Relativistic Dispersion Spectra across Lorentz boosted frames: Spurious modes and the enigma of causality

The analysis of excitation spectra in gradient-expanded relativistic fluid theories frequently leads to pathologies under Lorentz boosts. However, extracting the dispersion modes in a Lorentz boosted inertial frame can be nontrivial. Motivated by this problem, we develop a general framework for deriving the linearized dispersion spectra in Lorentz boosted frames using only information from the local rest-frame dispersion structure, particularly its mode-expansion coefficients. We observe that, under a Lorentz transformation from the local rest frame, additional boosted solutions may appear that conflict with the causality of the theory; we refer to these as "spurious modes." The key developments presented here are: (i) a convenient method for obtaining dispersion spectra across inertial frames, bypassing the traditional procedure of solving the boosted polynomial, and (ii) the establishment of a direct connection between mode conservation and the causality of a theory, supported by a detailed proof.