Causality criteria

By "causality of matter" one means its property not to admit superluminal excitations, i.e. excitations that propagate faster than the vacuum speed of light c. In discussing the propagation of small excitations, one has to distinguish between phase velocities ω_j/k, (1≦j≦g=number of dispersion branches), group velocities dω_j/dk, a front velocity v_f: = {Mathematical expression} and the propagation speed v_q:=(dp/dρ{variant})^1/2 of isotropic quasistatic (small) perturbations. We discuss some of their properties. In particular, the (maximal) speed v_s of small signals is not smaller than v_f, and equals v_f whenever the dispersion branches ω_j(k) behave reasonably at infinity of the complex k-plane. In essence stronger conditions guarantee v_q<v_f (in which case v_q≧c would imply superluminal behaviour).