  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.
