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Cosmological horizons, Noether charges and entropy

Laura Donnay, Gaston Giribet

TL;DR

This work extends the recently discovered infinite-dimensional horizon symmetry from black holes to cosmological horizons in de Sitter space by introducing Gaussian near-horizon boundary conditions and deriving an asymptotic symmetry algebra consisting of two Virasoro copies and supertranslations. The associated Noether charges are finite, conserved, and integrable, with the zero-mode reproducing the Gibbons-Hawking entropy via the relation S = A/(4G) and T = κ/(2π). The analysis is then applied to a broader set of spacetimes, including Schwarzschild-de Sitter and Taub-NUT-de Sitter, where the same charges compute the entropies of both black hole and cosmological horizons, highlighting a deep connection between horizon thermodynamics and asymptotic symmetries. These results generalize the membrane paradigm perspective to cosmological horizons and point toward a unified framework for horizon thermodynamics across diverse solutions.

Abstract

It has recently been shown that, in the vicinity of their event horizons, black holes exhibit an infinite-dimensional symmetry. This symmetry captures relevant physical information about the black hole, and in particular about its thermodynamics. Here, we show that the same holds in a cosmological setup. More precisely, we show that around the de Sitter cosmological horizon, an infinite set of diffeomorphisms preserving a sensible set of boundary conditions emerges. These boundary conditions are similar to those considered by Price and Thorne in the context of the membrane paradigm, and permit to accommodate interesting gravity solutions. As for other boundary conditions considered previously, they are preserved by an infinite-dimensional asymptotic isometry algebra that includes supertranslations and two copies of the Virasoro algebra. This symmetry has associated a set of Noether charges that turn out to be conserved, finite, and integrable. We derive these charges explicitly using the covariant formalism and analyze their physical meaning by evaluating them to the case of de Sitter space. In this case, the zero-mode of the charge is found to account for the Gibbons-Hawking entropy of the cosmological horizon. We then consider a much more general set of solutions, including asymptotically de-Sitter black holes and asymptotically Taub-NUT-de Sitter black holes.

Cosmological horizons, Noether charges and entropy

TL;DR

This work extends the recently discovered infinite-dimensional horizon symmetry from black holes to cosmological horizons in de Sitter space by introducing Gaussian near-horizon boundary conditions and deriving an asymptotic symmetry algebra consisting of two Virasoro copies and supertranslations. The associated Noether charges are finite, conserved, and integrable, with the zero-mode reproducing the Gibbons-Hawking entropy via the relation S = A/(4G) and T = κ/(2π). The analysis is then applied to a broader set of spacetimes, including Schwarzschild-de Sitter and Taub-NUT-de Sitter, where the same charges compute the entropies of both black hole and cosmological horizons, highlighting a deep connection between horizon thermodynamics and asymptotic symmetries. These results generalize the membrane paradigm perspective to cosmological horizons and point toward a unified framework for horizon thermodynamics across diverse solutions.

Abstract

It has recently been shown that, in the vicinity of their event horizons, black holes exhibit an infinite-dimensional symmetry. This symmetry captures relevant physical information about the black hole, and in particular about its thermodynamics. Here, we show that the same holds in a cosmological setup. More precisely, we show that around the de Sitter cosmological horizon, an infinite set of diffeomorphisms preserving a sensible set of boundary conditions emerges. These boundary conditions are similar to those considered by Price and Thorne in the context of the membrane paradigm, and permit to accommodate interesting gravity solutions. As for other boundary conditions considered previously, they are preserved by an infinite-dimensional asymptotic isometry algebra that includes supertranslations and two copies of the Virasoro algebra. This symmetry has associated a set of Noether charges that turn out to be conserved, finite, and integrable. We derive these charges explicitly using the covariant formalism and analyze their physical meaning by evaluating them to the case of de Sitter space. In this case, the zero-mode of the charge is found to account for the Gibbons-Hawking entropy of the cosmological horizon. We then consider a much more general set of solutions, including asymptotically de-Sitter black holes and asymptotically Taub-NUT-de Sitter black holes.

Paper Structure

This paper contains 18 sections, 66 equations, 1 figure.

Table of Contents

  1. Introduction
  2. de Sitter space
  3. Infinite-dimensional symmetry
  4. Horizon boundary conditions
  5. Asymptotic Killing vectors
  6. Noether charges
  7. Cosmological horizons
  8. de Sitter cosmological horizon
  9. Comparison with other boundary conditions
  10. Schwarzschild-de Sitter event horizons
  11. The black hole event horizon
  12. The cosmological horizon
  13. The Nariai limit
  14. Taub-NUT-de Sitter horizons
  15. The Schwarzschild-Taub-NUT black hole horizon
  16. ...and 3 more sections

Figures (1)

  • Figure 1: Diagram of de Sitter space in Kruskal coordinates, defined by $\hat{r}=\ell (1+uv)(1-uv)$ and $uv=-e^{-2\tau /\ell }$. These coordinates permit to see the maximal extension of the geometry, which shows the existence of two pair of causally disconnected static patches separated by a past and a future event horizon.