Solitons and singularities in relativistic ultrastiff fluids

A mathematical duality exists between massless scalar fields and relativistic fluids governed by an ultrastiff equation of state, in which the pressure equals the mass-energy density, and the sound speed equals $c$. This duality entails that certain solutions of the wave equation (a linear theory) can be mapped to solutions of ideal relativistic hydrodynamics (an inherently nonlinear theory). Leveraging this correspondence, we explore some interesting properties of ultrastiff fluids. Specifically, we demonstrate that all nonlinear sound waves in such media are solitons. Moreover, we prove that, in 3+1 dimensions, compactly supported configurations of an ultrastiff fluid inevitably evolve, in finite time, toward a singular state where the fluid's flow velocity exits the future lightcone. We also demonstrate that, prior to the formation of this singularity, first-order perturbation theory in the small parameter $c_s^{-2}-c^{-2}$ produces divergent results. As a consequence, a fluid whose speed of sound is, say, $0.995 c$ exhibits markedly different behavior near the singularity compared to a fluid with a speed of sound exactly equal to $c$.