Toward a holographic theory for general spacetimes

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.

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 $\mathrm{\Gamma }$ of the leaf, we can find a codimension-2 extremal surface $E(\mathrm{\Gamma })$ anchored to the boundary $\partial \mathrm{\Gamma }$ of $\mathrm{\Gamma }$, which is fully contained in the causal region $D_{\sigma }$ associated with $\sigma$.

Figure 3

Various FRW universes, $\mathrm{I},\mathrm{II},\mathrm{III},\dots$, have the same boundary area ${\mathcal{A}}_{*}$ at different times, $t_{*}(\mathrm{I}),t_{*}(\mathrm{II}),t_{*}(\mathrm{III}),\cdots$. Quantum states representing universes at these moments belong to Hilbert space ${\mathcal{H}}_{*}$, specified by the value of the boundary area.

Figure 4

A region $L(\gamma )$ of the leaf ${\sigma }_{*}$ is parametrized 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 $\gamma$ ($0\le \gamma \le \pi /2$) 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.

Figure 6

The value of $Q(\pi /2)$ as a function of $w$.

Figure 7

The shape of the extremal surfaces $E(\pi /2)$ for $w=-1$, $-0.98$, $-0.8$, 0, $1/3$, and 1. The horizontal axis is the cylindrical radial coordinate normalized by the apparent horizon radius, $\xi /{\xi }_{\mathrm{AH}}$, and the vertical axis is the Hubble time, $t_{\mathrm{H}}$.

Figure 8

A FRW universe whose dominant component changes from $w$ to $w^{\prime }$ at time $t_0$. Two surfaces depicted by orange lines are the latest extremal surface fully contained in the $w$ region (bottom) and the earliest extremal surface fully contained in the $w^{\prime }$ region (top), each anchored to the leaves at $t_{*}$ and $t_0$.

Figure 9

The ratio of the screen entanglement entropies, $R_w=R_{1w}(\pi /2)$, before and after the transition from a universe with the equation of state parameter $w$ to that with $w^{\prime }=1$, obtained from Figs. 6 and 7 using Eq. (58). The dot at $w=-1$ represents $R_{-1}=R_{1-1}(\pi /2)$ obtained in Eq. (59).

Figure 10

A steep potential (a) leading to the time evolution of the scalar field (b), the area of a leaf hemisphere (c), and the screen entanglement entropy (d). The same for a broad potential (e)–(h).

Figure 11

Possible structures of the Hilbert space ${\mathcal{H}}_{*}$ for a fixed boundary space $B$. In the direct-sum structure (left), each semiclassical spacetime in $D_{{\sigma }_{*}}$ has its own Hilbert space ${\mathcal{H}}_{*,w}$. The Russian-doll structure (right) corresponds to the scenario of spacetime equals entanglement; i.e. the entanglement entropies of the holographic degrees of freedom determine spacetime in $D_{{\sigma }_{*}}$. This implies that a superposition of exponentially many semiclassical spacetimes can lead to a different semiclassical spacetime.

Figure 12

If a black hole forms inside the holographic screen, future-directed ingoing light rays emanating orthogonally from the leaf ${\sigma }_{*}$ at an intermediate time may hit the singularity before reaching a caustic. While the diagram here assumes spherical symmetry for simplicity, the phenomenon can occur more generally.

Figure 13

To determine a state in the future, we need information on the “exterior” light sheet, the light sheet generated by light rays emanating from ${\sigma }_{*}$ in the $-k^a$ directions, in addition to that on the “interior” light sheet, i.e. the one generated by light rays emanating in the $+k^a$ directions.

Figure 14

In a universe beginning with a big bang, obtaining a future state requires a specification of signals from the big bang singularity, in addition to the information contained in the original state. In a FRW universe this is done by imposing spatial homogeneity and isotropy, which corresponds to selecting a fine-tuned state from the viewpoint of the big bang universe.