Causality and the AdS Dirichlet problem

TL;DR

This work analyzes the AdS Dirichlet problem with a finite timelike cut-off surface Σ_D to understand causality of information transfer. By examining gravitational fluctuations in Schwarzschild-AdS5 and in Rindler space, and performing both analytical (including WKB) and numerical studies of quasinormal modes, it shows that high-frequency, bulk-propagating modes travel on or within the induced light cones (v_UV = 1), indicating bulk causality is preserved despite long-wavelength superluminal hydrodynamic signals. Two exceptions arise from boundary gravitons: in AdS3 these gravitons can propagate acausally, and in Rindler space a pure-pressure boundary mode tied to Galilean symmetry yields acausal behavior unless additional boundary conditions are imposed. The results support causal bulk dynamics under Dirichlet boundaries and offer nuanced implications for the fluid/gravity correspondence and the black hole membrane paradigm, clarifying the role of UV versus IR propagation and the conditions under which boundary degrees of freedom affect causality.

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.

Figures (5)

• Figure 1: The quasinormal frequencies of the Rindler geometry in the scalar and tensor channels for two different values of the spatial momenta, ${\bf q} = 0$ (on the left) and ${\bf q} = 5$ (on the right). The absence of low lying hydrodynamic quasinormal modes is clear in the low momentum limit. Because the generic expression degenerates at $\bf q =0$, the left plot was in fact generated using ${\bf q} = 10^{-9}$.
• Figure 2: The quasinormal frequencies of the Rindler geometry in the vector channel for ${\bf q} = 0$ illustrating the low frequency shear mode.
• Figure 3: The asymptotic behavior of Rindler scalar (and tensor) channel quasinormal modes showing approach to the causal dispersion \ref{['rinduv']}.
• Figure 4: The lowest quasinormal mode in the scalar channel at low and intermediate frequency and momenta for various choices of Dirichlet surfaces: $u_D = 0$ (top), $u_D = 0.25$ (bottom). We show both the real and imaginary parts of the dispersion relation. For $\text{Re}({\mathfrak w})$ vs ${\mathfrak q}$ we indicate the prediction \ref{['dissound']} for the sound velocity at low frequency using a dashed line, while the dotted line is the predicted value for the high-frequency modes \ref{['sadsUV']}. Note that the ${\mathfrak q}=0$ sound mode propagates causally for the above values of $u_D$. As indicated in the text convergence to the asymptotic UV behaviour is in fact quite slow and hence we restrict to momenta ${\mathfrak q}\leq 5$. Finally, we also note that plots do not incorporate the red-shift scaling factor.
• Figure 5: The lowest quasinormal mode in the scalar channel at low and intermediate frequency and momenta for various choices of Dirichlet surfaces: $u_D = \frac{1}{\sqrt{2}}$ (top), where the sound mode starts to propagate on the light-cone of the Dirichlet surface and $u_D = 0.9$ (bottom) when the sound mode is acausal. The rest of the conventions are as in Fig. \ref{['f:sadssoundA']}.