Entangled universes

TL;DR

This work proposes and tests a generalized holographic entanglement entropy prescription for spacetimes with multiple asymptotic regions, including asymptotically Minkowski and AdS geometries. The central idea is that the entanglement between boundary components is captured by an HRT-like extremal surface γ_HRT(A) homologous to A, yielding S(A)=|γ_HRT(A)|/(4G_N) and defining entanglement wedges W(A) that support bulk reconstruction without requiring a boundary CFT dual. The authors substantiate the proposal through several lines of evidence: (i) adaptation of maximin/minimax and flow formulations to non-AdS boundaries; (ii) a constructive gluing mechanism that replaces Minkowski regions with AdS counterparts to recover standard HRT results; (iii) explicit computations in Brill-Lindquist multiboundary wormholes showing phase transitions and rich surface-topology (including index-1 bulge surfaces) that mirror AdS-like entanglement structures; and (iv) extensions to de Sitter asymptotics with orthodox/heterodox interpretations. They also outline tensor-network descriptions and potential dynamical scenarios for creating entangled universes, highlighting a broader framework for understanding entanglement and information flow in flat and cosmological spacetimes. Overall, the paper broadens holographic entanglement concepts beyond AdS and lays groundwork for flat-space holography, domain-wall techniques, and de Sitter network interpretations with practical computational tests in BL geometries.

Abstract

We propose a generalization of the RT and HRT holographic entanglement entropy formulas to spacetimes with asymptotically Minkowski as well as asymptotically AdS regions. We postulate that such spacetimes represent entangled states in a tensor product of Hilbert spaces, each corresponding to one asymptotic region. We show that our conjectured formula has the same general properties and passes the same general tests as the standard HRT formula. We provide further evidence for it by showing that in many cases the Minkowski asymptotic regions can be replaced by AdS ones using a domain wall. We illustrate the use of our formula by calculating entanglement entropies between asymptotic regions in Brill-Lindquist spacetimes, finding phase transitions similar to those known to occur in AdS. We construct networks of universes by gluing together Brill-Lindquist spaces along minimal surfaces. Finally, we discuss a variety of possible extensions and generalizations, including to universes with asymptotically de Sitter regions; in the latter case, we identify an ambiguity in the homology condition, leading to two different versions of the HRT formula which we call ``orthodox'' and ``heterodox'', with different physical interpretations.

Figures (45)

• Figure 1: Penrose diagrams for the spacetimes ${M_{\rm I}},{M_{\rm II}},{M_{\rm III}},{M_{\rm IV}}$ discussed in the main text. ${M_{\rm I}}$ is an AdS-Schwarzschild spacetime; the conformal boundaries $A,B$ are shown in blue, the exterior regions $W(A),W(B)$ in yellow, the bifurcation surface $\gamma_{\rm bif}$ in red, the future and past singularities in black, and the event horizons as dotted lines. ${M_{\rm II}}$ is a generic two-boundary asymptotically AdS wormhole; the causal shadow is shown in green, the entanglement wedges in yellow, and the HRT surface $\gamma_{\rm HRT}$ in red. (Of course, a truly generic spacetime does not have the spherical symmetry required to draw a Penrose diagram. Our purpose here is simply to illustrate the important qualitative features of the spacetime.) ${M_{\rm III}}$ is a Schwarzschild spacetime and ${M_{\rm IV}}$ is a generic two-sides asymptotically flat spacetime, with the same features shown as for ${M_{\rm I}}$ and ${M_{\rm II}}$.
• Figure 2: Gluing process to impose AdS boundary conditions on the Schwarzschild metric. On the top, the left and right Penrose diagrams are (identical) SAdS patches while the middle is Schwarzschild. The lighter regions are ''cut out" and after gluing across the branes (dotted lines) we obtain the metric in the second line.
• Figure 3: Case 1 for replacing asymptotic regions of Schwarzschild spacetime with patches of Schwarzschild-AdS spactime, requires $\mu_A > \mu_S$ for the gluing. The SAdS patches include the SAdS bifurcation surface. Including the original Schwarzschild bifurcation surface, the spacetime thus contains three candidate HRT surfaces, of which the smallest and hence the true HRT surface can be chosen to be the Schwarzschild one. The other two --- the SAdS bifurcation surfaces --- are constrictions for the two boundaries respectively, and the regions between them and the HRT surface is a python. Each python contains a bulge surface, an index-1 extremal surface whose area according to the python's lunch conjecture quantifies the complexity of the reconstructing observables in the python Brown:2019rox. Naively, the bulge surface is located at the domain wall, since that is extremal and a local maximum of the area among spherically-symmetric surfaces. However, it was shown in Arora:2024edk that this surface has index greater than 1, and the true bulge in such a circumstance lies in the vicinity of the domain wall but breaks the spherical symmetry.
• Figure 4: Case 2: Same as case 1 (Fig. \ref{['fig:caseA']}), except here the gluing requires $\mu_A < \mu_S$ and so the SAdS bifurcation surfaces are smaller than the Schwarzschild one, making them the HRT surfaces.
• Figure 5: Case 3: SAdS patches do not include the bifurcation surface, so the only extremal surface is the Scharzschild bifurcation surface, which is therefore the HRT surface.

Theorems & Definitions (5)

• Conjecture 1
• Conjecture 2
• Conjecture 3
• Lemma 1
• proof