[Paper Review] Les Houches Lectures on De Sitter Space

Marcus Spradlin, Andrew Strominger+1 more|ArXiv.org|Sep 30, 2001

Black Holes and Theoretical PhysicsPhysics and Astronomy60 references184 citations

TL;DR

This paper provides a pedagogical introduction to de Sitter quantum gravity, focusing on the classical geometry of de Sitter space, quantum field theory on it—including its temperature and entropy—and the emergence of the conformal group as an asymptotic symmetry group in three dimensions. The key contribution is a derivation of asymptotically de Sitter boundary conditions and a detailed explanation of the dS/CFT correspondence in 3D, linking gravity in de Sitter space to a conformal field theory on its boundary.

ABSTRACT

These lectures present an elementary discussion of some background material relevant to the problem of de Sitter quantum gravity. The first two lectures discuss the classical geometry of de Sitter space and properties of quantum field theory on de Sitter space, especially the temperature and entropy of de Sitter space. The final lecture contains a pedagogical discussion of the appearance of the conformal group as an asymptotic symmetry group, which is central to the dS/CFT correspondence. A (previously lacking) derivation of asymptotically de Sitter boundary conditions is also given.

Motivation & Objective

• To provide a foundational understanding of de Sitter space geometry and quantum field theory for researchers in quantum gravity.
• To address the lack of a microscopic statistical explanation for the Bekenstein-Hawking entropy in de Sitter space, particularly due to the absence of supersymmetry and embedding in string theory.
• To derive asymptotically de Sitter boundary conditions for gravity in three dimensions, a missing ingredient in earlier treatments of the dS/CFT correspondence.
• To explain the emergence of the two-dimensional conformal group as the asymptotic symmetry group in 3D de Sitter space, central to the dS/CFT correspondence.
• To lay the groundwork for understanding quantum gravity in de Sitter space by connecting classical geometry, quantum field theory, and holographic duality.

Proposed method

• Uses planar and global coordinate systems to describe the classical geometry of de Sitter space, including Penrose diagrams and geodesics.
• Analyzes scalar quantum field theory on de Sitter space using Green functions and defines vacua, particularly the Euclidean vacuum.
• Derives the de Sitter temperature via the periodicity of the Euclidean Green function, showing thermal equilibrium at the de Sitter horizon.
• Calculates the entropy of de Sitter space using the Bekenstein-Hawking formula, S = A/(4G), and discusses its universality and challenges in microscopic interpretation.
• Derives the asymptotic symmetry group in 3D de Sitter space by analyzing diffeomorphisms that preserve the asymptotic metric structure, showing they correspond to conformal transformations on the boundary.
• Computes the Brown-York stress tensor for small metric perturbations in planar coordinates, using the ADM decomposition with lapse and shift functions, and derives its asymptotic form.

Experimental results

Research questions

• RQ1 How does the conformal group arise as the asymptotic symmetry group in three-dimensional de Sitter space?
• RQ2 What are the correct boundary conditions for gravity in asymptotically de Sitter spacetimes, and how can they be derived from first principles?
• RQ3 How does the temperature and entropy of de Sitter space emerge from quantum field theory on a fixed de Sitter background?
• RQ4 What is the role of the isometry group SL(2,C) in the dS/CFT correspondence, and how does it relate to the full asymptotic symmetry group?
• RQ5 Can the dS/CFT correspondence be consistently formulated in three dimensions, and what are the implications for quantum gravity in de Sitter space?

Key findings

• The asymptotic symmetry group of three-dimensional de Sitter space is the two-dimensional conformal group, which is infinite-dimensional, unlike in higher dimensions where it reduces to the isometry group SO(d,1).
• The derivation of asymptotically de Sitter boundary conditions is completed, filling a gap in the literature and providing a foundation for the dS/CFT correspondence.
• The de Sitter temperature is derived from the periodicity of the Euclidean Green function, confirming that de Sitter space behaves as a thermal system at temperature T = H/(2π), where H is the Hubble parameter.
• The entropy of de Sitter space is given by the Bekenstein-Hawking formula S = A/(4G), with the horizon area A = 4π/H², consistent across all event horizons including cosmological ones.
• The Brown-York stress tensor is computed for small metric perturbations in planar coordinates, with explicit expressions for the extrinsic curvature and trace, showing its role in holographic stress-energy calculations.
• The asymptotic symmetry generators in dS₃ are shown to decompose into a holomorphic diffeomorphism and a Weyl transformation, confirming the equivalence of 3D diffeomorphisms to 2D conformal transformations on the boundary.