The Kinematic Route to Isorapidity

The second branch of the rotor hierarchy differentiates the four-velocity itself. The local rotor G = GφGr boosts the rest frame to Ug = G(cT̂)G†; its material derivative along the flow is the convective four-acceleration Ag = DUg/Dτ. Its time channel is the rate of the composed Lorentz factor, and its norm with Ug vanishes identically:

a0 = [Ag]T = c DΓ/Dτ,  [UgTAg]1̂ = u0a0 − u·a = 0  — exact in the rapidities

So isorapidity is zero power along the flow, and the spatial acceleration of an isorapid flow is purely transverse: the river turns but does not speed up. The bridge Ag = ΩτUg + UgΩτ† shows that this route and the Maurer–Cartan route of step 10 are one statement, a0 = 2eT. Choose the rapidities.

Rapidities feeding the rotor

Units: c = 1, rs = 2GM/c² = 1. Paper chart: χr = ψN − ψH = √(rs/r) − Hr/c, so vr = c tanh χr ≈ √(2GM/r) − Hr in slow flow. The galaxy-scale preset takes M ≈ 10¹¹ M⊙, R ≈ 2 kpc and a local Hz chosen so that rc ≈ 20R.

How much of the acceleration is along the flow

From the rotor to the acceleration in eight steps

BQ-Pauli level, following de Haas (2026), A Spin(1,3) Rapidity Theory for Galactic Rotation Curves, §2.2–2.5. Orthonormal legs (Îr, Ĵθ, K̂φ) and (σ̂r, σ̂θ, σ̂φ) in the order (I, J, K); time adjoint AT = −a0T̂ + a·K̂.

Velocity and acceleration in the disk

Check by explicit matrix multiplication

What this does and does not show

Rotor, four-velocity, convective acceleration, orthogonality and the power reading (eqs. 12–25) from de Haas (2026), A Spin(1,3) Rapidity Theory for Galactic Rotation Curves; the Maurer–Cartan route is step 10. Part of the BQ algebra models.