Towards the generalized gravitational entropy for spacetimes with non-Lorentz invariant duals

TL;DR

This work generalizes holographic entanglement entropy to spacetimes with non-Lorentzian duals by extending the Lewkowycz-Maldacena prescription and employing a fine-structure modular framework. It shows that the entangling surface is fixed by consistency between boundary and bulk causal structures rather than anchoring to the boundary, and that a null-geodesic regulation from the entangling surface determines the bulk extremal surface. The authors develop an intrinsic geometric prescription, validated in AdS$_3$/WCFT with CSS boundary conditions and in 3D flat space, and extend it to broader non-Lorentzian holographies, including Lifshitz-type duals. The approach yields explicit entanglement entropies via regulated extremal surfaces and provides a new extrapolate dictionary for flat holography, with potential implications for higher-dimensional and Lifshitz holography analyses plus broader quantum gravity applications.

Abstract

Based on the Lewkowycz-Maldacena prescription and the fine structure analysis of holographic entanglement proposed in arXiv:1803.05552, we explicitly calculate the holographic entanglement entropy for warped CFT that duals to AdS$_3$ with a Dirichlet-Neumann type of boundary conditions. We find that certain type of null geodesics emanating from the entangling surface $\partial\mathcal{A}$ relate the field theory UV cutoff and the gravity IR cutoff. Inspired by the construction, we furthermore propose an intrinsic prescription to calculate the generalized gravitational entropy for general spacetimes with non-Lorentz invariant duals. Compared with the RT formula, there are two main differences. Firstly, instead of requiring that the bulk extremal surface $\mathcal{E}$ should be anchored on $\partial\mathcal{A}$, we require the consistency between the boundary and bulk causal structures to determine the corresponding $\mathcal{E}$. Secondly we use the null geodesics (or hypersurfaces) emanating from $\partial\mathcal{A}$ and normal to $\mathcal{E}$ to regulate $\mathcal{E}$ in the bulk. We apply this prescription to flat space in three dimensions and get the entanglement entropies straightforwardly.

Figures (13)

• Figure 1: The blue solid line is the $\mathcal{E}_{\mathcal{A}}$ which is regulated from the spacelike geodesic $\mathcal{E}$. The red and green lines are the null geodesics $\gamma_{\pm}$ that connect the endpoints of $\mathcal{E}_{\mathcal{A}}$ and $\mathcal{A}$.
• Figure 2: The orange lines with arrows depict the trajectory of the modular flow in WCFT. Note that the modular flow can never pass through $\partial\mathcal{D}$, which is depicted by the two purple lines.
• Figure 3: The left figure gives an explicit diagram for a modular plane $\mathcal{P}(v_0)$. The blue line is $\mathcal{L}_{v_0}$ while the orange lines depict $\bar{\mathcal{L}}_{\bar{v}_0}^{r_0}$ with $\bar{v}_0=v_0\,,r> \frac{1}{l_u}$. The red line is $\bar{\mathcal{L}}_{v_0}$ with its turning point denoted as $\mathcal{E}(v_0)$. The other two black points are where $\mathcal{L}_{v_0}$ intersect with the red line at $v=\pm\infty$. The right figure is just the projection of the left figure to a flat plane.
• Figure 4: The causal decomposition for the WCFT, and the assignment of the imaginary parts for each region.
• Figure 5: This figure shows the causal decomposition of the bulk $\mathcal{M}$ and the boundary $\mathcal{B}$. The two surfaces that intersect at $\mathcal{E}$ are $\mathcal{N}_{\pm}$ and the brown line is the boundary interval $\mathcal{A}$.