What happens in hydrodynamic simulations of heavy-ion collisions when causality is violated?

TL;DR

This work probes how causality violations in Israel-Stewart-type relativistic viscous hydrodynamics influence both physical evolution and numerical stability. By comparing an analytical IS solution with MUSIC-based simulations, it links instability onset to the condition $v w \ge 1$ within the acoustic geometry defined by $G_{\mu\nu}=-u_\mu u_\nu + \frac{1}{w^2}(g_{\mu \nu}+u_\mu u_\nu)$ and the effective speed $w^2=c_s^2+\frac{\zeta}{\tau_\Pi(\varepsilon+P+\Pi)}$. The study classifies fluid cells as good, bad, or ugly based on $w$ and $v$, demonstrating exact agreement with analytic results in good/bad regions and revealing a frame-dependent instability in the ugly regime, which can be mitigated by initialization choices. The results highlight the practical need for careful initialization and numerical regulation in IS-based hydrodynamics to avoid unphysical outcomes in heavy-ion collision modeling.

Abstract

We summarize our recent investigations on how causality violations in Israel-Stewart-type relativistic viscous hydrodynamic simulations can give rise to both analytical and numerical instabilities. The classification of spacetime regions into causal and stable ("good"), acausal but stable ("bad"), and acausal and unstable ("ugly") is reviewed. We compare the predictions of the MUSIC hydrodynamic solver with an analytical solution, and demonstrate how the acausality-driven instabilities develop in a simple one-dimensional scenario.

Figures (3)

• Figure 1: Spacetime diagram illustrating the acoustic geometry of a fluid element in a "good" (left panel), "bad" (middle panel), and "ugly" (right panel) state. In the good state, the acoustic cone is contained within the lightcone ($0\leq w\leq 1$). In a bad state, the acoustic cone exits the lightcone ($w>1$), but still points to the future relative to the observer who is solving the equation ($vw<1$). In the ugly state, a portion of the acoustic cone points to the past ($vw\geq 1$), or there is no acoustic cone ($w^2<0$).
• Figure 2: Separation of fluid cells into good (green), bad (yellow), and ugly (red) in a simulation with only bulk viscosity (left panel) and both bulk and shear viscosities (right panel). The energy-momentum tensor is initialized using the IP-Glasma model in both cases.
• Figure 3: (Left panel) Numerical solution of a homogeneous, Israel-Stewart fluid with only bulk viscosity, with good (green), bad (yellow), and ugly (red) initial conditions, respectively, contrasted with the exact solution \ref{['solviamo']} (black). (Right panel) Ugly solution with zero-time-gradient initial conditions GavassinoSimulations2025bsn.