Causality and the AdS Dirichlet problem

TL;DR

This paper investigates the causality of information propagation in the AdS Dirichlet problem, showing that it is generally causal except in specific cases involving boundary gravitons, with implications for fluid/gravity duality and black hole physics.

Contribution

The study clarifies the causal structure of the AdS Dirichlet problem, identifying exceptions involving boundary gravitons and analyzing their impact on holographic and membrane paradigm frameworks.

Findings

Most cases exhibit causal information propagation on the Dirichlet hypersurface.

Boundary gravitons in AdS3 and Rindler space can violate induced light cone causality.

High-frequency dynamics remain causal despite superluminal hydrodynamic modes.

Abstract

The (planar) AdS Dirichlet problem has previously been shown to exhibit superluminal hydrodynamic sound modes. This problem is defined by bulk gravitational dynamics with Dirichlet boundary conditions imposed on a rigid timelike cut-off surface. We undertake a careful examination of this set-up and argue that, in most cases, the propagation of information between points on the Dirichlet hypersurface is nevertheless causal with respect to the induced light cones. In particular, the high-frequency dynamics is causal in this sense. There are however two exceptions and both involve boundary gravitons whose propagation is not constrained by the Einstein equations. These occur in i) AdS$_3$, where the boundary gravitons generally do not respect the induced light cones on the boundary, and ii) Rindler space, where they are related to the infinite speed of sound in incompressible fluids. We discuss implications for the fluid/gravity correspondence with rigid Dirichlet boundaries and for the black hole membrane paradigm.