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

Abstract: The analysis of linearized perturbations in relativistic field theories involving derivative corrections frequently leads to pathologies that, although generic, may not always be manifest, like in a local rest frame scenario. Hence, analyzing such theories conclusively in a more general state of equilibrium, like in a Lorentz boosted inertial frame, is imperative, albeit extracting the boosted modes could be quite nontrivial. Motivated by this problem, in this study, we develop a general framework for deriving the dispersion spectra in Lorentz-boosted inertial frames solely from the information of the local rest frame dispersion coefficients. Apart from the well-behaved modes, which are observed to follow an exact mapping across the boost, we show that in some cases, unphysical ``spurious" modes might turn up, which diverge at zero boost limit and thus lack any rest-frame analogue. Finally, we investigate the conflict between the existence of these spurious modes and the causality for a given relativistic theory. The key developments provided here are (i) a convenient way for obtaining the dispersion spectra across inertial frames, bypassing the traditional method of solving the boosted polynomial, and (ii) exploring the direct connection between the mode conservation and the causality of a theory with detailed proof. Due to the general approach, these results can find direct applications for a range of other effective theory formulations, like gauge theory plasmas, strongly coupled systems, and heavy quark effective theory.

• The paper introduces a new parametric framework that maps local rest frame dispersion spectra to arbitrary boosted frames using Lorentz transformations.
• It demonstrates that spurious modes emerge from non-injective mappings, serving as precise indicators of causality violations in effective field theories.
• Applications are highlighted in relativistic hydrodynamics and kinetic theory, providing actionable insights for numerical stability and theory refinement.

Relativistic Dispersion Spectra across Lorentz Boosted Frames: Framework, Spurious Modes, and Causality

Introduction and Motivation

The analysis of linearized perturbations in relativistic field theories is tightly connected to constraints of stability and causality, which are foundational for the admissibility of effective theories. This work presents a comprehensive and general framework for relating the dispersion spectra of perturbations in arbitrary inertial (Lorentz-boosted) frames to their local rest frame (LRF) properties. The ramifications are profound for hydrodynamics, kinetic theory, and broader classes of effective theories, especially those with higher-derivative corrections where direct boosted analysis is nontrivial. The study introduces an efficient mapping of dispersion spectra, clarifies the emergence of unphysical (spurious) modes under Lorentz transformations, and rigorously explores their connection to the breakdown of causality.

General Framework: Mapping Dispersion Spectra across Frames

The paper constructs a systematic parametric approach for deriving the entire spectrum of modes in a boosted frame using only the known LRF dispersion relations, thereby removing the need to solve the boosted dispersion polynomial directly. The primary result is that every LRF mode $W^i(\tilde k)$ can be mapped into the boosted frame via a specific parametric relation involving the Lorentz transformation of frequency and momentum:

$$\omega = \gamma \left[ W^i(p) - v p \right], \quad k = \gamma \left[ p - v W^i(p) \right],$$

where $p$ is a complex parameter. Modes in the boosted frame are generated by eliminating $p$ between these expressions, which produces all physically allowed modes, both hydrodynamic and non-hydrodynamic, as well as possible spurious solutions.

Crucially, for every LRF mode, the framework prescribes a unique mode in the boosted frame if and only if the mapping is one-to-one; many-to-one mappings signal the possibility of spurious modes.

Spurious Modes: Origin, Analytic Structure, and Diagnostics

Spurious modes are found to arise from the multi-valued nature of the Lorentz transformation in the complex momentum plane, specifically when the mapping from the rest frame to the boosted frame ceases to be injective. These modes do not have an LRF analogue and are characterized by divergence as the boost velocity $v\to 0$, i.e.:

$$\omega_{\text{spurious}}(k) \sim \frac{1}{v^\alpha}, \quad \alpha>0,\quad v\rightarrow 0.$$

This divergence is incompatible with any physically admissible rest-frame mode and signals a breakdown of the effective field theory description.

The emergence of spurious modes precisely tracks elevated polynomial degree in $\omega$ upon boost: for rest frame polynomials of degree $M$, the boosted polynomial $P_v(\omega,k,v)$ can have degree $M_v > M$. The surplus $$\omega = \gamma \left[ W^i(p) - v p \right], \quad k = \gamma \left[ p - v W^i(p) \right],$$0 solutions are spurious and physically unconnected to LRF excitations. The analysis establishes that these unphysical roots are both necessary and sufficient signals of acausality.

Causality Constraints and Theoretical Consistency

The work offers a rigorous connection between the absence of spurious modes and the satisfaction of causality—as formalized in (Heller et al., 2022) and (Hoult et al., 2023)—in both small-$$\omega = \gamma \left[ W^i(p) - v p \right], \quad k = \gamma \left[ p - v W^i(p) \right],$$1 and large-$$\omega = \gamma \left[ W^i(p) - v p \right], \quad k = \gamma \left[ p - v W^i(p) \right],$$2 limits. Specifically, LRF dispersion modes consistent with causality are free of poles or essential singularities at finite $$\omega = \gamma \left[ W^i(p) - v p \right], \quad k = \gamma \left[ p - v W^i(p) \right],$$3, and have at most linear large-$$\omega = \gamma \left[ W^i(p) - v p \right], \quad k = \gamma \left[ p - v W^i(p) \right],$$4 growth with subluminal or luminal front velocity. The appearance of spurious boosted roots is shown to be incompatible with these requirements, as their analytic structure necessarily involves either a non-analyticity (pole/essential singularity) in the small-$$\omega = \gamma \left[ W^i(p) - v p \right], \quad k = \gamma \left[ p - v W^i(p) \right],$$5 expansion or asymptotically superluminal growth at large $$\omega = \gamma \left[ W^i(p) - v p \right], \quad k = \gamma \left[ p - v W^i(p) \right],$$6.

The discussion is formalized by showing that, for a generic LRF mode with $$\omega = \gamma \left[ W^i(p) - v p \right], \quad k = \gamma \left[ p - v W^i(p) \right],$$7 as $$\omega = \gamma \left[ W^i(p) - v p \right], \quad k = \gamma \left[ p - v W^i(p) \right],$$8, spurious roots only occur if $$\omega = \gamma \left[ W^i(p) - v p \right], \quad k = \gamma \left[ p - v W^i(p) \right],$$9, directly violating causality bounds (Heller et al., 2022, Hoult et al., 2023).

Explicit Construction and Analytical Results

The framework results in general recursive algorithms for constructing boosted modes:

• Non-spurious modes are recovered uniquely by expanding both $p$0 and the Lorentz-transformed polynomial in powers of $p$1 and $p$2, allowing all boosted frame expansion coefficients to be written in terms of those from the LRF.
• For hydrodynamic and non-hydrodynamic branches, closed-form recursive formulae express boosted frame coefficients, displaying an explicit $p$3-suppression structure: higher-order corrections in the boosted frame become less relevant at high velocities, an effect with deep implications for the stability of highly boosted flows.
• Two archetype examples are analyzed: the Maxwell-Cattaneo (Müller-Israel-Stewart shear) mode—where the mapping produces only physical boosted roots, and relativistic Navier-Stokes—where the mapping generates an unphysical, divergent (spurious) root signaling the causal pathology of first-order dissipative hydrodynamics.

Practical and Theoretical Implications

Theoretical Implications

This methodology provides an operational, analytic test for causality directly at the level of dispersion relations, independent of direct equation-solving in the boosted frame. It formalizes and generalizes the diagnosis of causality breakdowns discussed in (Hoult et al., 2023), tightens the correspondence between mode conservation under boosts and causality, and paves the way for the systematic construction of higher-order and nonlinear theories that are compatible with Lorentz invariance and causal propagation.

Phenomenological and Computational Consequences

1. Numerical Hydrodynamics: The framework is directly applicable to numerical codes in ultra-relativistic flows (heavy-ion collisions, astrophysics), where spurious roots contaminate spectral analyses and cause unstable time evolution. The ability to diagnose and excise these modes using only LRF data is an immediate advance for simulation stability and fidelity.
2. Design of Causal Algorithms: Since $p$4-suppression naturally suppresses higher-order terms at high velocities, only leading contributions matter in near-luminal frames—informing the construction of stable and causal truncated theories relevant for far-from-equilibrium dynamics and large-gradient expansions.
3. Breakdown Diagnostics: The presence of spurious boosted roots can serve as a computable criterion pinpointing the breakdown of the hydrodynamic or effective theory regime, and signals when new higher-order terms or distinct degrees of freedom are required for consistency.
4. Broader Applicability: The mapping and causality diagnostic extend to kinetic theory, gauge theories, higher-derivative gravity, and condensed matter analogues where collective excitations in moving backgrounds play a dominant role.

Future Directions

Natural extensions include the classification of all effective theories conserving mode number under boosts, the application of the diagnostic to nonlinear and full nonlinearized theories (including quantum corrections), and the geometric interpretation of the many-to-one map singularities in the context of the analytic properties of effective actions.

Additionally, embedding these analytic criteria into live numerical simulations would enable automatic flagging and suppression of unphysical instabilities, improving the reliability of predictions in high-energy, condensed matter, and astrophysical contexts.

Conclusion

The paper provides a rigorous, systematic framework quantifying the transformation of dispersion spectra under Lorentz boosts in effective field theories. The parametric mapping eliminates the need for direct, often intractable, solution of boosted dispersion polynomials, and enables the full analytic and numerical characterization of both physical and spurious excitations. The presence of spurious boosted roots is conclusively shown to be a direct signature of causality violation, applicable across the breadth of effective theory realizations. The results are foundational for future developments in stable, causal relativistic hydrodynamics, kinetic theory, and beyond.

References:

• "Rigorous Bounds on Transport from Causality" (Heller et al., 2022)
• "Causality and classical dispersion relations" (Hoult et al., 2023)
• "Field redefinition and its impact in relativistic hydrodynamics" (Bhattacharyya et al., 2024)
• "Causality and stability in relativistic hydrodynamic theory - a choice to be endured" (Bhattacharyya et al., 2024)
• "Why do newer degrees of freedom appear in higher-order truncated hydrodynamic theory?" (Mitra, 2024)