Linear causality and stability constraints on relativistic second-order magnetohydrodynamics

The gist: This work constructs a theoretical framework for relativistic second-order magnetohydrodynamics based on entropy current analysis, investigating linear causality and stability constraints in an anisotropic system.

Ideal RMHD from entropy analysis

The foundational evolution laws are established by following the modern symmetry-based formulation of relativistic magnetohydrodynamics, which comprises the conservation of energy-momentum ∂µTµν = 0, particle number ∂µNµ = 0, and magnetic flux conservation ∂µF˜µν = 0 <ref:2606.00649#pg2> The theory focuses on a parity-even plasma, excluding P-odd/CP-odd fluctuations such as the chiral magnetic effect <ref:2606.00649#pg4> The thermodynamic consistency is anchored in the local thermodynamic structure governed by the entropy density s (ϵ, n, Bµ) via the first law of thermodynamics <ref:2606.00649#pg6> To resolve anisotropy, a projection tensor Ξµν is utilized to parameterize zeroth-order constitutive relations in terms of longitudinal pressure p∥ and transverse pressure p⊥ <ref:2606.00649#pg8> The ideal level demands that the entropy production vanishes, imposing algebraic constraints such as T s + µn + BµHµ =ϵ + p⊥ <ref:2606.00649#pg9>

First-Order Derivative Corrections of RMHD

Dissipative processes are accounted for by decomposing the energy-momentum tensor into ideal and first-order dissipative components, including anisotropic bulk viscous pressures Π⊥ and Π∥ The macroscopic evolution of these dissipative degrees of freedom is governed by modified conservation laws derived from the divergence of the non-equilibrium entropy current The construction of linear constitutive laws is constrained by Curie’s Principle, which dictates that dissipative fluxes only couple to thermodynamic forces of the same tensorial rank and parity The resulting stability conditions require the associated transport matrices to be positive semi-definite, leading to constraints like ζ⊥ ≥ 0, ζ∥ ≥ 0, and ζ⊥ζ∥−1/4ζ× + ζ’×2 ≥ 0

Second-Order Derivative Corrections of RMHD

The inherent pathologies of relativistic Navier-Stokes (NS) theory necessitate a transition to a second-order transient framework, adopting the Israel–Stewart (IS) approach to restore hyperbolicity by promoting dissipative fluxes to dynamical variables The IS entropy current SµIS is constructed by including quadratic terms in the dissipative fluxes Qµ, constrained by the requirement that entropy must be maximized at equilibrium, leading to the stability condition uµQµ ≤ 0 The final evolution equations for the dissipative fluxes are obtained by enforcing ∂µSµIS ≥ 0, resulting in magnetized analogues of Israel-Stewart equations characterized by additional anisotropic channels

Linear mode analysis

The analysis is conducted in a background rest frame with a constant magnetic field along the z-axis, decomposing the system into magneto-sonic, Alfvén, and charge diffusion sectors using a secular matrix equation M19×19 = 0 The small-k expansion reveals leading order solutions corresponding to hydrodynamic manifolds and gapped modes associated with relaxation times The large-k expansion probes the short-wavelength asymptotics, where propagating branches scale linearly with k, and non-hydrodynamic relaxation branches remain finite in the short-wavelength limit

Exact Solutions Analysis

Numerical solutions are benchmarked against analytical approximations obtained from both small-k and large-k expansions to quantify the domains of validity The comparison confirms a two-scale structure where the small-k expansion captures the infrared hydrodynamic behavior, while the large-k expansion captures short-wavelength propagation/relaxation structure relevant to causality

Angle analysis

The angular dependence of the spectrum is investigated numerically by comparing exact roots with asymptotic solutions from small-k and large-k expansions For the magneto-sonic and Alfvén sectors, the small-k expansion deteriorates rapidly as θ approaches π/2, whereas the large-k expansion remains robust over the entire angular interval The charge-diffusion sector shows its mismatch near θ = 0 in the large-k regime

Causality analysis based on numerical spectrum

Causality is diagnosed by requiring the asymptotic signal (group) velocity at short wavelengths to not exceed the speed of light, vg ≤ 1 The study shows that causality constraints are intrinsically sector-dependent: the magneto-sonic sector is controlled by coupled effects of anisotropic bulk, shear, and relaxation channels The Alfvén sector exhibits smoother but still correlated constraints, and the charge-diffusion sector reduces to a telegraph-type condition that the asymptotic propagation speed remains subluminal

Conclusion

The restoration of causality in second-order RMHD is governed by the interplay between anisotropic transport coefficients and corresponding relaxation times in each sector The results indicate that the magneto-sonic sector is affected by several coupled dissipative channels, the Alfvén sector exhibits more regular but still correlated constraints, and the charge-diffusion sector reduces to a telegraph-type requirement that the charge front velocity remain subluminal Overall, this study provides a systematic entropy-current construction of secondorder RMHD together with a mode-resolved analysis of its linear causal structure

Acknowledgments

Duan She’s research is funded by the Startup Research Fund of Henan Academy of Sciences (No. 231820058), the High-level Achievements Reward and Cultivation Projects (No. 20252320001), the Basic Research Fund of Henan Academy of Sciences (No.20260620005), and the Key Laboratory of Quark and Lepton Physics Contracts No. QLPL2025P01

A Covariant tensor basis for magnetized RMHD

The covariant basis fixes the time-like fluid velocity uµ and a unit space-like vector bµ specifying the local magnetic-field direction The magnetic induction four-vector is written as Bµ = Bbµ, with B ≡ p−BµBµ The two projectors used throughout the paper are ∆µν = gµν − uµuν, Ξµν = gµν − uµu ν + b µb ν

Matching and frame choices in magnetized hydrodynamics

The matching prescription ensures that nonequilibrium corrections are not counted twice as shifts of the thermodynamic variables The frame choice in the body of the paper is chosen because it keeps energy flow and transverse magnetic field correction absent, while retaining the particle diffusion and induction channels needed for the linear spectrum analyzed in the main text

Explicit form of entropy in second-order

The explicit form of Qµ is given by a sum involving scalars A, B, vectors Cλ, Dλ, and tensors Fλν and Gλν The tensor structures are defined explicitly in Appendix C and Appendix D

D Explicit form of matrixes

The explicit forms of the matrices A10×10, B6×6 and C3×3 are listed in Appendix D The coupling terms MAC, MBC, MCA and MCB are defined by specific elements in the matrix structure

E Large-k expansion solution

The analysis of the large-k expansion solutions involves solving secular equations for various coefficients aA, bA, cA and dA Necessary conditions for stability, such as Im ωA−1 = 0 and Im ω′A0 < 0, are derived from the Routh–Hurwitz stability criterion The explicit forms of these coefficients are detailed in Appendix E

References

[1] A. Bzdak and V. Skokov, Event-by-event fluctuations of magnetic and electric fields in heavy ion collisions, Physics Letters B 710 (2012) 171–174

[2] W.-T. Deng and X.-G. Huang, Event-by-event generation of electromagnetic fields in heavy ion collisions, Physical Review C 85 (2012) 044907

[3] K.

1. A relativistic second-order magnetohydrodynamics (RMHD) solver can be constructed that ensures hyperbolic evolution by incorporating the relaxation dynamics of dissipative fluxes, as described in Section 2, ensuring the non-negativity of entropy production to derive a self-consistent set of relaxation equations for the dissipative fluxes.

2. The AI system can perform linear mode analysis to decompose the excitation spectrum into magnetosonic, Alfvén, and charge-diffusion sectors and derive asymptotic dispersion relations in both the long-wavelength (small-k) and short-wavelength (large-k) regimes, validating them against exact numerical roots.

3. The system can dynamically map the admissible causal domain by evaluating the large-k behavior of the propagating branches alongside the damping properties of non-hydrodynamic modes, thereby delineating constraints governed by the interplay between anisotropic transport coefficients and relaxation times.

4. The AI can diagnose causality and stability directly from numerical dispersion relations, testing for subluminal propagation via the requirement that the asymptotic signal (group) velocity at short wavelengths does not exceed the speed of light in Eq. (7.1).

5. The system can generate a causal constraint map by extracting the maximal asymptotic speed vmax and plotting it against dissipative ratios, identifying regions where the theory is causal versus acausal, as shown in Figs. 8–10.

6. The AI can perform angular dependence analysis to assess formula validity: for magneto-sonic and Alfvén sectors, it can identify where the small-k expansion becomes less accurate as θ approaches π/2, while for the charge-diffusion sector, it identifies where the large-k approximation is mainly affected near θ → 0.

7. The system can verify stability using Routh–Hurwitz criteria by checking necessary conditions such as "Im b'A 0, e'A > 0, which ensures that all relaxation poles and non-hydrodynamic branches must remain in the lower-half complex ω-plane."