Black holes and black branes in Lifshitz spacetimes

TL;DR

This work constructs analytic charged black holes and black branes in asymptotically Lifshitz spacetimes with arbitrary $z$ and dimensions, within an Einstein-dilaton-Maxwell framework with $N$ U(1) fields. By solving the field equations, the authors obtain explicit analytic solutions for the metric, scalar, and gauge fields, and they compute thermodynamic quantities (temperature, entropy, mass, charges, and chemical potentials) that satisfy a first-law relation. They map the phase structure in both grand-canonical and canonical ensembles, revealing distinct behaviors across $1\le z<2$, $z=2$, and $z>2$, including Hawking-Page-type transitions, extremal black holes, and electric instabilities for certain regions. The results illuminate how Lifshitz scaling and the dynamical exponent shape the thermodynamics of charged holographic systems and may guide holographic interpretations and extensions to string-theory embeddings and transport properties.

Abstract

We construct analytic solutions describing black holes and black branes in asymptotically Lifshitz spacetimes with arbitrary dynamical exponent z and for arbitrary number of dimensions. The model considered consists of Einstein gravity with negative cosmological constant, a scalar, and N U(1) gauge fields with dilatonic-like couplings. We study the phase diagrams and thermodynamic instabilities of the solution, and find qualitative differences between the cases with 1<= z<2, z=2 and z>2.

Figures (10)

• Figure 1: Diagram showing the relations between theories with one gauge field less, with boundary topology $\mathbb{R}_t\times S^{d-1}$ (BH, after black hole) and $\mathbb{R}_t\times \mathbb{R}^{d-1}$ (BB, after black brane). The square represents asymptotically AdS spaces. The diagram continues indefinitely to the right and to the left up to $U(1)^0$, consisting on the Schwarzschild-AdS solution. The framed quantities indicate how many charges are there in the black hole.
• Figure 2: Diagram showing the relations between theories with one gauge field less and the AdS limit for black brane solutions of our model. Circles represent asymptotically Lifshitz spacetimes whereas squares correspond to asymptotically AdS ones. The diagram continues indifenitely to the right and to the left up to $U(1)^0$. The framed quantities indicate how many charges contribute to give the Reissner-Nordström-like factor in $b(r)$.
• Figure 3: Diagram showing the relations between theories with one gauge field less and the AdS limit for black hole solutions. Circles represent asymptotically Lifshitz spacetimes whereas squares correspond to asymptotically AdS ones. The diagram continues indifenitely to the right and to the left up to $U(1)^0$. The framed quantities correspond to the number of charges contributing to give the RN-like factor of $b$ .
• Figure 4: Relations between the different specific cases arising from action \ref{['eq.action']}. See previous captions for symbolism. The web of relations for the cases showed is incomplete for the sake of clarity, but can be completed with the previous diagrams.
• Figure 5: Sketch of the phase diagrams obtained here for different values of the dynamic exponent $z$. See text for details.