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A microscopic model of de Sitter spacetime with an observer

Damiano Tietto, Herman Verlinde

TL;DR

The paper presents a microscopic quantum-mechanical model of de Sitter holography with an observer, deriving a total entropy formula for a 3D Schwarzschild–de Sitter patch plus the observer and matching it to the DSSYK spectral density. By introducing an observer, the authors obtain a maximal entropy $S_{ m max}(ψ)=S_{ m GH}(ψ)+β_{ m dS}E(ψ)$ and show that the DSSYK entropy satisfies $S_{ m SYK}(ψ)=S_0+S_{ m max}(ψ)$ with $λ=8π G_N$, effectively linking the deficit-angle parameter to the DSSYK thermodynamics. This yields a de Sitter interpretation of the DSSYK spectral density and explains the appearance of two inverse temperatures, connected via $β_{ m GH}=(2π/ψ)β_{ m max}$, aligning physical and fake temperatures within the dual picture. The result strengthens the gravity–SYK correspondence in low dimensions and motivates further exploration of 2D gravity reductions as natural duals.

Abstract

We introduce a simple microscopic quantum mechanical model of low-dimensional de Sitter holography with an observer. Using semiclassical gravity and elementary thermodynamic considerations, we derive a formula for the total entropy of a 3D Schwarzschild-de Sitter universe with an observer. We then match this entropy formula with the exactly known spectral density of the double scaled SYK model. Our result gives a de Sitter interpretation of the appearance of two notions of temperature in DSSYK.

A microscopic model of de Sitter spacetime with an observer

TL;DR

The paper presents a microscopic quantum-mechanical model of de Sitter holography with an observer, deriving a total entropy formula for a 3D Schwarzschild–de Sitter patch plus the observer and matching it to the DSSYK spectral density. By introducing an observer, the authors obtain a maximal entropy and show that the DSSYK entropy satisfies with , effectively linking the deficit-angle parameter to the DSSYK thermodynamics. This yields a de Sitter interpretation of the DSSYK spectral density and explains the appearance of two inverse temperatures, connected via , aligning physical and fake temperatures within the dual picture. The result strengthens the gravity–SYK correspondence in low dimensions and motivates further exploration of 2D gravity reductions as natural duals.

Abstract

We introduce a simple microscopic quantum mechanical model of low-dimensional de Sitter holography with an observer. Using semiclassical gravity and elementary thermodynamic considerations, we derive a formula for the total entropy of a 3D Schwarzschild-de Sitter universe with an observer. We then match this entropy formula with the exactly known spectral density of the double scaled SYK model. Our result gives a de Sitter interpretation of the appearance of two notions of temperature in DSSYK.

Paper Structure

This paper contains 9 sections, 51 equations, 3 figures.

Table of Contents

  1. Introduction
  2. 3D Schwarzschild-de Sitter thermodynamics
  3. DSSYK partition function and thermodynamics
  4. De Sitter thermodynamics with an observer
  5. Matching de Sitter and DSSYK thermodynamics
  6. De Sitter interpretation of SYK spectral density
  7. De Sitter interpretation of the two temperatures
  8. Concluding remarks
  9. Relation between localized energy and the deficit angle

Figures (3)

  • Figure 1: We consider a Schwarzschild-de Sitter spacetime with an observer located at the center. The total entropy of the combined system is the sum of the entropies of the SdS spacetime and of the observer.
  • Figure 2: We view the combined system of the SdS static patch and the observer as an entangled bound state. The total entropy of the combined system is the sum of the entropies of the two subsystems.
  • Figure 3: The entropy $S_{\rm max}(\psi)$ (blue) of the combined system of the SdS static patch and the observer is the sum of the Gibbons-Hawking entropy $S_{\rm GH}(\psi)$ (red) and the entropy $S_{\rm obs}(\psi) = \beta_{\rm dS} E(\psi)$ (green) that the observer of energy $E(\psi)$ has extracted from the ambient SdS spacetime.