Incompressible Fluids of the de Sitter Horizon and Beyond

TL;DR

This paper investigates linear and nonlinear deformations of four-dimensional de Sitter space, revealing connections to incompressible fluid dynamics and analyzing the spectrum of perturbations near horizons and boundaries.

Contribution

It introduces a novel approach linking deformations of de Sitter space to incompressible Navier-Stokes solutions and explores their spectral properties at horizons and boundaries.

Findings

Deformations near horizons relate to incompressible fluid equations.

Finite deformations are characterized by Navier-Stokes solutions.

Spectrum analysis connects de Sitter perturbations to black hole analogs.

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$_4$.