The central dogma and cosmological horizons

TL;DR

The paper investigates whether the black hole central dogma extends to cosmological horizons in de Sitter space. It shows that naively importing AdS/CFT-like extremization fails due to the minimax nature of the de Sitter bifurcation surface and resulting entanglement wedge inconsistencies. To resolve this, the author proposes anchoring extremal surfaces to the horizon and performing a two-sided extremization, yielding sensible entropy behavior: vanishing total entropy, $S = A/(4G)$ for a single static patch, and island-like transitions at half-horizon size, along with a de Sitter version of the Hartman-Maldacena transition. The discussion emphasizes horizon-horizon interactions and the need for a more complete holographic understanding of the horizon theories and their dynamics.

Abstract

The central dogma of black hole physics -- which says that from the outside a black hole can be described in terms of a quantum system with exp$(\text{Area}/4G_N)$ states evolving unitarily -- has recently been supported by computations indicating that the interior of the black hole is encoded in the Hawking radiation of the exterior. In this paper, we probe whether such a dogma for cosmological horizons has any support from similar computations. The fact that the de Sitter bifurcation surface is a minimax surface (instead of a maximin surface) causes problems with this interpretation when trying to import calculations analogous to the AdS case. This suggests anchoring extremal surfaces to the horizon itself, where we formulate a two-sided extremization prescription and find answers consistent with general expectations for a quantum theory of de Sitter space: vanishing total entropy, an entropy of $A/4G_N$ when restricting to a single static patch, an entropy of a subregion of the horizon which grows as the region size grows until an island-like transition at half the horizon size when the entanglement wedge becomes the entire static patch interior, and a de Sitter version of the Hartman-Maldacena transition.

Figures (11)

• Figure 1: Two different stereographic projections. The one on the left foliates the sphere by lower-dimensional spheres from north to south, which get projected onto the plane as spheres of increasing radius. The one on the right foliates the sphere by lower-dimensional spheres from west to east, which get projected onto the plane as two sets of spheres surrounding two points. The sphere passing through the north and south poles gets mapped to a lower-dimensional plane. The bottom line shows a direct view of the parameterization of the plane, radial coordinates on the left and owl coordinates on the right.
• Figure 2: The de Sitter region is replaced by a pair of quantum-mechanical theories, each described by a Hilbert space of $e^{A/4G}$ degrees of freedom.
• Figure 3: The fine-grained entropy for regions specified in the exact quantum-mechanical description (left side of equality) would equal the generalized entropy for the regions specified in the semiclassical spacetime (right side of equality) if the minimax surface shown was an acceptable quantum extremal surface.
• Figure 4: The QES is given as the degenerate surface which sits right on the cutoff surface.
• Figure 5: We represent wedge holography as a series of modifications to the familiar example of the top line Almheiri:2019yqkAlmheiri:2019psyAlmheiri:2019qdqChen:2020uacChen:2020hmv. On the top left we have the spacetime for the eternal black hole in AdS$_d$ coupled to two flat space baths. There is a holographic CFT$_d$ everywhere in the spacetime. The top middle represents the microscopic description, where we have a holographic CFT$_{d-1}$ (represented by a red dot) coupled to a $d$-dimensional bath hosting the holographic CFT$_d$, and this joint system's thermofield double partner. The top right illustrates the emergent $(d+1)$-dimensional description (the thermofield double part is not represented for simplicity). In the middle line we have the same representations, except we now have a finite bath and an additional holographic CFT$_{d-1}$ (represented by a blue dot) which is also coupled to the bath CFT$_d$ (and their thermofield double partners) Geng:2021iyqBalasubramanian:2021xcm. The Penrose diagram is identified at the left and right. The final line shrinks the bath size to zero, so the interacting CFT's sit right on top of one another Penington:2019kkiGeng:2020fxl. Notice in this case the boundary dual has spacetime to both sides. In our de Sitter analogy this will be the primary feature we are interested in, as our "boundary dual" is on the cosmic horizon and has spacetime to both sides. The analog of the doubly emergent AdS$_{d+1}$ will not play a role.