Causality determination in the region between necessary and sufficient conditions

Determine whether solutions of the Baier–Romatschke–Son–Starinets–Stephanov (BRSSS) one-dimensional boost-invariant hydrodynamic equations are causal when their trajectories satisfy all necessary but not all sufficient nonlinear causality conditions, i.e., establish causality or acausality of solutions in the intermediate region where necessary conditions hold but some sufficient conditions fail.

Background

The paper derives necessary and sufficient inequalities for nonlinear causality of one-dimensional expanding viscous fluids governed by BRSSS. Trajectories in phase space can lie in regions where all necessary but not all sufficient conditions are met, leaving the causality status undecided by these criteria alone.

This ambiguity matters for practical modeling: some initial conditions may evolve through this intermediate region, and without a definitive criterion one cannot certify whether the evolution respects relativistic causality.

References

With only these necessary and sufficient conditions of nonlinear causality, it is not yet determined in between whether it is causal or not: All necessary conditions are satisfied, but some of the sufficient conditions are not satisfied.

— Constraint on initial conditions of one-dimensional expanding fluids from nonlinear causality (2412.02405 - Hoshino et al., 3 Dec 2024) in Section 3.2 (Constraint on the inverse Reynolds number)