Incompressible Fluids of the de Sitter Horizon and Beyond

TL;DR

The paper investigates how Einstein gravity in four-dimensional de Sitter space encodes horizon dynamics as an incompressible fluid on the two-sphere. By imposing conformal Dirichlet boundary conditions on a timelike surface near the cosmological horizon and matching conditions on spacelike slices near $\mathcal{I}^+$, the authors show that linear and nonlinear perturbations reduce to the incompressible Navier–Stokes equation on $S^2$ with $\nu=1$ (or $\nu_{sf}=-1$ on spacelike slices). They compute discrete fluid modes, examine the flow of dispersion relations as the cutoff surface moves from horizon toward the worldline, and relate spectra to quasinormal modes. The spacelike-slice analysis reveals a tower of modes matching quasinormal modes and, via analytic continuation, maps to massless topological AdS$_4$ black holes, highlighting a deep dS/AdS relationship in horizon holography.

Abstract

There are (at least) two surfaces of particular interest in eternal de Sitter space. One is the timelike hypersurface constituting the lab wall of a static patch observer and the other is the future boundary of global de Sitter space. We study both linear and non-linear deformations of four-dimensional de Sitter space which obey the Einstein equation. Our deformations leave the induced conformal metric and trace of the extrinsic curvature unchanged for a fixed hypersurface. This hypersurface is either timelike within the static patch or spacelike in the future diamond. We require the deformations to be regular at the future horizon of the static patch observer. For linearized perturbations in the future diamond, this corresponds to imposing incoming flux solely from the future horizon of a single static patch observer. When the slices are arbitrarily close to the cosmological horizon, the finite deformations are characterized by solutions to the incompressible Navier-Stokes equation for both spacelike and timelike hypersurfaces. We then study, at the level of linearized gravity, the change in the discrete dispersion relation as we push the timelike hypersurface toward the worldline of the static patch. Finally, we study the spectrum of linearized solutions as the spacelike slices are pushed to future infinity and relate our calculations to analogous ones in the context of massless topological black holes in AdS$_4$.

Figures (8)

• Figure 1: Penrose diagram of de Sitter space indicating the various static patches and future/past diamonds.
• Figure 2: Penrose diagram of de Sitter space indicating constant $\rho$ (red) and $\tau$ (gray diagonal) slices.
• Figure 3: Our boundary conditions for the linearized modes are such that the induced metric on the $\rho=1$ slice is unchanged and there is no incoming flux from the past horizon.
• Figure 4: Flow of frequency spectrum $i\omega \ell$ vs $l$ as we move away from $r_c \approx 0$ toward the cosmological horizon.
• Figure 5: Flow of frequency spectrum $i\omega \ell$ vs $l$ as we approach $r_c=\ell$. Points obeying the fluid dispersion relation are represented in blue.