Action, Mass and Entropy of Schwarzschild-de Sitter black holes and the de Sitter/CFT Correspondence

TL;DR

This work addresses the challenge of defining conserved charges in asymptotically de Sitter spacetimes by extending the AdS boundary counterterm method to de Sitter space. It derives the de Sitter counterterms up to $d\le 8$ (and up to $d\le 9$ in the introduction) and uses them to obtain finite on-shell actions and a holographic stress tensor, enabling a mass definition for Schwarzschild–de Sitter black holes and a Gibbs-Duhem–based entropy outside the cosmological horizon. A key result is that inflationary coordinates render the action finite in all $d$, while covering coordinates introduce a linear divergence in odd $d$ that cannot be canceled by polynomial boundary invariants, highlighting coordinate-dependent subtleties in the dS/CFT picture. The SdS analysis shows the mass approaches dimensionally dependent constants with signs depending on parity of $d$, and the entropy decreases with increasing mass, providing further evidence for a dS/CFT-like correspondence with interpretive caveats. Overall, the paper strengthens the case for a dS/CFT framework while clarifying limitations and open interpretive issues.

Abstract

We investigate a recent proposal for defining a conserved mass in asymptotically de Sitter spacetimes that is based on a conjectured holographic duality between such spacetimes and Euclidean conformal field theory. We show that an algorithm for deriving such terms in asymptotically anti de Sitter spacetimes has an asymptotically de Sitter counterpart, and derive the explicit form for such terms up to 9 dimensions. We show that divergences of the on-shell action for de Sitter spacetime are removed in any dimension in inflationary coordinates, but in covering coordinates a linear divergence remains in odd dimensions that cannot be cancelled by local terms that are polynomial in boundary curvature invariants. We show that the class of Schwarzschild-de Sitter black holes up to 9 dimensions has finite action and conserved mass, and construct a definition of entropy outside the cosmological horizon by generalizing the Gibbs-Duhem relation in asymptotically dS spacetimes. The entropy is agreement with that obtained from CFT methods in $d=2$. In general our results provide further supporting evidence for a dS/CFT correspondence, although some important interpretive problems remain.