On the stability of gravity with Dirichlet walls

TL;DR

This paper investigates the stability of gravity in the presence of Dirichlet walls, revealing novel frequency-dependent boundary conditions and instabilities, yet confirming stability in many configurations relevant to holography and black hole physics.

Contribution

It introduces a new analysis of linearized gravity with Dirichlet walls, highlighting frequency-dependent boundary conditions and stability properties across various spacetimes.

Findings

Frequency-dependent boundary conditions for spin-2 fields.

Existence of instabilities outside spherical Dirichlet walls.

Stability of flat walls and black holes near Dirichlet boundaries.

Abstract

Dirichlet walls -- timelike boundaries at finite distance from the bulk on which the induced metric is held fixed -- have been used to model AdS spacetimes with a finite cutoff. In the context of gauge/gravity duality, such models are often described as dual to some novel UV-cufoff version of a corresponding CFT that maintains local Lorentz invariance. We study linearized gravity in the presence of such a wall and find it to differ significantly from the seemingly-analogous case of Dirichlet boundary conditions for fields of spins zero and one. In particular, using the Kodama-Ishibashi formalism, the boundary condition that must be imposed on scalar-sector master field with harmonic time dependence depends explicitly on their frequency. That this feature first arises for spin-2 appears to be related to the second-order nature of the equations of motion. It gives rise to a number of novel instabilities, though both global and planar Anti-de Sitter remain (linearly) stable in the presence of large-radius Dirichlet cutoffs. The instabilities arise on the outside of spherical Dirichlet walls, and also inside sufficiently large such spherical walls in de Sitter space. We analyze both inside and outside of flat and spherical walls in Minkowski, de Sitter, and anti-de Sitter space, as well as in certain black hole spacetimes and find stability for cases not mentioned above. In particular, we find no linear instabilities in the presence of flat walls. We also find evidence supporting the conjecture that neutral black holes are repelled by Dirichlet walls.