CL as an Equi-Redshift Flow

The constant-Lagrangian (CL) condition DΓ/Dt = 0 is stated at the rapidity level, with Γ_rotor = cosh χ_r cosh χ_φ. In the Galilean-coframe metric with a radial and an azimuthal river its slow-flow counterpart has a coordinate-free meaning:

γ_rs, the river’s Lorentz factor relative to the static Killing observers: γ_rs = −u·u_s/c² = c/√(−g_tt) = Γ_rotor + O(χ⁴)

The river observers are geodesics with conserved Killing energy, so CL means, wherever the radial flow is nonzero, that the river flows along level surfaces of the redshift factor √(−g_tt): v_r² + v_φ² = K. This is exact within the metric; the constant K is a closure. A slowly changing H leaves the analysis intact. Choose the parameters.

Rivers and disk

Matching radius R/r_s

Evaluation radius r/r_s

Units: c = 1, r_s = 2GM/c² = 1. Radial river v_r = √(2GM/r) − Hr (paper chart, Hubble term only when switched on); Keplerian azimuthal river v_φ = √(GM/r); CL river v_φ = √(K − v_r²) for r ≥ R, rigid inside R.

γ_rs along the disk

γ_rs − 1 is the fractional clock slowing of the river relative to the static observers. With the Keplerian azimuthal river it falls as r grows; the CL river keeps it constant beyond R: the river flows along an equi-redshift level.

Streamlines over the redshift levels

Rotor Γ and metric γ_rs at the evaluation radius

From the Killing vector to CL in seven steps

Metric of the gravity-rotor page, paper chart: ds² = −(c² − v_r² − v_φ²)dt² − 2v_rdr dt − 2v_φr s_θdφ dt + dr² + r²dθ² + r²s_θ²dφ², with Ω = v_φ/(r s_θ).

Check with numbers

What this does and does not show

CL condition, isorapidity Γ = cosh χ_r cosh χ_φ, the 3GM/R profile and the residual Π = Γ₀Ḣr from de Haas (2026), The Gravitational Rapidity Potential and Relativistic Gravitational Fluid Dynamics. Part of the BQ algebra models.