  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$_4$.
