BDNK Formulation in Relativistic Hydrodynamics

• BDNK formulation is a first-order relativistic hydrodynamics approach that retains inertial (time-derivative) effects to ensure causality and stability.
• It employs a controlled hydrodynamic frame and specific transport coefficients to overcome the acausality and instability issues of earlier models like Eckart and Landau–Lifshitz.
• The framework bridges kinetic theory and macroscopic simulations, enabling precise modeling of heavy-ion collisions, neutron star dynamics, and related phenomena.

The BDNK (Bemfica–Disconzi–Noronha–Kovtun) formulation represents a modern, mathematically consistent approach to relativistic hydrodynamics that incorporates first-order viscous corrections. It distinguishes itself by constructing causal, stable, and strongly hyperbolic dynamical equations through a careful choice of constitutive relations and hydrodynamic frame, overcoming longstanding pathologies of earlier first-order dissipative models such as those due to Eckart and Landau–Lifshitz. BDNK theory has proven particularly significant for both theoretical understanding and numerical modeling of relativistic viscous flows in contexts ranging from heavy-ion collisions to neutron star dynamics.

1. Theoretical Foundations and Motivation

Early first-order relativistic dissipative hydrodynamics, exemplified by the formulations of Eckart (1940) and Landau–Lifshitz, were plagued by fundamental physical failures: acausality (the possibility of superluminal signal propagation) and linear instabilities about equilibrium. These issues arise from the structure of the constitutive relations, particularly the replacement of time derivatives by spatial gradients in the Chapman–Enskog expansion, which eliminates memory (inertial) effects and leads to parabolic dissipation equations with infinite propagation speed.

In contrast, the BDNK formulation retains time-derivative (inertial) terms explicitly in the constitutive relations, as originally articulated in “Nonlinear Causality of General First-Order Relativistic Viscous Hydrodynamics” [Phys. Rev. D 100, 104020 (2019)]. By reframing the hydrodynamic variables and introducing an appropriate “hydrodynamic frame,” BDNK achieves a first-order theory that is nonlinearly causal, strongly hyperbolic, and locally well-posed both at the level of principal part analysis and under nonlinear evolution (Pandya et al., 2021, Pandya et al., 2022).

2. Constitutive Structure, Frame Freedom, and Causality

The BDNK energy-momentum tensor for a relativistic viscous fluid can be written as

$$T^{\mu\nu} = (\epsilon + \mathcal{A})\, u^\mu u^\nu + (P + \Pi)\, \Delta^{\mu\nu} + \mathcal{Q}^\mu u^\nu + u^\mu \mathcal{Q}^\nu - 2\eta\, \sigma^{\mu\nu}$$

where

• $u^\mu$ is the fluid four-velocity ($u^\mu u_\mu = -1$)
• $\epsilon$ is the energy density, $P$ is the pressure
• $\Delta^{\mu\nu} = g^{\mu\nu} + u^\mu u^\nu$ projects orthogonal to $u^\mu$
• $$\sigma^{\mu\nu} = \Delta^{\mu\alpha} \Delta^{\nu\beta} \left[\nabla_\alpha u_\beta + \nabla_\beta u_\alpha - \frac{2}{3} g_{\alpha\beta} \nabla_\lambda u^\lambda\right]/2$$ is the shear tensor
• $\eta$ is the shear viscosity
• The “script” variables $\mathcal{A}$, $\Pi$, and $\mathcal{Q}^\mu$ encode first-order corrections to the equilibrium variables and fluxes, each parameterized by transport coefficients and involving first derivatives of hydrodynamic variables.

Frame freedom is essential: BDNK exploits a controlled redefinition of $u^\mu$, $T$, and related variables so that the coefficients in the constitutive relations ensure causality and stability. The transport coefficients—“relaxation times” and gradient-coupling coefficients—must satisfy a set of nonlinear inequalities that enforce strongly hyperbolic dynamics (see, e.g., causality conditions in (Pandya et al., 2022, Redondo-Yuste, 25 Nov 2024)).

Unlike traditional approaches, the BDNK frame allows for a finite relaxation time in the evolution of dissipative variables: $$\mathcal{E} = \epsilon + \tau_\epsilon \left[ u^c \nabla_c \epsilon + (\epsilon+P)\nabla_c u^c \right]$$ with similar expressions for the pressure correction and heat flux. The relaxation times control how rapidly the system returns to Navier–Stokes values, and their values directly impact the characteristic speeds of the theory.

3. Microscopic Derivation and Kinetic Theory Origin

A central achievement of BDNK is its systematic derivation from relativistic kinetic theory using a moment expansion of the Boltzmann equation (Rocha et al., 2022). Instead of eliminating time derivatives via conservation laws as in Chapman–Enskog, BDNK projects the Boltzmann equation onto an orthogonal basis, keeping inertial terms. The first-order deviation function for the single-particle distribution is

$$\phi_p = \tilde{\mathcal{S}}_p\, \theta + \tilde{\mathcal{V}}_p\, p^{\langle \mu \rangle} \nabla_\mu \alpha + \mathcal{T}_p\, p^{\langle \mu} p^{\nu \rangle} \sigma_{\mu\nu}$$

where $\theta \equiv \nabla_\mu u^\mu$ (expansion), $\sigma_{\mu\nu}$ (shear), and $\nabla_\mu \alpha$ (thermal potential gradient). Transport coefficients (e.g., $\eta$, $\kappa$, $\chi$, $\lambda$) are computed via moments of the collision operator, employing a modified relaxation time approximation to maintain conservation laws for arbitrary matching conditions.

This procedure provides explicit, microscopic expressions for all BDNK coefficients, ensures physical consistency, and confirms the absence of instability or acausal propagating modes in linear response (Rocha et al., 2022, Rocha et al., 2023).

4. Numerical Analysis, Shock Resolution, and Critical Behavior

Practical application of BDNK in numerical simulations demonstrates its stability for smooth flows and controlled behavior for shocks if the hydrodynamic frame is chosen appropriately (Pandya et al., 2021, Pandya et al., 2022, Keeble et al., 28 Aug 2025). For example:

• In smooth initial data scenarios, straightforward finite difference and conservative finite volume methods yield stable, convergent solutions without the need for complex stabilization schemes otherwise employed for Israel–Stewart (MIS) theory.
• For smooth shockwaves, if the maximum characteristic speed equals or exceeds the speed of light, arbitrarily strong shocks can be resolved smoothly.
• In the classical shock-tube problem, even though mathematical well-posedness for discontinuous initial data has not been established, high-resolution shock-capturing schemes plus careful evolution of perfect fluid components produce solutions matching those of MIS and ideal hydrodynamics in the appropriate limits.

A distinctive feature is the transient violation of the weak energy condition observed at high viscosity in regions of strong gradients. Such behavior is transient and relaxes towards a universal attractor, reflective of hydrodynamic attractors predicted in kinetic theory and confirmed in Bjorken flow-like scenarios.

However, in highly out-of-equilibrium regimes (large Knudsen number, large viscosity, or steep gradients), both BDNK and first-order Navier–Stokes equations can display pathologies such as divergences or runaway solutions, indicating the breakdown of first-order hydrodynamics and the possible need for a transition to a higher-order or second-order transient theory (Rocha et al., 2023).

5. Comparison With Israel–Stewart and Other Theories

Whereas second-order theories like Israel–Stewart (MIS) achieve causality and stability by introducing independent dynamical variables for the dissipative currents and their corresponding relaxation-type evolution equations, BDNK retains a minimal set of hydrodynamic degrees of freedom. Instead, BDNK constitutive relations extend first order in gradients, utilizing time derivatives that are not eliminated via conservation laws:

• BDNK PDEs are second order in time, reflecting memory effects inherent to physical dissipation, but do not include extra (non-hydrodynamic) fields.
• The underlying set of equations forms a strictly hyperbolic system, with causality and stability directly linked to the choice of frame and transport coefficients.
• In the hydrodynamic regime, the spectrum of BDNK coincides with that of MIS-type theories (for sound and shear modes), though the latter also contain explicit transient (non-hydrodynamic) modes (Bhattacharyya et al., 2023).
• BDNK serves as a limit of more general, higher-order (e.g., divergence-type or second-order) formulations when the non-hydrodynamic degrees of freedom decouple.

6. Numerical Implementation and Practical Considerations

The BDNK framework, due to the absence of auxiliary fields, describes dissipative relativistic hydrodynamics using balance laws and a systematic first-order PDE structure. Typical numerical discretizations include:

• Flux-conservative finite difference or finite volume schemes with central-upwind or shock-capturing fluxes, enabling smooth transition to perfect-fluid (Euler) solvers as transport coefficients vanish (Pandya et al., 2022).
• For strong shocks or regions of discontinuity, non-oscillatory ENO/WENO reconstruction is essential to minimize numerical artifacts.
• For smooth flows, the hyperbolic nature and principal part diagonalizability permit stable and convergent evolution using standard explicit time integration as long as the time step satisfies a CFL-like condition based on the highest characteristic speed.

For discontinuous initial data, especially for the classical shock-tube problem, the mathematical well-posedness of the BDNK PDEs (even in a weak sense) is not established. In practice, special numerical treatment of the inviscid terms is required (Pandya et al., 2021).

A critical implementation issue is the preservation of certain differential constraints introduced during first-order reductions. Recent investigations suggest that these constraints are generally not preserved by the pure evolution equations and may grow exponentially from small violations, presenting a challenge to strong hyperbolicity in the conformal case (Fantini et al., 6 Jun 2025). This suggests the need for enhancement either through constraint-damping techniques or alternative reduction strategies.

7. Applications and Physical Implications

BDNK theory has rapidly found applications in:

• Simulations of heavy-ion collisions, where relativistic viscosity and heat conduction play crucial roles in the thermalization of quark–gluon plasma. The theory’s agreement with MIS at low viscosity and divergence in the high-gradient regime points to its validity envelope and the need for careful modeling near the boundary of hydrodynamic applicability.
• Astrophysical contexts such as neutron star mergers and gravitational wave emission, where evolution with strong gradients and out-of-equilibrium effects can be rigorously and causally tracked up to the breakdown scale of first-order hydrodynamics.
• Studies of hydrodynamic attractors and universal behavior in far-from-equilibrium’s relaxation to equilibrium, supported by the BDNK framework’s uniquely predictive power in the presence of large initial perturbations not handled by classical models.

BDNK’s programmatic role in modern hydrodynamics is summarized in the following comparison:

Aspect	Eckart / Landau–Lifshitz	MIS / Second-order	BDNK
Causality	No	Yes	Yes
Stability	No	Yes	Yes
Degrees of Freedom	Minimal	Auxiliary fields needed	Minimal
Order in Gradients	First	Second	First (with time derivatives)
Well-posedness	No	Yes	Yes (for suitable frame)

8. Limitations and Future Directions

Several open issues and limitations remain:

• Well-posedness for arbitrary discontinuous initial data (“shock tube” type problems) is not fully established for BDNK.
• Constraint preservation in first-order reductions is a technical challenge, motivating new analytic or numerical stabilization strategies (Fantini et al., 6 Jun 2025).
• At sufficiently high viscosity or large gradients, BDNK solutions can transiently violate energy conditions or develop singular gradients, marking the breakdown of the first-order gradient expansion and signaling the requirement for higher-order (transient, divergence-type) corrections (Keeble et al., 28 Aug 2025).
• In non-relativistic limits, BDNK reduces to Navier–Stokes with modified Fourier law for heat flux, and imposes combined causality-induced bounds on transport coefficients, e.g., for $\eta/s$ (R et al., 2023).

Recent developments seek to further clarify the domain of validity, extend BDNK to include fluctuating hydrodynamics and stochastic effects, and assess the implications of frame transformations for practical computations.

9. Summary

The BDNK formulation marks a theoretically robust and practically versatile milestone in relativistic viscous hydrodynamics. It provides a first-order model that is causal, stable, and strongly hyperbolic by exploiting a careful hydrodynamic-frame choice and retaining inertial effects in the constitutive relations. Numerical evidence demonstrates stable, convergent evolution for smooth data and successful shock resolution with proper frame selection. The approach unifies and clarifies the relationship between microscopic (kinetic theory) and macroscopic (fluid dynamics) descriptions, enabling advanced simulations of high-energy and astrophysical systems. Outstanding issues in mathematical well-posedness and constraint evolution are active research topics, highlighting the ongoing development of the formulation and its foundational place in the paper of dissipative relativistic fluids.