Toward a Holographic Theory for General Spacetimes

TL;DR

The paper develops a holographic framework for general spacetimes grounded on a holographic screen, positing that spacetime geometry emerges from quantum entanglement and presenting two competing Hilbert-space pictures: direct-sum versus spacetime-equals-entanglement. It analyzes FRW universes to show that screen entanglement entropies obey a volume law and reduce to AdS/CFT in the appropriate limit, introducing a FRW dictionary with $S(\gamma)=\frac{1}{4}\|E(\gamma)\|$ and $Q(\gamma)=S(\gamma)/(V(\gamma)/4)$, and it discusses dynamics during transitions. The work then contrasts the two Hilbert-space structures, explores bulk reconstruction and exterior-region information, and examines time evolution, arguing for a strengthened covariant entropy bound and the necessity of nonlinear/state-dependent operators in the spacetime-equals-entanglement picture. Finally, it addresses the challenge of selecting a quantum-gravity state and the landscape of vacua, proposing scenarios for the multiverse that connect to observable cosmology and holographic principles.

Abstract

We study a holographic theory of general spacetimes that does not rely on the existence of asymptotic regions. This theory is to be formulated in a holographic space. When a semiclassical description is applicable, the holographic space is assumed to be a holographic screen: a codimension-1 surface that is capable of encoding states of the gravitational spacetime. Our analysis is guided by conjectured relationships between gravitational spacetime and quantum entanglement in the holographic description. To understand basic features of this picture, we catalog predictions for the holographic entanglement structure of cosmological spacetimes. We find that qualitative features of holographic entanglement entropies for such spacetimes differ from those in AdS/CFT but that the former reduce to the latter in the appropriate limit. The Hilbert space of the theory is analyzed, and two plausible structures are found: a direct sum and "spacetime equals entanglement" structure. The former preserves a naive relationship between linear operators and observable quantities, while the latter respects a more direct connection between holographic entanglement and spacetime. We also discuss the issue of selecting a state in quantum gravity, in particular how the state of the multiverse may be selected in the landscape.

Figures (14)

• Figure 1: For a fixed semiclassical spacetime, the holographic screen is a hypersurface obtained as the collection of codimension-2 surfaces (labeled by $\tau$) on which the expansion of the light rays emanating from a timelike curve $p(\tau)$ vanishes, $\theta = 0$. This way of erecting the holographic screen automatically deals with the redundancy associated with complementarity. The ambiguity of choosing $p(\tau)$ reflects a large freedom in fixing the redundancy associated with holography.
• Figure 2: The congruence of past-directed light rays emanating from $p_0$ (the origin of the reference frame) has the largest cross sectional area on a leaf $\sigma$, where the holographic theory lives. At any point on $\sigma$, there are two future-directed null vectors orthogonal to the leaf: $k^a$ and $l^a$. For a given region $\Gamma$ of the leaf, we can find a codimension-2 extremal surface $E(\Gamma)$ anchored to the boundary $\partial \Gamma$ of $\Gamma$, which is fully contained in the causal region $D_\sigma$ associated with $\sigma$.
• Figure 3: Various FRW universes, ${\rm I}, {\rm II}, {\rm III}, \cdots$, have the same boundary area ${\cal A}_*$ at different times, $t_*({\rm I}), t_*({\rm II}), t_*({\rm III}), \cdots$. Quantum states representing universes at these moments belong to Hilbert space ${\cal H}_*$ specified by the value of the boundary area.
• Figure 4: A region $L(\gamma)$ of the leaf $\sigma_*$ is parameterized by an angle $\gamma: [0, \pi]$. The extremal surface $E(\gamma)$ anchored to its boundary, $\partial L(\gamma)$, is also depicted schematically. (In fact, $E(\gamma)$ bulges into the time direction.)
• Figure 5: The value of $Q(\gamma)$ as a function of $\gamma$ ($0 \leq \gamma \leq \pi/2$) for $w = -1$ (vacuum energy), $-0.98$, $-0.8$, $0$ (matter), $1/3$ (radiation), and $1$. The dotted line indicates the lower bound given by the flat space geometry, which can be realized in a curvature dominated open FRW universe.

Theorems & Definitions (2)

• Theorem 1
• proof