Causality constraints and anisotropic states

TL;DR

This work shows that anisotropy in relativistic hydrodynamics fundamentally alters the analytic structure of dispersion relations by introducing a continuum of collisions between hydrodynamic modes at complex wavevectors when a branch point sits at the origin. Through simple examples and a holographic calculation in a boosted ${\cal N}=4$ SYM plasma, the authors demonstrate that convergence radii become angle-dependent and may be constrained by causality, independent of the isotropic case. The paper derives a comprehensive set of upper bounds on the radii of convergence and on anisotropic transport coefficients, formulated via several complex variables and Reinhardt domain concepts, and highlights how these bounds can be inverted to constrain transport in anisotropic systems. The results generalize previous isotropic causality bounds, enable systematic exploration of anisotropic hydro, and point toward a holographic and geometric framework (e.g., expansion surfaces $\Sigma_r$ and critical surfaces $\Sigma_0$) to understand the limits of hydrodynamic descriptions in realistic, directionally dependent media.

Abstract

We investigate the effects of anisotropy on dispersion relations and convergence in relativistic hydrodynamics. In particular, we show that for dispersion relations with a branch point at the origin (such as sound modes), there exists a continuum of collisions between hydrodynamic modes at complex wavevector. These collisions are then explicitly demonstrated to be present in a holographic plasma. We lay out a criterion for when the continuum of collisions affects the convergence of the hydrodynamic derivative expansion. Finally, the radius of convergence of hydrodynamic dispersion relations in anisotropic systems is bounded from above on the basis of compatibility with microscopic causality.

Figures (11)

• Figure 1: A plot of the two first-order sound modes as they increase with $k_x$, written in terms of unitless quantities $\mathfrak{w} = \omega/(2 \pi T_0)$, $\mathfrak{q}_x = k_x/(2 \pi T_0)$, and with values $v_s = 1/\sqrt{3}$ and $\Gamma = 2/(2 \pi T_0)$. The branch point at the origin (${\mathfrak{q}}_x=0, {\mathfrak{w}}=0$) is marked with a black dot, while the branch point that sets the radius of convergence of the Puiseux series (${\mathfrak{q}}_x = \frac{1}{\sqrt{3}}, {\mathfrak{w}}=-\frac{i}{3}$) is marked with a red star.
• Figure 2: On the left is a plot of the critical surface for the boosted shear mode with $v_0 = \frac{1}{\sqrt{3}}$, and $D = 4/(2 \pi T_0)$. We work with dimensionless parameters $\mathfrak{w} = \omega/(2 \pi T_0)$, $\mathfrak{q}_x = k_x/(2 \pi T_0)$, and $\mathfrak{q}_z = k_z/(2 \pi T_0)$. The closest set of critical points to the origin, marked in black and with distance from the origin given analytically by equation \ref{['eq:boosted_shear_radii_analytical']}, set the radius of convergence and are given as a curve by equation \ref{['eq:boundary_curve_boosted_shear']}. The radii depend on the relative size of $|\mathfrak{q}_x|$ and $|\mathfrak{q}_z|$. On the right is a plot of the radius of convergence of the boosted shear mode as a function of $\theta$ when $v_0 = \frac{1}{\sqrt{3}}$, and $D = 4/(2 \pi T_0)$.
• Figure 3: Surfaces of critical points for the boosted sound mode. There are surfaces, $\Sigma_0$, that cut in sharply to reach the origin, leading to a non-commutativity of limits.
• Figure 4: Plot of the critical surfaces for ${\mathfrak b} = 2$, ${\mathfrak c}=1$ and ${\cal V}_A=1/\sqrt{2}$. The critical surfaces are plotted in the space of $|{\mathfrak{q}}_z|$ and $|{\mathfrak{q}}_x|$ as usual. The critical surfaces $\Sigma_0$ reach the origin (as expected), and are ultimately responsible for the non-commutativity of the small-$|{\bf k}|$ and the $\theta \to \frac{\pi}{2}$ limits..
• Figure 5: A survey of the hydrodynamic and lowest-lying non-hydrodynamic quasinormal modes of the shear and sound channels. The top left panel shows the first few modes of the shear channel varying as $v_0$ runs from $-0.8$ to $0.8$. The bottom left panel shows the hydrodynamic shear mode in more detail. The top right panel shows the first few modes of the sound channel varying as $v_0$ runs from $-0.8$ to $0.8$. The bottom right panel shows the hydrodynamic sound modes in more detail. Both plots are at ${\mathfrak{q}}_x = {\mathfrak{q}}_z = 0.1$.