BDNK Framework: Relativistic Viscous Hydrodynamics

• BDNK is a first-order relativistic viscous hydrodynamics theory that introduces time-like derivatives to ensure causality, stability, and strong hyperbolicity.
• Its formulation exploits hydrodynamic frame freedom and matching conditions to tightly constrain transport coefficients for physical viability.
• The framework is applied in heavy-ion collision and astrophysical simulations using robust numerical methods for shock regularization and addressing steep gradient limitations.

The BDNK (Bemfica–Disconzi–Noronha–Kovtun) framework is a first-order relativistic viscous hydrodynamics theory designed to address the causality, stability, and mathematical consistency issues that plague standard first-order models such as Eckart and Landau–Lifshitz. By introducing suitable time-derivative terms and parameterizing frame freedom in the constitutive relations, BDNK achieves a causal, stable, and strongly hyperbolic system of equations established through rigorous mathematical analysis. This framework is structurally distinct from Müller–Israel–Stewart (MIS) type transient theories, as BDNK includes only evolution equations for the conserved currents, while the dissipative corrections are incorporated through first-order derivative expansions, with specific constraints on transport coefficients to maintain physical viability.

1. Theoretical Motivation and Framework Structure

Early attempts at relativistic viscous hydrodynamics—specifically the classic Navier–Stokes formulation (Eckart, Landau–Lifshitz)—were found to be acausal and unstable in the relativistic regime. These issues arise because their constitutive relations depend solely on space-like gradients, leading to parabolic equations and unbounded signal speeds. The BDNK framework addresses these problems by retaining explicit time-like derivatives in the first-order constitutive relations, thereby promoting the system to hyperbolic type and enforcing finite signal propagation speed.

In the BDNK theory, the energy–momentum tensor $T^{\mu\nu}$ and (optionally) the baryon current $J^\mu$ are written as

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

where $\epsilon$ is the energy density, $P$ the equilibrium pressure, $u^\mu$ the fluid four-velocity, $\Delta^{\mu\nu} = g^{\mu\nu} + u^\mu u^\nu$ the transverse projector, $\eta$ the shear viscosity, and $\sigma^{\mu\nu}$ the traceless shear tensor. The terms $\mathcal{A}$, $\Pi$, and $\mathcal{Q}^\mu$ encode first-order corrections involving both time-like and space-like derivatives; their precise form depends on the hydrodynamic frame choice.

The inclusion of these time-like terms allows the system to be written in a first-order, strongly hyperbolic form, provided the associated transport coefficients (including $\lambda$, $\chi$, and others encoding the non-equilibrium response) satisfy nonlinear inequalities. The mathematical analysis ensures the principal symbol is diagonalizable with real eigenvalues—meaning the equations genuinely propagate information at finite speeds and are suitable for a well-posed initial value problem (Pandya et al., 2021, Keeble et al., 28 Aug 2025).

2. Hydrodynamic Frames, Matching Conditions, and Transport Coefficient Constraints

A defining feature of BDNK is its systematic exploitation of the hydrodynamic frame ambiguity: the freedom to redefine nonequilibrium fluid variables (temperature, velocity, chemical potential, etc.) out of equilibrium. This is parameterized through matching conditions—typically denoted by parameters like $q$, $s$, and $z$—which select which moments of the kinetic distribution function are to be identified with equilibrium variables. For example, Landau matching sets energy flow corrections to zero, $z=1$; Eckart matching fixes the particle flow, $z=0$.

The constitutive relations for dissipative currents (specializing to an ultra-relativistic scalar theory) are given by

$$\begin{aligned} \xi &= \frac{12}{g\beta^2}(q-1)(s-1), \ \chi &= \frac{36}{g\beta^3}(q-2)(s-2), \ \varkappa &= \frac{12}{g\beta^2}z, \ \lambda &= \frac{48}{g\beta^3}(z-1), \ \eta &= \frac{48}{g\beta^3}, \end{aligned}$$

where $\beta=1/T$ and $g$ is set by the microscopic theory (Rocha et al., 2023, Brito et al., 2023). The frame choice thus directly determines the values of the transport coefficients, with explicit exclusion of the so-called “exotic” Eckart frame (e.g. $z=0$) for stability.

The BDNK equations’ causality and stability are guaranteed by choosing frame parameters such that inequalities like $(q+s-4)(z-4)\geq 0$ and $z>1$ (among others) are satisfied. These conditions ensure all physical modes are damped (no exponential growth) and none propagate superluminally (Brito et al., 2023, Roy et al., 2023).

3. Causality, Stability, and Analytic Structure

Causality (no signal propagates faster than light) and stability (no exponentially growing modes) are both tightly connected to the analytic structure of the linearized equations. The transverse and longitudinal sector dispersion relations in BDNK are of hyperbolic (mixed hyperbolic-parabolic) type, contrasting with the strictly parabolic, acausal, and unstable profiles of relativistic Navier–Stokes. In the sound channel, the BDNK dispersion relation is quartic in frequency, as opposed to the cubic structure of MIS theory (Heydari et al., 22 Apr 2024).

A critical mathematical tool for characterizing stability is the Liénard–Chipart criterion, a refined variant of the Routh–Hurwitz test. For the polynomial $$f(\Upsilon) = a_0 \Upsilon^N + a_1 \Upsilon^{N-1} + \dots + a_N$$ (with $\Upsilon = i\omega$), this yields sufficient and necessary conditions for all roots to have positive real part, preventing dynamical instabilities:

• All coefficients $a_n > 0$,
• Selected principal minors of the Hurwitz matrix (odd/even depending on convention) must be positive (Brito et al., 2023).

Local univalence (single-valuedness) of the hydrodynamic mode expansions is generally preserved in the shear channel for all physically acceptable transport coefficients. In the sound sector, local univalence coincides with (but does not strictly determine) the stability and causality domain for BDNK transport parameters in the “intermediate momentum” region (Heydari et al., 22 Apr 2024).

4. Reformulation and Frame Changing: BDNK vs. MIS-type Theories

When the BDNK energy–momentum tensor—originally expressed in an arbitrary (generalized) frame—is rewritten in the Landau frame via a field redefinition, the shift in velocity and temperature ($u^\mu = \hat{u}^\mu + \delta u^\mu$, $T = \hat{T} + \delta T$) necessarily involves an infinite gradient expansion. The BDNK theory remains local in its own (general) frame but becomes nonlocal when recast into the Landau frame; this nonlocality can be absorbed by introducing an infinite tower of “non-fluid” degrees of freedom, akin to the regulator fields in second-order MIS-type theories (Bhattacharyya et al., 2023).

In the linear regime, all hydrodynamic (gapless) modes are equivalently reproduced in either formulation, but non-hydrodynamic (gapped or transient) modes depend on the method of frame change or summation. This highlights the deep connection between frame choice, the order of the derivative expansion, and local vs. nonlocal evolution.

5. Numerical Schemes, Shock Regularization, and Regime of Validity

Flux-conservative and finite-volume numerical implementations have been developed for BDNK in conformal and nonconformal (zero chemical potential) scenarios (Clarisse et al., 18 Oct 2025, Pandya et al., 2022). These schemes employ high-resolution shock-capturing (e.g. Kurganov–Tadmor) or finite-difference methods on structured grids (including multi-block cubed-sphere coordinates for curved domains) (Keeble et al., 28 Aug 2025).

A central empirical result is that, in the regime of validity defined by small Knudsen numbers ($K\mathrm{n}_u$, $K\mathrm{n}_T \ll 1$)—termed the “hydrodynamic-frame–robust regime”—BDNK solutions show very weak dependence on frame parameters and robustly regularize incipient shock discontinuities that are singular in the Euler equations. In these domains, the BDNK evolution remains convergent and well-posed. However, for sufficiently large viscosity or steep gradients, the first-order expansion breaks down (evidenced by loss of numerical convergence and violation of energy or causality bounds), indicating the limits of the theory’s applicability (Keeble et al., 28 Aug 2025, Clarisse et al., 18 Oct 2025).

A summary table organizes key features:

Aspect	BDNK	Classic Navier–Stokes	MIS (Transient)
Order in gradients	First (with time-like)	First (space-like only)	Second (with regulator)
Causality/stability	Achievable (w/ constraints)	Always acausal/unstable	Achievable (w/ constraints)
Hyperbolicity	Strong (frame-dependent)	Parabolic	Strong (w/ relaxation)
Frame dependence	Explicit — essential	None (but ambiguous)	Fixed (usually Landau)
Shock regularization	Yes (frame-robust regime)	No (shock formation)	Yes

6. Applications: Heavy-Ion Collisions, Neutron Stars, and Fluctuations

The BDNK theory is positioned as a physically justified, practical alternative to MIS theory for systems where first-order dissipative corrections are important but additional regulator fields are unwarranted or unavailable. Notable applications include:

• Heavy-ion collisions: Numerical and analytic studies in Bjorken flow demonstrate BDNK can capture hydrodynamic attractors and transient phenomena in regimes with moderate gradients. For larger gradients, it shares limitations with Navier–Stokes in failing to remain physically viable, indicating that second-order or full kinetic treatments become necessary (Rocha et al., 2023, Roy et al., 2023).
• Neutron star mergers and oscillations: BDNK has been applied to nonlinear evolution and mode analysis of viscous neutron stars, with explicit extraction of quasi-normal mode frequencies and decay rates. The framework allows direct tracking of how physical viscosity affects damping without introducing spurious or acausal behavior, provided parameter constraints are satisfied (Redondo-Yuste, 25 Nov 2024, Shum et al., 18 Sep 2025).
• Thermal and stochastic fluctuations: A mathematically consistent BDNK theory of fluctuations has been constructed using the Martin–Siggia–Rose (MSR) approach, revealing that noise-induced long-range correlations in frame-dependent primary fields (e.g., chemical potential, velocity) do not contaminate conserved density correlations, owing to the unique structure of non-hydrodynamic modes (Gavassino et al., 9 Feb 2024).

7. Limitations, Constraint Instabilities, and Outlook

Despite its robust hyperbolicity at the level of the principal symbol, recent investigations have shown that the first-order reductions used to prove well-posedness in Sobolev spaces generate differential constraints that may not be robustly propagated during evolution. In particular, for conformal fluids, the associated auxiliary variables' constraints can experience exponential growth, leading to a breakdown of the strong formulation, especially in numerical schemes (Fantini et al., 6 Jun 2025). This points to the need for alternative reductions, constraint-damping techniques, or direct analysis of the original second-order system to sharpen mathematical existence claims in practical computations.

Furthermore, while BDNK regularizes shocks in the frame-robust, low-Knudsen number regime, there is evidence that finite-time singularities (discontinuities) can still form from smooth initial data if the first-order expansion becomes inapplicable due to large gradients or high viscosity (Keeble et al., 28 Aug 2025). Therefore, careful diagnostic monitoring and, if necessary, switching to higher-order or kinetic theory treatments are required when exploring far-from-equilibrium dynamics.

In summary, the BDNK framework provides a mathematically and physically consistent first-order relativistic viscous hydrodynamics theory, with a precisely delineated regime of applicability and explicit, testable criteria for stability and causality. Its flexibility in frame choice allows direct connection to kinetic theory and various matching prescriptions, while its strong-hyperbolic structure supports robust numerical modeling in regimes relevant for high-energy nuclear and astrophysical phenomena. Nonetheless, ongoing work is required to resolve constraint propagation issues and to determine the theory's efficacy at the limits of the hydrodynamic regime.