Must ultrabaric matter be superluminal?

We examine the question of whether or not special relativity requires that the pressure must be less than the energy density of matter. To do this, we study a model of matter consisting of a classical one-dimensional lattice of point particles interacting via a potential satisfying the three-dimensional Klein-Gordon equation. Despite the fact that for this model the pressure $p$ can exceed the energy density $\rho c^2$, giving rise to an adiabatic sound speed $c_s={\left(\frac{\mathrm{dp}}{d\rho }\right)}^{\frac{1}{2}}>c$, and in the low-frequency limit to a group velocity $\frac{d\omega }{\mathrm{dk}}>c$ and phase velocity $\frac{\omega }{k}>c$, for this type of lattice model, the formally calculated speed $c_s$ is not a signal speed and we find that the true signal propagation speed $v_{\mathrm{signal}}<c$. Thus special relativity alone does not guarantee that $p<\mathrm{}\rho c^2$. We briefly discuss other constraints on $p\left(\rho \right)$, none of which seem sufficiently rigorous to rule out the possibility that $p>\mathrm{}\rho c^2$ at high densities. The significance of the present result for the upper mass limit of neutron stars and the existence of black holes is also considered.