed) line gives the surface integral used to define the energy given at the bottom time slice. $r_{-}$ will be taken to approach the horizon (or 0), and $r_+$ will be taken to go to infinity, and further $\Delta x^i$ are both held fixed, with their product $\Delta x^1\times \Delta x^2=V_2$.
• Figure 3: Above we graph the exponents of the modes at infinity $r^{\alpha_i(z)}$ as a function of z. The (solid) black line ($-z-2$) represents the proper "energy" mode. The (long-dashed) red line $(-2)$ represents the inhomogeneous mode induced by the $S^2$. The line $(-2)$, and all its non-linear descendants, will be universal for backgrounds with the sphere. All other curves that are above the black line $(-z-2)$ represent infinite energy modes, and curves below this line represent modes with $0$ contribution to the energy of the background. The top (short-dashed) magenta curve is $\left(\frac{-z}{2}-1+\frac{\gamma(z)}{2}\right)$ (recall this mode has coefficient ${\mathcal{C}}_1$), which always represents an infinite energy mode. The bottom (dash-dot) blue curve is $\left(\frac{-z}{2}-1-\frac{\gamma(z)}{2}\right)$ (recall this mode has coefficient ${\mathcal{C}}_2$). This curve represents a finite energy mode for $z>2$ and an infinite energy mode for $1\leq z\leq 2$: the value $z=2$ gives the curve where this curve crosses the line $-z-2$.
• Figure 4: Graphs of $\log_{10}\left(T(r_0)\right)$ for $z=4,3,2.01$ colored red (solid), green (dashdot) and blue (dashed) respectively for the $\sigma=1$ case. Note the feature developing for smaller values of $z$.
• Figure 5: On the left is graphs of $\log_{10}\left(T(r_0)\right)$ for $z=1.1$ colored red (dash-dot), and the temperature for $z=1$ the normal AdS black hole with temperature $T=\frac{3r_0^2+1}{4\pi r_0}$ in green (solid). On the right, we plot $\log_{10}\left(T(r_0)\right)$ for $z=1.74, 1.761, 1.78$ in blue (dashed), green (solid), and red (dashdot) respectively. All graphs are for the $\sigma=1$ case.