Charged Dilatonic AdS Black Branes in Arbitrary Dimensions

TL;DR

This work constructs and analyzes static, charged dilatonic black brane solutions in $(n{+}2)$-dimensional Einstein–Maxwell–dilaton gravity with negative cosmological constant, focusing on asymptotically AdS spacetimes that can host Lifshitz-like regions in the bulk. The authors derive the general non-extremal solution, characterized by mass density $m$, electric charge density $q$, and dilatonic charge density $\mu_\phi$, and establish extremality conditions $m \ge \mu_\phi$ and $|\hat q| \le q_{\text{Lif}}$, with near-horizon geometries that are Lifshitz or Lifshitz–Schwarzschild and smoothly connected to AdS in the UV. They provide large-$R$ and near-horizon expansions, study the thermodynamics, and compare to AdS–RN to highlight distinct extremal structures and holographic implications for dual CFTs at finite temperature and chemical potential. The results extend Lifshitz–AdS analyses to arbitrary dimensions, clarifying how horizon data encode dilatonic charge and supporting holographic RG flows from AdS UV to Lifshitz IR fixed points.

Abstract

We study electromagnetically charged dilatonic black brane solutions in arbitrary dimensions with flat transverse spaces, that are asymptotically AdS. This class of solutions includes spacetimes which possess a bulk region where the metric is approximately invariant under Lifshitz scalings. Given fixed asymptotic boundary conditions, we analyze how the behavior of the bulk up to the horizon varies with the charges and derive the extremality conditions for these spacetimes.

Figures (6)

• Figure 1: $k$ (left) and $\frac{R_\text{\sc h}}{R_\text{sch}}$ (right) plotted as functions of $\frac{\mu_\phi}{m}$ for $n = 2$ to 5. The parameter $k$ starts at $(n + 1)$ when $\mu_\phi = 0$ and goes to zero as $\mu_\phi \to m$. Correspondingly, the ratio $\frac{R_\text{\sc h}}{R_\text{sch}}$ starts at one for all $n$ at $\mu_\phi = 0$ and reaches the extremal value of the ratio when $m = \mu_\phi$ (dashed vertical line).
• Figure 2: $\sqrt{A(R)}, \sqrt{B(R)}$ and $\phi(R)$ for $n = 2$, $\alpha = 1$, $k = 2.4$ ($\mu_\phi = 0.5\,m$) and $\hat{q} = 0.5\,q_{\text{\tiny Lif}}$. The curves for $\sqrt{A(R)}$ and $\phi(R)$, from left to right, correspond to $m = 0.01$, $0.1$, $1.0$, $10$ and $100$, respectively. The curves for $\sqrt{B(R)}$ follow the same colour pattern, and for each mass, $\sqrt{B(R)}$ ends at the corresponding value of $R_\text{\sc h}$ shown by the same coloured dotted line. The square roots of $A(R)$ and $B(R)$ were plotted to show their linear (asymptotic AdS) nature as $R$ becomes large. The coefficients $A_0$ and $B_0$\ref{['asymp-AdS-bndy-cond:R']} are both very close to one in all the cases shown in the figure.
• Figure 3: The functions $B_0^{-n}\sqrt{A(R)}, B_0^{-1}\sqrt{B(R)}$ and $\phi(R)$ in the bulk for $n = 2$ and $\alpha = 1$. The plots for $B_0^{-n}\sqrt{A(R)}$ and $\phi(R)$, from left to right, correspond to $m = 0.01$, $0.1$, $1.0$, $10$ and $100$, respectively, while the curves for $B_0^{-1}\sqrt{B(R)}$ follow the same colour pattern. The functions $A(R)$ and $B(R)$ were divided by the normalization constants $A_0$ and $B_0^2$\ref{['asymp-AdS-bndy-cond:R']} and their square roots were plotted to show their linear (asymptotic AdS) nature as $R$ becomes large. Compared with the non-extremal case (figure \ref{['figure:non-ext-asymp']}), the functions asymptote to AdS much slowly.
• Figure 4: Comparing $A(w), B(w)$ and $\phi(w)$ near the horizon (solid) with the corresponding global Lifshitz solutions (dotted). We have set $n = 2$ and $\alpha = 1$ ($q_{\text{\tiny Lif}} = 2$). The three plots in each sub-figure correspond to $m = 0.1$, $m = 1.0$ and $m = 10$, respectively, with the corresponding values $R_{\text{\sc h}, \text{ext}} = 0.37$, $R_{\text{\sc h}, \text{ext}} = 0.79$ and $R_{\text{\sc h}, \text{ext}} = 1.7$ for the horizon radii.
• Figure 5: Comparing $A(w), B(w)$ and $\phi(w)$ near the horizon (solid) with the corresponding Lifshitz-Schwarzschild solutions (dotted). We have set $n = 2$, $\alpha = 1$ ($q_{\text{\tiny Lif}} = 2$) and have chosen $k = 2.4$ ($\mu_\phi = 0.5\,m$). The three plots in each sub-figure correspond to $m = 0.1$, $m = 1.0$ and $m = 10$, respectively, with the corresponding values $R_\text{\sc h} = 0.5$, $R_\text{\sc h} = 1.1$ and $R_\text{\sc h} = 2.3$ for the horizon radius. $B(R)$ and $\phi(R)$ abruptly halts at the respective values based on the boundary conditions chosen; in particular, $\phi_{c, \text{nh-{\sc ls}}} = -0.5$ for all value of the mass, as can be verified from the plot.