Lorentzian geometry of relaxation

We show that relativistic theories with purely relaxational excitation spectra, such as kinetic theory and transient hydrodynamics, naturally endow the dispersion plane $\{iω,ik\}$ with a Lorentzian geometric structure analogous to that of the Minkowski plane $\{t,x\}$. In this picture, timelike future-directed, timelike past-directed, and spacelike directions correspond respectively to relaxation-like, unstable-like, and evanescent-like modes. Under mild structural assumptions on the underlying theory, causality constrains dispersion relations to follow spacelike trajectories on the plane. This geometric viewpoint recasts longstanding problems in relativistic matter physics as elementary geometric ones that can often be solved graphically. As applications, we derive universal constraints on dispersion relations, deviations from time dilation, the observer dependence of spectral hierarchies, the regime of validity of hydrodynamics in boosted frame, the maximal allowed diffusivity and viscosity of relativistic media, and the presence of non-hydrodynamic branch cuts in kinetic theory.