Causes of Sound Faster than Light in Classical Models of Ultradense Matter

A variety of classical theories of ultradense matter give phase and group velocities ($c_s$) for long-wavelength sound which exceed the speed of light in vacuum. This sound speed, which depends only upon the volume dependence of the total relativistic energy density, can reflect a lack of causality (signals propagate faster than $c$), or a possibility of amplification for some higher-frequency sound waves (the medium is not truly in its lowest energy state) which then cancel low-frequency effects outside the light cone. The significance of $c_s>c$ then depends upon the dynamical behavior of the system for higher frequencies. From the exact solution for a sound wave of arbitrary frequency propagating in a one-dimensional lattice of point sources of a neutral vector-meson field, it is shown that $c_s>c$ results from a breakdown of the analog of the Kramers-Kronig relation whenever the computed self-energy of a source exceeds its renormalized mass. The negative bare mass of source particles contributes the possibility of exponentially growing particle accelerations which can lower the energy of the system indefinitely. The response of the lattice can remain causal despite $c_s>c$ when such runaway modes are included in Green's functions. When they are suppressed, noncausality accompanies $c_s>c$. Classical nonlinear field theories which give $c_s>c$ are shown to be noncausal.