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Relativistic hydrodynamics

Stable and causal frames for BDNK hydrodynamics

A constructive route to relativistic fluid equations that keep disturbances stable and signals within the light cone.

Jay Armas · University of Amsterdam & Niels Bohr Institute

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Causal propagation within a light coneA spacetime diagram with time increasing upwards. Two example signal trajectories lie inside the future light cone, illustrating propagation slower than light.ctxLIGHT CONEsignal trajectories|v| ≤ c
Illustration of causal propagation. The trajectories are schematic, not numerical results from the paper.
6 familiesExplicit prescriptions for stable and causal frames.
3 conformal limitsFamilies that remain finite at the conformal endpoint.
Neutral → chargedA map extending neutral constructions to charged fluids.

The central idea

Choose the frame. Keep the fluid’s physics.

Relativistic fluids describe matter in settings ranging from nuclear collisions to astrophysics. Adding viscosity and charge transport makes these descriptions more realistic, but familiar first-order equations can develop instabilities or allow signals to travel faster than light.

The Bemfica–Disconzi–Noronha–Kovtun (BDNK) formulation uses the freedom to define hydrodynamic variables away from equilibrium to address this problem. In this paper, I give explicit families of frame choices for prescribed equations of state, viscosities and conductivity, within the stated thermodynamic and transport conditions. The physical inputs remain fixed while the remaining frame coefficients are chosen to ensure linear stability and causal characteristic speeds.

01 / CONSTRUCTION

From inequalities to usable frames

Simplified Routh–Hurwitz criteria and telegrapher factorisation turn complicated stability conditions into tractable constructions. Six explicit families provide starting points for numerical implementations.

02 / GENERALISATION

From neutral to charged fluids

A common uplift maps neutral frame choices to charged ones, introducing a telegrapher factor for the charge sector while preserving the prescribed physical transport.

03 / PROPAGATION

Control the characteristic speeds

Adjustable frame parameters give different characteristic structures. With positive shear viscosity and conductivity, family F6 has strictly positive, strictly subluminal comoving characteristic speeds.

A numerical example

Propagation and damping, mode by mode

These plots show the sound, shear and charge modes of the charged F6 family in a moving frame, with boost v = 0.6 and a propagation angle of 60° relative to the boost. The left panel shows the real frequencies; the right shows the imaginary frequencies, which are non-positive throughout the sampled range.

Real and imaginary frequencies of sound (red), shear (blue) and charge (green) modes for charged F6 at boost 0.6 and angle 60 degrees.
Charged F6 example from the BDNK Mode Explorer Mathematica calculation. Red: sound; blue: shear (two coincident polarizations); green: charge. Dashed lines in the left panel indicate the light-cone slopes ±k; causality concerns the asymptotic characteristic speeds. With the convention exp(−iωt + ik·x), Im ω ≤ 0 corresponds to non-growing modes. This finite momentum scan illustrates the construction and is not a proof for all momenta or equations of state. Click the plots to enlarge.
Example inputs

The illustrative pressure is p(T, μ) = T⁴ + μ²T²/2 + μ⁴/24 + m²T²/2, at T₀ = μ₀ = m = 1, with η = 1/5, ζ = 1/10 and conductivity 𝒟 = 1/20. Frame parameters: Amplitude = 2, Beta = 1/2 and AlphaQ = 2. The momentum range is 0 ≤ k/T₀ ≤ 10, sampled over 120 intervals using 50-digit arithmetic.

Research resource

AI research protocol

A structured protocol for AI-assisted searches for stable and causal hydrodynamic frames. It brings together the construction strategy, polynomial factorisation and checks against the original equations.

The method offers a route to exploring other hydrodynamic theories, with each extension requiring its own physical constraints and verification.

Put the method to work

Get the AI research protocolView and download the protocol on GitHub.
  1. Specify the equation of state and physical transport; impose entropy constraints.
  2. Reduce the stability criteria to analytically tractable conditions.
  3. Construct neutral families and check any conformal limits.
  4. Uplift neutral seeds using charged telegrapher factorisation.
  5. Verify the original determinant, arbitrary boosts and degenerate boundaries.