Causality and Stability of First-Order Relativistic Spin Hydrodynamics with Conserved Charges

Technical Summary: Causality and Stability of First-Order Relativistic Spin Hydrodynamics with Conserved Charges

Problem Statement
Relativistic spin hydrodynamics has emerged as a necessary extension of conventional hydrodynamics to describe the spin polarization of particles in the quark-gluon plasma (QGP) created in non-central heavy-ion collisions. While recent studies have analyzed the causality and stability of first-order spin hydrodynamic theories, these analyses have almost exclusively focused on charge-neutral systems ($n=0$). However, in scenarios involving finite baryon density—such as forward rapidities, the RHIC Beam Energy Scan, and future facilities like FAIR and NICA—conserved-charge dynamics are critical. It remains unclear whether the inclusion of particle-number conservation merely adds hydrodynamic degrees of freedom or fundamentally alters the pathological features (acausality and instability) inherent in first-order spin hydrodynamics. This paper addresses the gap in understanding how conserved-charge dynamics modifies the mode spectrum, causality, and stability of first-order relativistic spin hydrodynamics.

Methodology
The authors investigate the linear stability and causality of first-order relativistic spin hydrodynamics within the canonical formulation, adopting the counting scheme where the spin chemical potential $\omega_{\mu\nu}$ is treated as a first-order gradient correction ($O(\partial)$). The analysis is performed in the Landau frame.

1. Theoretical Framework: The study begins with the conservation equations for particle number ($N^\mu$), energy-momentum ($T^{\mu\nu}$), and total angular momentum ($J^{\lambda\mu\nu}$). Constitutive relations are derived using entropy current analysis, incorporating particle diffusion, shear and bulk viscosity, and spin-orbit/spin-vorticity couplings.
2. Linear Perturbation Analysis: Small perturbations are introduced around a homogeneous, static, irrotational equilibrium state. The perturbations include fluctuations in energy density ($\delta e$), particle number density ($\delta n$), fluid velocity ($\delta u^i$), spin density components ($\delta S^{0i}, \delta S^{ij}$), and the diffusion current.
3. Dispersion Relations: The linearized equations of motion are cast into a matrix form. By solving the characteristic equation ($\det M = 0$), the authors derive the complete dispersion relations $\omega(k)$ for all perturbation modes.
4. Asymptotic Analysis: The stability and causality conditions are evaluated in two limits:
   • Long-wavelength limit ($k \to 0$): To determine the stability of relaxation modes and the damping of propagating modes.
   • Short-wavelength limit ($k \to \infty$): To test the causality condition, specifically ensuring that the phase velocity and group velocity do not exceed the speed of light and that the frequency-to-wavenumber ratio remains bounded.

Key Contributions and Results

• Modification of the Sound Sector: The inclusion of particle-number conservation introduces a new hydrodynamic degree of freedom (particle number density) and couples charge fluctuations to energy and momentum. This alters the acoustic mode. The sound velocity is no longer determined solely by the conventional thermodynamic sound speed ($c_s$) but also depends on the particle-number response coefficient ($c_n$) and background density. Furthermore, the damping rate of the sound mode acquires additional diffusive contributions involving the particle diffusion coefficient ($\kappa$) and thermodynamic susceptibilities ($\chi_n, \chi_e$).
• Spin Relaxation Robustness: The analysis reveals that the non-propagating spin relaxation modes (associated with the coupling between the spin tensor and the antisymmetric part of the energy-momentum tensor) retain the same structure as in the charge-neutral case. The particle-number degree of freedom does not qualitatively change the intrinsic spin relaxation mechanism.
• Emergence of New Non-Hydrodynamic Modes: The study identifies new modes arising exclusively from particle-number conservation. Specifically, a mode derived from the block matrix $M_1$ (Eq. 55) exhibits behavior where the frequency grows without bound in the short-wavelength limit ($|\omega/k| \to \infty$).
• Causality Violation: The newly identified particle-number-induced mode violates the causality condition in the short-wavelength limit. This demonstrates that the inclusion of particle-number conservation can itself be a source of acausal behavior in first-order theories.
• Persistence of Instability: The study confirms that the instability inherent in first-order spin hydrodynamics (specifically the condition $\lambda' < 0$ required for stability in the long-wavelength limit, which contradicts the entropy positivity condition $\lambda > 0$) is not resolved by including conserved charges. The fundamental pathology of the first-order theory persists.

Significance and Claims
The paper claims that these results reveal a nontrivial interplay between spin and conserved-charge dynamics. The primary significance lies in demonstrating that:

1. Conserved-charge dynamics significantly modifies the propagating sector and introduces new stability constraints involving diffusion and thermodynamic susceptibilities.
2. Particle-number conservation does not cure the instability of first-order spin hydrodynamics; rather, it introduces new non-hydrodynamic modes that can violate causality.
3. A physically consistent description of relativistic spin fluids at finite density likely requires extensions beyond the first-order framework (analogous to the Müller-Israel-Stewart theory in conventional hydrodynamics) to ensure both causality and stability.

The authors conclude that their work provides a necessary constraint on relativistic spin hydrodynamic theories at finite density and sets the stage for future investigations into second-order formulations and alternative pseudogauge choices.