On the stability of gravity with Dirichlet walls
Dirichlet walls - time-like boundaries at a 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-cutoff 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 conditions that must be imposed on the scalar-sector master field with harmonic time dependence depend 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 spherical walls in de Sitter space. We analyze both the 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 the 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.