On the stability of gravity with Dirichlet walls

TL;DR

<3-5 sentence high-level summary>The paper analyzes linearized gravity in spacetimes with Dirichlet walls, using the Kodama–Ishibashi master-field formalism to translate the wall into sector-specific boundary conditions. Tensor and vector gravitational perturbations satisfy standard, positive-definite eigenvalue problems and are generally stable across Minkowski, AdS, and dS backgrounds, while scalar perturbations feature a frequency-dependent Robin boundary condition that can induce instabilities, particularly outside spherical cavities and inside sufficiently large de Sitter cavities. For flat walls, the spectra indicate linear stability, and in AdS the exterior scalar modes can become unstable for certain wall radii, whereas interior regions tend to be stable. The work also provides evidence that neutral black holes are repelled by Dirichlet walls and discusses implications for holography and nonlinear dynamics, outlining directions for future study of charged cases and nonlinear stability.</_paper_summary>

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.

Figures (11)

• Figure 1: External Minkowski tensor and vector modes are stable: Spectra of frequencies for the tensor (left) and vector (right) modes for $r> r_D$, $K=1$, $M=\lambda=0$, $n=4$. These are obtained by solving $e_T(\hat{\omega}) = 0$ and $e_V(\hat{\omega}) = 0$ respectively, with $\ell_I =$ 2, 3, 4, 5 (blue circles, yellow squares, green diamonds, red triangles).
• Figure 2: External Minkowski scalars are unstable: Spectrum of frequencies (left panel) for scalar modes with $r> r_D$, $K=1$, $M=\lambda=0$, $n=2$. These are given by the solutions of $e_S(\hat{\omega})=0$, with $\ell_S =$ 2, 3, 4, 5 (blue circles, yellow squares, green diamonds, red triangles). On the right panel we plot the imaginary part of the unstable modes as a function of $\ell_S$.
• Figure 3: The imaginary part of the frequency of the unstable scalar mode outside of a spherical cavity in global AdS$_4$ for $\ell =2,3,4$ (blue circles, yellow squares, green diamonds).
• Figure 4: The imaginary part of the frequency of the unstable scalar mode between two cavities in flat space for $\ell =2,3,4$ (blue circles, yellow squares, green diamonds).
• Figure 5: Unstable internal dS scalar modes: The data points are the value of the location of the wall $\rho_D$ at which the first unstable modes appear as a function of angular momentum $\ell_S$, while the dashed line is $r_{crit}/L_{AdS}$. Results are shown for $n=2$ (left) and $n=3$ (right).