On the reconstruction of Lifshitz spacetimes

TL;DR

The paper investigates reconstructing Lifshitz spacetimes from boundary data using three holographic diagnostics: differential entropy (hole-ography), causal wedges, and entanglement wedges. The Lifshitz background introduces two key complications relative to AdS: (i) light rays with transverse momentum fail to reach the boundary, and (ii) the Lifshitz boundary is degenerate, complicating entanglement-based reconstructions; using the Lifshitz metric $ds^2 = L^2\left(-\frac{dt^2}{u^{2z}} + \frac{dx^2}{u^2} + \frac{du^2}{u^2}\right)$ with $z\ge1$, the authors examine how each reconstruction program fares. They find that (i) differential entropy can reconstruct only a subset of time-varying bulk curves, with certain curves non-reconstructible due to turning light rays; (ii) the causal wedge degenerates when the radial cut-off is removed, making a nondegenerate wedge dependent on a cut-off; and (iii) the entanglement wedge in Lifshitz generally does not close with extremal surfaces alone and requires including boundary-emitted light-sheets, signaling a departure from the AdS/HRT paradigm. These results indicate that naive RT/HRT prescriptions must be revised for Lifshitz holography and motivate exploring non-relativistic boundary structures (e.g., Newton–Cartan frameworks) or alternative bulk reconstruction schemes.

Abstract

We consider the reconstruction of a Lifshitz spacetime from three perspectives: differential entropy (or "hole-ography"), causal wedges and entanglement wedges. We find that not all time-varying bulk curves in vacuum Lifshitz can be reconstructed via the differential entropy approach, adding a caveat to the general analysis of \cite{Headrick:2014eia}. We show that the causal wedge for Lifshitz spacetimes degenerates, while the entanglement wedge requires the additional consideration of a set of boundary-emanating light-sheets. The need to include bulk surfaces with no clear field theory interpretation in the differential entropy construction and the change in the entanglement wedge formation both serve as warnings against a naive application of holographic entanglement entropy proposals in Lifshitz spacetimes.

Figures (14)

• Figure 1: Boundaries $\ell^2 = u^{2(z-1)}$ below which the radial effective potential for null geodesics $V_{\textrm{eff}}^{(0)}(u)< 0$ for different values of $z$: 1 (blue), 3/2 (yellow), 2 (green), 3 (red) and 10 (purple).
• Figure 2: The causal wedge $\blacklozenge_{{\cal A}}$ in Poincaré AdS$_3$. We fix the interval width $2a=2$ and also show the causal information surface $\Xi_{\cal A}$ (given by \ref{['eq:futurewedgebdyAdS']} with $t=0$).
• Figure 3: Regulated boundary domain of dependence $\lozenge_{\cal A}^\varepsilon$ in Lifshitz spacetime with $z=3/2$. We fix the interval width $2a=2$ and plot three different values of the cut-off $\varepsilon$: $0.5$, $0.2$ and $0.1$ (from the outside to the centre) in black. We show $\lozenge_{\cal A}$ for Poincaré AdS$_3$ in gray for comparison.
• Figure 4: The causal wedge $\blacklozenge_{{\cal A}}^\varepsilon$ in a cut-off Lifshitz spacetime with $z=2$. We fix the interval width $2a=2$ and also the cut-off $\varepsilon=0.5$. The central yellow section is built from light rays sent from the future- and past-most tips of the regulated boundary domain of dependence $\lozenge_{\cal A}^\varepsilon$ (Type I), whereas the outer blue sections are built from light rays sent from its edges with $|\ell|=|\ell_\star|$ (Type II).
• Figure 5: Examples of the causal information surface $\Xi_{\cal A}$ in Lifshitz spacetime with $z=2$. We fix the interval width $2a=2$ and find the surface for five different values of the cut-off $\varepsilon$: $0.05$, $0.1$, $0.25$, $0.5$ and $1$, from left to right. For each curve, the central yellow section is built from Type I light rays, whereas the outer blue sections are built from Type II light rays, as in figure \ref{['fig:causalwedge']}. Recall that $|\ell_\star|=\varepsilon^{z-1}$. Black dots mark the radial extent $u_{\Xi}$ in each case via \ref{['eq:radialextent']}.