Disturbing moving fluids

TL;DR

The paper addresses how relativistic boosts affect the excitation spectra of moving fluids under a broad Onsager-like symmetry, showing that many spectral features become Lorentz invariant when described in a covariant first-order framework ${\mathbb{E}}^\mu \partial_\mu \Psi = -{\sigma \!\!\sigma} \Psi$. By establishing local conservation laws, analyzing the nonhydrodynamic spectrum at zero wavenumber, and bounding the collision operator with variational inequalities, the authors derive a rigorous bound for boosted spectra: if the rest-frame spectrum lies in $a \le i\omega \le b$, then in a frame moving at speed $v$ the boosted spectrum satisfies $\dfrac{a(1-v)}{\gamma} \le i\omega' \le \dfrac{b}{\gamma(1-v)}$ (with refinements incorporating signaling speed $w$). These results imply persistence of gaps and convergence radii for hydrodynamics across inertial frames, and they reveal that time-dilation intuition can fail in systems with finite signal speeds, while reducing to the familiar nonrelativistic limit when $w\ll1$. The framework applies to kinetic theory, transient hydrodynamics, and related models, and it clarifies when quasi-hydrodynamic descriptions remain well-defined under boosts. The supplementary material extends the analysis to covariant derivations and broader kinetic settings, highlighting the generality and limitations of the symmetry-based bounds.

Abstract

In Newtonian physics, the excitation spectrum of a fluid is the same in all reference frames, up to a trivial shift. In special relativity, this is no longer the case. Relativity of simultaneity causes different inertial observers to measure markedly different excitation spectra, with stability being the only property known to be Lorentz invariant in all causal theories. Here, we show that, under a certain Onsager-like symmetry principle (which applies to kinetic theory and transient hydrodynamics), it is possible to place rigorous bounds on phase velocities, eigenmode convergence radii, spectral gaps, and equilibration rates in any inertial frame, using only information about the rest frame spectrum at zero wavenumber. The conventional intuition coming from time dilation is also shown to lead to generically wrong predictions, but becomes accurate if the fluid is non-relativistic in the rest frame.

Figures (5)

• Figure 1: Splitting of a single non-hydrodynamic frequency $i\omega\,{=}\,1$ into multiple frequencies $i\omega'$ induced by a boost of speed $v$. The black curves show the bounds \ref{['theboundoni']}, with $a=b=1$, and they delimit the allowed region. The dashed curves refer to Cattaneo's eigenfrequency $i\omega'_1$, see equation \ref{['cattaniamo']}, with $w=0$ (blue), $0.9$ (magenta) and $1$ (red); the blue curve also captures the modes $i\omega'_{2,3}$. The yellow region covers the continuum of modes \ref{['RTAmo']} of RTA.
• Figure 2: Bounds \ref{['theboundoni2']} on the nonhydrodynamic spectrum of a moving fluid, assuming a single rest-frame nonhydrodynamic frequency $i\omega = 1$, and rest-frame signaling speed $w=0$ (blue), $0.5$ (magenta), $0.97$ (red), and $1$ (black dashed). For $w=0$, the admissible region collapses to the time--dilation line $i\omega' = 1/\gamma$, while for $w=1$ the result reduces to \ref{['theboundoni']}, which is the only case in which $i\omega'$ may diverge as $v \to 1$.
• Figure S1: Nonhydrodynamic modes at zero wavenumber for a medium moving along the $x$ direction with velocity $v$. The blue curves show the individual relaxation rates $i\omega'$ as functions of $v$, while the dashed curves delimit the region permitted by Theorem 2. Each panel corresponds to a distinct randomly generated model with $D\,{=}\,5$, $N\,{=}\,1$, and the indicated value of $w$.
• Figure S2: Nonhydrodynamic modes at zero wavenumber for a medium moving along the $x$ direction with velocity $v$. The blue curves display the individual relaxation rates $i\omega'$ as functions of $v$, while the dashed curves indicate the region allowed by Theorem 2. Each panel corresponds to a distinct randomly generated model of the form \ref{['isotropizziamo']}. In the upper panel, the coefficients $\sigma_n$ are chosen as completely independent random variables. In the lower panel, the $\sigma_n$ are random variables subject to the constraints $\sigma_1=\sigma_3$ and $\sigma_2=\sigma_4$.
• Figure S3: Non-hydrodynamic spectrum of model \ref{['quasuzzo']}. A large population of modes has minimal relaxation rate $1$, and maximal propagation speed $1/2$. At finite boosts, these modes populate the yellow region, bounded from below by the curve $(1-|v|/2)/\gamma$ (dashed). In addition, there are two fully decoupled quasi-hydrodynamic modes, with rest-frame relaxation rates $i\omega_{\pm}=\lambda\ll 1$, and which propagate exactly at the speed of light. At sufficiently high boosts, one of these modes experiences an arbitrarily large blueshift, and thus decays faster than the rest of the spectrum.

Theorems & Definitions (8)

• Theorem 1
• proof
• Corollary 1
• Corollary 2
• Corollary 3
• Theorem 2
• proof
• Definition 1