Causal Theories of Relativistic Hydrodynamics

• The topic defines a framework that incorporates extra dynamical fields or infinite gradient expansions to cure acausality and instability in high-energy fluid dynamics.
• It contrasts traditional Navier–Stokes formulations with modern causal approaches that redefine hydrodynamic frames to ensure proper signal propagation.
• The methodologies rely on mathematical criteria like Hurwitz stability and Lienard–Chipart to establish reliable models for astrophysical phenomena and heavy-ion collisions.

Causal theories of relativistic hydrodynamics are frameworks for describing dissipative fluid dynamics that avoid unphysical signal propagation (acausality) and pathologies such as instability under small perturbations, especially in strongly relativistic and out-of-equilibrium regimes. These theories address the inadequacies of the first-order (Navier–Stokes–type) formulations by systematically incorporating either additional dynamical degrees of freedom, higher-order gradient corrections, or carefully redefined fluid variables. The interplay between causality, stability, and the hydrodynamic frame lies at the mathematical and physical core of these modern approaches.

1. Breakdown of First-Order Relativistic Navier–Stokes Theory

The relativistic Navier–Stokes (NS) formulation describes dissipative currents (such as the shear stress $\pi^{\mu\nu}$ and particle diffusion current $\nu^\mu$) as functions of local spatial gradients: $$\nu^\mu = z\,\frac{3}{g\,\beta^2}\,\nabla^\mu \alpha,\qquad \pi^{\mu\nu} = \frac{48}{g\,\beta^3}\,\sigma^{\mu\nu}$$ ($z$, $g$, $\beta$ parameterize microphysics/interaction strength). Here, only space-like derivatives are present, leading to parabolic equations whose mode structure (e.g., for transverse perturbations, $\Omega = \frac{3}{4} i \bar{\eta} \kappa^2$) is purely diffusive. This causes two problems, independent of the matching conditions:

• Acausality: Signal propagation at arbitrarily high (actually infinite) speeds.
• Instability in Boosted Frames: When linear perturbations are analyzed in Lorentz-boosted backgrounds, exponential growth and unphysical modes appear.

Significance: These problems have historically necessitated the search for improved, causal theories of relativistic hydrodynamics (Brito et al., 2023, Hoult, 26 Sep 2025).

2. The Role of Hydrodynamic Frames

Hydrodynamic fields such as fluid velocity $u^\mu$ and temperature $T$ admit different non-equilibrium definitions known as hydrodynamic frames. The Landau and Eckart frames are canonical choices, tying the flow velocity to either energy or particle number flow, respectively. However:

• If the fluid variables are strictly defined via such matching conditions (e.g., in the Landau frame, $T^{\mu\nu} u_\nu = -\epsilon u^\mu$), and the theory includes only a finite number of gradient corrections, then acausality is unavoidable (Mitra, 22 Aug 2025, Bhattacharyya et al., 26 Jul 2024).
• Causal and stable theories with only fluid variables (such as BDNK-type theories) require the abandonment of the standard Landau/Eckart matching and the adoption of a generalized frame in which the variables are not tied directly to the stress-energy tensor (Brito et al., 2023, Bemfica et al., 2019).

Context: The frame dependence is not a mere technicality—it determines which variables are physical, what information is tracked non-perturbatively, and how stability/causality are implemented in both the equations and in numerical algorithms (Bhattacharyya et al., 26 Jul 2024).

3. Extended Degree of Freedom and Infinite Derivative Expansion

To cure acausality and achieve robust stability:

• Müller–Israel–Stewart (MIS) and Transient Theories: These introduce new dynamical fields (e.g., the independent shear-stress tensor $\pi^{\mu\nu}$) and postulate relaxation-type equations,

$$\tau_\pi D \pi^{\mu\nu} + \pi^{\mu\nu} = 2\eta\, \sigma^{\mu\nu} + \cdots$$

With $\tau_\pi$ the relaxation time, this approach “integrates in” the infinite series of time derivatives needed to restore causality, replacing second (or higher) order spatial derivatives by evolution equations for additional, physically motivated variables (Hoult, 26 Sep 2025, Tsumura et al., 2015).

• Infinite Gradient Expansion: Alternatively, working in a generalized frame, one can formally retain only fluid variables, but at the cost of an infinite sum over gradient corrections:

$$\pi^{\mu\nu} = -2\eta \sum_{n=0}^\infty [-\tau_\pi D]^n \sigma^{\mu\nu}$$

The closed form, $$\pi^{\mu\nu} = -2\eta (1+\tau_\pi D)^{-1}\sigma^{\mu\nu}$$, represents an all-orders resummation (Mitra, 22 Aug 2025). Any finite truncation (no matter how high the order) leads to acausal propagation.

Key Table: Relation between approach, variables, and the treatment of derivatives: | Approach | Variables | Derivatives | |----------------------------|-------------------------------|-----------------------| | Landau/Eckart frame + finite gradients | $T$, $u^\mu$ | Finite order | | General frame (BDNK) | $T$, $u^\mu$ (frame redefined) | All-order (formally summed) | | Transient (MIS/extended) | $T$, $u^\mu$, $\pi^{\mu\nu}$ | Finite order for each variable |

Implication: One cannot simultaneously have a theory (i) defined solely in terms of $T$, $u^\mu$, (ii) in Landau/Eckart frame, and (iii) with a finite number of derivative corrections, without encountering acausality. Either infinite nonlocal corrections (all-order gradients) or new dynamical fields (“integrated in” in the sense of effective field theory) are essential (Mitra, 22 Aug 2025, Bhattacharyya et al., 26 Jul 2024).

4. Causality and Stability Constraints: Mathematical Tools

Dispersion Relations and Stability: The linear stability of a relativistic hydrodynamic theory is encoded in the dispersion relations for small perturbations. These are typically polynomials in $i\omega$: $$f(\Upsilon) = \sum_{j=0}^{N} a_j\,\Upsilon^j, \qquad \Upsilon \equiv i\,\omega.$$ The requirement of Hurwitz stability (all roots $\text{Re}(\omega)<0$) ensures damping of all modes. The Lienard–Chipart criterion provides an efficient algorithm: all coefficients $a_j > 0$ and a specified subset of principal minors must be positive (Brito et al., 2023). The imposition of such algebraic constraints on the characteristic polynomials derived from the linearized equations provides design rules for permitted transport parameters and matching conditions.

Causality (Characteristic Velocities): In the large-$k$ limit, the group velocities obtained from the dispersion relations,

$$\lim_{k \to \infty} \left| \frac{\partial\,\mathrm{Re}\,\omega}{\partial k}\right| \le 1,$$

must not exceed the speed of light. These requirements exclude entire classes of matching parameters (e.g., the extreme Eckart choice $z=0$ in BDNK) and frame choices that otherwise lead to pathologies (Brito et al., 2023, Bemfica et al., 2019).

5. Microscopic and Effective Theories

Fundamental derivations emphasize the kinetic origins and the organizational structure for the effective macroscopic theories.

• Kinetic Theory Approaches: Chapman–Enskog iterations and renormalization-group methods produce the causal equations, clarify the origin of new dynamical degrees of freedom, and furnish microscopic expressions for transport coefficients and relaxation times (Tsumura et al., 2015, Panday et al., 9 Apr 2024).
• Quantum Effective Field Theory (SK/MSR): Consistent thermodynamic fluctuations and stochastic hydrodynamics can be constructed via the Schwinger–Keldysh or Martin–Siggia–Rose formalisms, but only if the underlying macroscopic theory is both causal and stable. In some BDNK-like first-order settings, such stochastic extensions can be pathological unless second-order corrections (or auxiliary fields) are properly included (Jain et al., 2023, Mullins et al., 2023).

6. Broader Impact and Applications

The rigorous formulation of causal, stable relativistic hydrodynamics underpins the reliability of numerical simulations and the physical interpretation of high-energy astrophysical phenomena, heavy-ion collisions, and compact objects. The systematics revealed by recent work provide:

• Predictive parameter space: Allowed windows for transport coefficients and matching parameters, guided by algebraic stability and causality criteria.
• Framework for extensions: Generalization to chiral fluids with anomalies (Abboud et al., 2023), charged plasmas (relativistic MHD) (Hoult, 26 Sep 2025), superfluid systems, and even anisotropic and magnetized plasmas (Bemfica et al., 2023).
• Unified picture: The connection between hydrodynamic frames, degrees of freedom, and all-order derivative expansions, as well as the equivalence (at the effective theory level) between “integrating in” extra variables and “integrating out” infinite gradient corrections.

7. Future Directions and Outstanding Challenges

Research continues into both the formal structure and phenomenological impact of these theories:

• Transseries and attractor solutions at large gradients and far-from-equilibrium regimes (Weickgenannt, 2022).
• Numerical algorithms for solving the resulting equations with stability, in realistic simulations of quark–gluon plasma or relativistic jets (Takamoto et al., 2011, Pandya et al., 2022).
• Rigorous demonstration of nonlinear causality conditions, well-posedness, and mathematical uniqueness in broader classes of systems (Bemfica et al., 2019, Hoult, 26 Sep 2025).
• Fluctuations and stochastic effects at both first and second order, and their role in capturing effective macroscopic noise consistent with microscopic physics (Jain et al., 2023, Kumar et al., 2013).

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In summary, the modern understanding of causal theories of relativistic hydrodynamics hinges on the recognition that causality and stability are fundamentally entwined with hydrodynamic frame choice, derivative expansion structure, and the possible emergence of new dynamical degrees of freedom. The precise, algebraic constraints emerging from linear stability and causality analyses (such as those based on the Lienard–Chipart and Routh–Hurwitz conditions) now guide the construction of physically viable dissipative fluid models, both at the microscopic and effective field theory levels. These frameworks form the basis for reliable description and simulation of high-energy, far-from-equilibrium fluid systems in modern physics (Mitra, 22 Aug 2025, Bhattacharyya et al., 26 Jul 2024, Brito et al., 2023, Hoult, 26 Sep 2025).