Dispersion Relations Alone Cannot Guarantee Causality

We show that linear superpositions of plane waves involving a single-valued, covariantly stable dispersion relation $\omega (k)$ always propagate outside the light cone unless $\omega (k)=a+bk$. This implies that there is no notion of causality for individual dispersion relations since no mathematical condition on the function $\omega (k)$ (such as the front velocity or the asymptotic group velocity conditions) can serve as a sufficient condition for subluminal propagation in dispersive media. Instead, causality can only emerge from a careful cancellation that occurs when one superimposes all the excitation branches of a physical model. This happens automatically in local theories of matter that are covariantly stable. Hence, we find that the need for nonhydrodynamic modes in relativistic fluid mechanics is analogous to the need for antiparticles in relativistic quantum mechanics.

Figure 1

The principle of causality. If a perturbation has initial support inside a region of space $\mathcal{R}$ (blue segment), then it cannot propagate outside the set $J^{+}(\mathcal{R})$ called the “causal future of $\mathcal{R}$” [6], or “future light cone of $\mathcal{R}$” [5] (red region).

Figure 2

Path of integration for the proof of Theorem 1.