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On extremal surfaces and de Sitter entropy

K. Narayan

TL;DR

The paper investigates extremal surfaces in the static patch of de Sitter space, focusing on the future and past universes and identifying connected timelike codimension-2 surfaces that stretch from the future boundary $I^+$ to the past boundary $I^-$. In four dimensions, these surfaces develop a divergent area whose coefficient precisely matches the de Sitter entropy, and in a minimal limit pass through the bifurcation region, resembling rotated Hartman-Maldacena-type constructions from AdS. The author argues for a possible dS/CFT interpretation of $dS_4$ as dual to two ghost-CFTs on $I^+$ and $I^-$ in an entangled state, supported by toy models of ghost-spins showing positive-norm, nontrivial entanglement in appropriate sectors. Together, these results hint at a holographic-like entanglement structure underlying de Sitter entropy and motivate further exploration of ghost-CFT duals and their entanglement properties. The work connects geometric extremal surfaces to entropy in de Sitter space and broadens the discussion of holographic interpretations in non-unitary, ghost-like dual theories.

Abstract

We study extremal surfaces in the static patch coordinatization of de Sitter space, focussing on the future and past universes. We find connected timelike codim-2 surfaces on a boundary Euclidean time slice stretching from the future boundary $I^+$ to the past boundary $I^-$. In a limit, these surfaces pass through the bifurcation region and have minimal area with a divergent piece alone, whose coefficient is de Sitter entropy in 4-dimensions. These are reminiscent of rotated versions of certain surfaces in the $AdS$ black hole. We close with some speculations on a possible $dS/CFT$ interpretation of 4-dim de Sitter space as dual to two copies of ghost-CFTs in an entangled state. For a simple toy model of two copies of ghost-spin chains, we argue that similar entangled states always have positive norm and positive entanglement.

On extremal surfaces and de Sitter entropy

TL;DR

The paper investigates extremal surfaces in the static patch of de Sitter space, focusing on the future and past universes and identifying connected timelike codimension-2 surfaces that stretch from the future boundary to the past boundary . In four dimensions, these surfaces develop a divergent area whose coefficient precisely matches the de Sitter entropy, and in a minimal limit pass through the bifurcation region, resembling rotated Hartman-Maldacena-type constructions from AdS. The author argues for a possible dS/CFT interpretation of as dual to two ghost-CFTs on and in an entangled state, supported by toy models of ghost-spins showing positive-norm, nontrivial entanglement in appropriate sectors. Together, these results hint at a holographic-like entanglement structure underlying de Sitter entropy and motivate further exploration of ghost-CFT duals and their entanglement properties. The work connects geometric extremal surfaces to entropy in de Sitter space and broadens the discussion of holographic interpretations in non-unitary, ghost-like dual theories.

Abstract

We study extremal surfaces in the static patch coordinatization of de Sitter space, focussing on the future and past universes. We find connected timelike codim-2 surfaces on a boundary Euclidean time slice stretching from the future boundary to the past boundary . In a limit, these surfaces pass through the bifurcation region and have minimal area with a divergent piece alone, whose coefficient is de Sitter entropy in 4-dimensions. These are reminiscent of rotated versions of certain surfaces in the black hole. We close with some speculations on a possible interpretation of 4-dim de Sitter space as dual to two copies of ghost-CFTs in an entangled state. For a simple toy model of two copies of ghost-spin chains, we argue that similar entangled states always have positive norm and positive entanglement.

Paper Structure

This paper contains 9 sections, 50 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The static patch and de Sitter entropy
  3. The future and past universes and extremal surfaces
  4. An $S^{d-1}$ equatorial plane
  5. Features of $S^{d-1}$ equatorial plane extremal surfaces
  6. The $w=const$ slice
  7. Discussion
  8. Reviewing extremal surfaces in Poincare $dS$
  9. Reviewing ghost-spins and ghost-spin chains

Figures (3)

  • Figure 1: Penrose diagram of de Sitter space in static coordinates: $N$ and $S$ are the Northern and Southern hemispheres. $F$ and $P$ are future and past universes.
  • Figure 2: Timelike extremal surfaces in the $(\tau,w)$-plane stretching from $I^+$ to $I^-$. These are akin to rotated versions of the surfaces of Hartman-Maldacena in the $AdS$ black hole. The red timelike surface intersects the horizons. The limiting blue surface is almost null (almost hugging the horizons) and passes through the bifurcation region: it has minimal area.
  • Figure 3: Timelike extremal surfaces in the $(\tau,\theta)$-plane stretching from ${\cal I}^+$ to ${\cal I}^-$. The limiting blue surface stretches from one hemisphere ($\theta={\pi\over 2}$) at $I^+$ to another equivalent one at $I^-$.