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Uniqueness of the extremal Schwarzschild de Sitter spacetime

David Katona, James Lucietti

Abstract

We prove that any analytic vacuum spacetime with a positive cosmological constant in four and higher dimensions, that contains a static extremal Killing horizon with a maximally symmetric compact cross-section, must be locally isometric to either the extremal Schwarzschild de Sitter solution or its near-horizon geometry (the Nariai solution). In four-dimensions, this implies these solutions are the only analytic vacuum spacetimes that contain a static extremal horizon with compact cross-sections (up to identifications). We also consider the analogous uniqueness problem for the four-dimensional extremal hyperbolic Schwarzschild anti-de Sitter solution and show that it reduces to a spectral problem for the laplacian on compact hyperbolic surfaces, if a cohomological obstruction to the uniqueness of infinitesimal transverse deformations of the horizon is absent.

Uniqueness of the extremal Schwarzschild de Sitter spacetime

Abstract

We prove that any analytic vacuum spacetime with a positive cosmological constant in four and higher dimensions, that contains a static extremal Killing horizon with a maximally symmetric compact cross-section, must be locally isometric to either the extremal Schwarzschild de Sitter solution or its near-horizon geometry (the Nariai solution). In four-dimensions, this implies these solutions are the only analytic vacuum spacetimes that contain a static extremal horizon with compact cross-sections (up to identifications). We also consider the analogous uniqueness problem for the four-dimensional extremal hyperbolic Schwarzschild anti-de Sitter solution and show that it reduces to a spectral problem for the laplacian on compact hyperbolic surfaces, if a cohomological obstruction to the uniqueness of infinitesimal transverse deformations of the horizon is absent.
Paper Structure (8 sections, 8 theorems, 56 equations, 1 figure)

This paper contains 8 sections, 8 theorems, 56 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Extremal Schwarzschild (anti)-de Sitter solutions
  3. Transverse deformations of static extremal horizons
  4. Einstein equations near an extremal horizon
  5. Horizon geometry and first order transverse deformations
  6. Second and higher order transverse deformations
  7. Ricci tensor in Gaussian null coordinates
  8. Symmetric traceless 2-tensors on hyperbolic surfaces

Key Result

Theorem 1

Let $(M, g)$ be an analytic spacetime that obeys the $d\ge 4$-dimensional vacuum Einstein equation with cosmological constant $\Lambda>0$ and contains a static degenerate Killing horizon with a maximally symmetric compact cross-section. Then $(M, g)$ is locally isometric either to the extremal Schwa

Figures (1)

  • Figure 1: Penrose diagrams for $(a)$ extremal Schwarzschild-dS lake_effects_1977podolsky_structure_1999 and $(b)$ extremal hyperbolic Schwarzschild-AdS solutions mann_topological_1997.

Theorems & Definitions (14)

  • Theorem 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 2
  • ...and 4 more