Nomura, Salzetta, Sanches, and Weinberg (Phys. Rev. D 95, 086002 (2017)) proposed a holographic theory for general (non-AdS, cosmological) spacetimes, formulated on holographic screens and guided by the conjecture that entanglement among holographic degrees of freedom determines semiclassical geometry. They left the theory's defining structure open: which of two possible Hilbert-space organizations is realized (a direct sum of Hilbert spaces for distinct semiclassical sectors, versus a single Hilbert space in which entanglement patterns select the geometry, "spacetime equals entanglement"); whether the strengthened covariant entropy bound — the entropy on the union of the interior and exterior light sheets of a leaf is bounded by the leaf area over 4 — holds in full generality; what corrections to the area-law extremal-surface entropy restore a manifestly monotonic coarse-grained entropy during far-from-equilibrium processes such as reheating; the precise form of the (generally nonlinear, state-dependent) time evolution and operator map; how much of the bulk can be reconstructed from a single leaf; and how the state of the full quantum-gravitational system is selected in cosmology and the string landscape. The six issues below remain open in the literature.

Background

The holographic principle conjectures that the number of independent degrees of freedom describing a spacetime region scales with the area of its boundary rather than its volume, in units of the Planck area. Its precise formulation is Bousso's covariant entropy bound: the entropy fluxing through a light sheet — a null hypersurface generated by light rays orthogonal to a spacelike surface, terminating when the rays focus — is bounded by the area A\mathcal{A} of that surface divided by 4 (in Planck units). The best-understood realization is the AdS/CFT correspondence (Maldacena), in which quantum gravity in an asymptotically anti-de Sitter (AdS) spacetime is dual to a conventional quantum field theory on a fixed boundary. In AdS/CFT, the entanglement entropy of a boundary region is given by the Ryu–Takayanagi/HRT prescription: it equals one quarter of the area of the bulk extremal surface (the codimension-2 surface of stationary area anchored to the region's boundary), with quantum corrections captured by quantum extremal surfaces, which extremize the generalized entropy (surface area over 4 plus the bulk matter entanglement entropy); surfaces of this type underlie the entanglement-island and replica-wormhole resolution of the black hole information paradox (an entanglement island is a bulk region whose entropy is counted as part of the entropy of the distant radiation; replica wormholes are the multiply-connected spacetime saddles that implement this counting in the gravitational path integral).

The source paper addresses spacetimes without AdS asymptotics, in particular cosmological ones. There, the natural encoding surface is a holographic screen (Bousso–Engelhardt): a codimension-1 hypersurface foliated by marginally trapped surfaces called leaves, each of which has two orthogonal null directions such that the light rays in one direction have zero expansion. Nomura, Salzetta, Sanches, and Weinberg conjectured that the holographic theory lives on such screens, that the entanglement entropies of the holographic degrees of freedom determine the semiclassical spacetime (so that the entanglement entropy of a leaf region Γ\Gamma is one quarter of the area ∥E(Γ)∥\|E(\Gamma)\| of the extremal surface E(Γ)E(\Gamma) anchored to ∂Γ\partial\Gamma), and they cataloged the entanglement structure of Friedmann–Lemaître–Robertson–Walker (FRW) cosmologies — the homogeneous, isotropic solutions of Einstein's equations that model the expanding universe at large scales. Analyzing the Hilbert space, they found two consistent organizations. In the direct-sum structure, each distinct semiclassical spacetime has its own Hilbert space, and observables and time evolution remain linear, state-independent operators. In the "spacetime equals entanglement" (SEE) structure, all states of a fixed boundary space belong to one Hilbert space, and a state represents a given spacetime precisely when its entanglement entropies match that geometry — which forces the operators describing excitations and evolution to be nonlinear and state-dependent, since a superposition of semiclassical spacetimes can represent yet another spacetime.

The authors showed that their picture implies a strengthened covariant entropy bound: the entropy on the union of the interior and exterior light sheets of a leaf σ∗\sigma_* is bounded by A∗/4\mathcal{A}_*/4, i.e. ln⁡dim⁡H∗=A∗/4\ln\dim\mathcal{H}_*=\mathcal{A}_*/4, half of what the ordinary (per-sheet) covariant bound would allow; they noted that a realistic counterexample would force adding extra A∗/4\mathcal{A}_*/4 degrees of freedom on the screen, and that no known system requires this. They also analyzed how a holographic state on one leaf fails to determine the future state autonomously (signals from a past singularity or past null infinity must be specified), the corrections needed to the area-law entropy during rapid bulk processes such as reheating so that coarse-grained entropy is monotonic, and — for the full quantum-gravitational system — how the state is selected. In the eternally inflating multiverse of the string landscape (a superposition of bubble universes in metastable vacua), they contrasted a scattering state, which requires separate initial conditions and faces the Boltzmann-brain problem (the risk that randomly fluctuated observers dominate the measure, emptying the theory of predictions), with a normalizable bound state satisfying the constraint H∣Ψ~⟩=0H|\tilde\Psi\rangle=0 (the Wheeler–DeWitt-type condition that the full state be annihilated by the generator of holographic time) with spacetime-normalizable coefficients ∑i∣ci∣2<∞\sum_i|c_i|^2<\infty, which would in principle fix the state once the theory is given. All six of these issues were explicitly left open.

Problem Statement

Determine the defining structure of the holographic theory for general spacetimes (spacetimes without asymptotically AdS regions, including cosmological ones), as posed by Nomura, Salzetta, Sanches, and Weinberg. The defining structure is fixed by the following determinations, which are to be mutually consistent within a single specified theory that reproduces AdS/CFT and semiclassical gravity in the appropriate limits (item 4 applies only if the spacetime-equals-entanglement organization is realized):

  1. Which Hilbert-space organization is realized: a direct sum of Hilbert spaces associated with distinct semiclassical spacetimes, preserving linear, state-independent observables and time evolution; or the "spacetime equals entanglement" organization, in which all states with a fixed boundary space lie in a single Hilbert space and a state represents a spacetime exactly when its entanglement entropies match that geometry.
  2. Whether the strengthened covariant entropy bound — the entropy on the union of the interior and exterior light sheets of a leaf is bounded by A/4\mathcal{A}/4 (equivalently ln⁡dim⁡H∗=A∗/4\ln\dim\mathcal{H}_*=\mathcal{A}_*/4 for the Hilbert space of a leaf σ∗\sigma_*) — is valid in full generality, or whether extra degrees of freedom on the holographic screen are required in some cases.
  3. What corrections — quantum and/or nonextremal-surface — to the leading area-law prescription S(Γ)=∥E(Γ)∥/4S(\Gamma)=\|E(\Gamma)\|/4 for the entropy of a leaf region Γ\Gamma are necessary during rapid, far-from-equilibrium bulk processes such as reheating (the post-inflationary epoch in which the inflaton's energy is converted into thermal radiation) or phase transitions, so that the coarse-grained entropy is manifestly monotonic, and how large the residual transient decreases of S(Γ)S(\Gamma) can be once these corrections are included.
  4. If the spacetime-equals-entanglement organization is the one realized, the precise form of the nonlinear, state-dependent time-evolution law and operator map that yields effective linear semiclassical dynamics within each branch while remaining microscopically consistent.
  5. How complete bulk reconstruction from a single leaf is: which bulk operators or regions necessarily require data from other boundary spaces or explicit bulk time-evolution operators, and how those additional boundary labels are organized.
  6. How the state (selection condition) of the full quantum-gravitational system is fixed in cosmological and landscape contexts: in particular, whether the multiverse is a normalizable bound state satisfying H∣Ψ~⟩=0H|\tilde\Psi\rangle=0 with spacetime-normalizable coefficients ∑i∣ci∣2<∞\sum_i|c_i|^2<\infty, or a scattering-type state requiring separate specification of initial conditions.

References

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