Stability and causality criteria in linear mode analysis: Stability means causality

Causality and stability are fundamental requirements for the differential equations describing predictable relativistic many-body systems. In this work, we investigate the stability and causality criteria in linear mode analysis. We discuss the updated stability criterion in $3+1$-dimensional systems and introduce the improved sufficient criterion for causality. Our findings clearly demonstrate that stability implies causality in linear mode analysis. Furthermore, based on the theorems present in this work, we conclude that if updated stability criterion and improved causality criterion are fulfilled in one inertial frame of reference (IFR), they hold for all IFR.