A paucity of bulk entangling surfaces: AdS wormholes with de Sitter interiors

TL;DR

This work analyzes planar AdS-dS-wormholes that satisfy the null energy condition and possess two asymptotically AdS boundaries connected by an inflating interior. The authors prove that no real wormhole-spanning codimension-2 extremal surfaces exist, implying vanishing leading-order mutual information for finite subregions on opposite boundaries and raising questions about the correct holographic entropy prescription. To address ill-defined cases, they explore regulated spacetimes and propose extended prescriptions (overline{HHRT} and overline{maximin}) as well as the potential role of complex extremal surfaces, drawing support from de Sitter analyses. The results reveal a delicate interplay between energy conditions, symmetry, and inflation in holographic entanglement, with implications for the holographic description of cosmological interiors and the need for refined entropy functionals.

Abstract

We study and construct spacetimes, dubbed planar AdS-dS-wormholes, satisfying the null energy condition and having two asymptotically AdS boundaries connected through a (non-traversable) inflating wormhole. As for other wormholes, it is natural to expect dual descriptions in terms of two disconnected CFTs in appropriate entangled states. But for our cases certain expected bulk entangling surfaces used by the Hubeny-Rangamani-Takayanagi (HRT) prescription to compute CFT entropy do not exist. In particular, no real codimension-2 extremal surface can run from one end of the wormhole to the other. According to HRT, the mutual information between any two finite-sized subregions (one in each CFT) must then vanish at leading order in large $N$ -- though the leading-order mutual information per unit area between the two CFTs taken as wholes may be nonzero. Some planar AdS-dS-wormholes also fail to have plane-symmetric surfaces that would compute the total entropy of either CFT. We suggest this to remain true of less-symmetric surfaces so that the HRT entropy is ill-defined and some modified prescription is required. It may be possible to simply extend HRT or the closely-related maximin construction by a limiting procedure, though complex extremal surfaces could also play an important role.

Figures (13)

• Figure 1: A sample conformal diagram for an AdS-dS-wormhole. The surface labeled $\Xi$ (blue in color version) is a putative wormhole-spanning surface (which we will show cannot exist if the spacetime obeys the null energy condition). The surface $\Sigma$ (red in color version) is an achronal surface that approaches close to $\mathscr{I}_\mathrm{dS}$ and thus has large volume element. The dashed lines indicate the boundary of the past of the dS-like part $\mathscr{I}_\mathrm{dS}$ of the conformal boundary. The wormhole shown has a right/left $\mathbb{Z}_2$ reflection symmetry. The explicit wormholes of section \ref{['sec:cutpaste']} will share this symmetry, though it is not needed for our general arguments. The edges of $\mathscr{I}_\mathrm{dS}$ are marked $\mathcal{E}$.
• Figure 2: Our cut-and-paste AdS-dS-wormholes. The two types of regions are pasted together along null shells, indicated by the dotted lines labeled $A$, which are taken to lie along (parts of the) Killing horizons of the patches I and II. The dashed lines labeled $H_\mathrm{Cauchy}$ are Killing horizons of patch I and are Cauchy horizons of the full spacetime; the dashed lines labeled $H^-$ are the past event horizons. The two patches labeled I are isometric under a left/right reflection. \ref{['subfig:cutpastethreshold']}: A case where the edges $\mathcal{E}$ of $\mathscr{I}_\mathrm{dS}$ lie on the past event horizons of $\mathscr{I}_\mathrm{AdS}$. \ref{['subfig:cutpastegeneral']}: A less extreme case where $\mathscr{I}_\mathrm{dS}$ lies below the past event horizon.
• Figure 3: Conformal diagrams for the spacetimes from which we cut our (shaded) regions I and II. The dashed lines on both diagrams correspond to the Killing horizons at $r = r_+$. For simplicity we do not show the relative bending between the singularity and boundary.
• Figure 4: Firing in matter from the AdS boundaries modifies the cut-and-paste wormholes of figure \ref{['fig:cutpaste']}. The spacetimes shown are based on figure \ref{['subfig:cutpastethreshold']}, though corresponding results also hold for figure \ref{['subfig:cutpastegeneral']}. \ref{['subfig:Vaidyashell']}: Patch I is replaced by an AdS-Vaidya metric representing pressureless null dust (shaded) falling in from $\mathscr{I}_\mathrm{AdS}$. This adds a future singularity that cuts off part of the Cauchy horizon $H_\mathrm{Cauchy}$. \ref{['subfig:doublepatched']}: One can remove the Cauchy horizon completely by firing in a thin null shell $B$ (light gray lines beneath dashed lines) along $H^-$. The shell further divides region I into subregions Ia and Ib on either side. This shell cannot be pressureless (see footnote \ref{['note:pressure']}) and is not a simple limit of the Vaidya case shown at top.
• Figure 5: Surfaces of $\theta_R=0$ (dashed lines; red in color version) and $\theta_L=0$ (dotted lines; blue in color version) for the AdS-dS-wormholes shown in figure \ref{['fig:Vaidyapatching']}. Note that since affine parameters diverge at $\mathscr{I}_\mathrm{dS}$ and $\mathscr{I}_\mathrm{AdS}$, the Raychaudhuri equation guarantees that $\theta_R$, $\theta_L$ both vanish on these surfaces. We take the ingoing matter to consist of null shells (solid gray lines). \ref{['subfig:thetazerodoublepatched']}: The spacetime of figure \ref{['subfig:doublepatched']}. Null shells with non-zero pressure are fired in along the past horizons of $\mathscr{I}_\mathrm{AdS}$; this fine-tuning leads the $\theta_R = 0$, $\theta_L = 0$ surfaces to overlap along portions of these past horizons. \ref{['subfig:thetazeroVaidya']}: When the incoming shells are displaced to the future the surfaces $\theta_R = 0$, $\theta_L = 0$ no longer intersect in $M$ and total entropy surfaces do not exist in $M$. Here the shell may be chosen pressureless so that this case is a simple limit of figure \ref{['subfig:Vaidyashell']}. A version in which this new null shell is smoothed out is shown in figure \ref{['fig:thetazeroVaidya']}.