The Stretched Horizon Limit

Abstract

We consider four-dimensional general relativity with a positive cosmological constant, $Λ$, in the presence of a boundary, $Γ$, of finite spatial size. The boundary is located near a cosmological event horizon, and is subject to boundary conditions that fix the conformal class of the induced metric, and, $K$, the trace of the extrinsic curvature along $Γ$. The proximity of $Γ$ to the horizon is controlled by the dimensionless parameter ${K}{Λ^{-\frac{1}{2}}}$. We provide an exhaustive analysis of linearised gravitational perturbations for the setup. This is performed both for a $Γ$ encasing a portion of the static patch that ends just before the cosmological horizon (pole patch), as well as a $Γ$ containing only the region near the cosmological horizon (cosmic patch). In the pole patch, we uncover a layered hierarchy of modes: ordinary normal modes, a novel type of boundary gapless mode, and boundary soft modes of frequency $ω\approx \pm 2πi T_{\text{dS}}$, with $T_{\text{dS}}$ the horizon temperature. Minkowskian behaviour is recovered only for angular momenta $l \gtrsim {K}{Λ^{-\frac{1}{2}}}$ which can be made parametrically large, thus attenuating previously found growing modes. In the cosmic patch, we uncover sound and shear fluid-dynamical modes that we interpret in terms of a conformal fluid with shear viscosity over entropy density ratio $\tfracη{s} = \tfrac{1}{4π}$ and vanishing bulk viscosity $ζ=0$. The fluid dynamical sector is shown to admit a non-linear treatment. We describe a scaling regime in which the stretched horizon gravitational dynamics is dictated by a universal Rindler geometry, independent to the details of the infilling horizon. We briefly discuss quantitative features that distinguish cosmological and black hole horizons away from the Rindler regime.