Acausality-driven instabilities in transient relativistic viscous hydrodynamics

TL;DR

This work investigates the interplay between acausality and instability in Israel-Stewart–type relativistic viscous hydrodynamics. It introduces a framework based on the acoustic propagation speed $w$ and fluid speed $v$, distinguishing regimes as causal, acausal-stable, and acausal-unstable, and presents a new analytic, space-homogeneous benchmark with an exact $v(t)$ solution to validate numerical solvers. The authors demonstrate that instabilities arise when $vw\ge 1$ (and in the elliptic case when $w^2<0$), and show that a MUSIC solver reproduces the analytic results in stable regimes but develops numerical runaway in the unstable regime. A (2+1)D bulk-only test with realistic initial conditions reveals acausal regions can emerge during evolution, with a second-order term $\delta_{\Pi\Pi}$ greatly mitigating these pathologies, highlighting the sensitivity of causality to model details. Overall, the paper provides diagnostic tools and a concrete benchmark for validating relativistic dissipative hydrodynamics codes and clarifies the limits of causality and well-posedness across transient theories, with BDNK remaining robustly well-posed.

Abstract

We investigate non-linear instabilities stemming from superluminal propagation of information in Israel-Stewart-like models of relativistic viscous fluid dynamics. In relativity, the characteristic speed of propagation of information, $w$, and the speed of the fluid, $v$, allow us to differentiate between regimes of the hydrodynamic equations that are acausal but stable ($w>1$), unstable ($v^{2} w^{2} \geq 1$), and covariantly ill-posed ($w^{2} \leq 0$). As an analytical benchmark, we present a new solution that illustrates these distinct regimes. We compare this analytical solution to the result of a numerical relativistic viscous fluid dynamics solver, and confirm that the analytical result can be recovered numerically in the stable regime, whether causal or acausal. The onset of numerical instabilities is further found to occur in the regime predicted by the analytical solution.

Figures (6)

• Figure 1: Spacetime diagrams illustrating the difference between the acausal but stable regime (left panel) and the unstable regime (right panel). The red cone is the acoustic cone in tangent space, which is larger than the lightcone (yellow) due to acausality. In the stable case, signals (green) propagate to the future, and dissipation works as usual. In the unstable case, some signals (green) travel to the past, and they experience reversed dissipation, which causes non-equilibrium displacements to spontaneously blow up. For illustration, we took $w=2$ (working with bulk viscosity alone), and a fluid velocity $v=0.4$ in the stable case, and $0.6$ in the unstable case. The brown arrows represent the four-velocity vectors of the fluid.
• Figure 2: Graph of the function $v(t)$ obtained by inverting eq. \ref{['elboss']}. This is an analytical solution of Israel-Stewart theory with only bulk. Around $v\sim 0.732$, the system exits the causal region (good), and enters the acausal but stable region (bad). Nothing serious happens. However, around $v_c \sim 0.838$, the fluid enters the acausal and unstable region (ugly), and the solution bifurcates. The new solution branch is pathological, since $\Pi$ keeps getting more and more negative, so the fluid accelerates to the speed of light. Note that, when $v\rightarrow 1$, the ratio $\Pi/P$ saturates to $-4$, which is a finite value.
• Figure 3: Comparison between the analytical solution in eq. \ref{['elboss']} and numerical solutions with initial conditions with different causality/stability status. (Left Panel) Exact initial condition. (Right Panel) "Zero-time-gradient" initial conditions. See text for details.
• Figure 4: Causality-stability status (panels (a), (c) and (e)) and temperature field ((b), (d) and (f) panels) along the hydrodynamic evolution for the initial (panels (a) and (b)) and later ((c) and (d); (e) and (f)) times for a pure-bulk fluid. In the temperature field plots, the red lines represent the boundaries of the ugly regions of the $vw > 1$ kind. A minimal Israel-Stewart model ('Minimal IS', eq. \ref{['eq:min_IS-model']}) is employed in the center panels and the Israel-Stewart model with a second order transport coefficient ('IS with $\delta_{\Pi \Pi}$', eq. \ref{['eq:w2-2nd-IS']}) in the lower panels.
• Figure 5: Error estimators as a function of normalized time for simulations with analytical pre-initial conditions. (Left panel) $E_{1}(t)$ (see eq. \ref{['eq:e1']}). (Right panel) $E_{2}(t)$ (see eq. \ref{['eq:e2']}).