BQ-Dirac Kinematic Route to Isorapidity

The second branch of the rotor hierarchy differentiates the four-velocity itself. At the Dirac level the local rotor boosts the rest frame by one similarity, U_g = Q_g(cβ₀)Q_g⁻¹ with Q_g = diag(G, (G^†)⁻¹), and its material derivative along the flow is one commutator with the Dirac Maurer–Cartan form:

A_g = DU_g/Dτ = [Ω_τ, U_g], {U_g, A_g} = D(U_g²)/Dτ = 0, a₀ = c DΓ/Dτ

The orthogonality of velocity and acceleration is the derivative of the Clifford square U_g² = −c²𝟙, exact in the rapidities. So isorapidity, DΓ/Dτ = 0, 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 BQ-Pauli version is the kinematic route. Choose the rapidities.

Rapidities feeding the rotor

Bulge radius R/r_s

Evaluation radius r/R

Units: c = 1, r_s = 2GM/c² = 1. Paper chart: χ_r = ψ_N − ψ_H = √(r_s/r) − Hr/c, so v_r = c tanh χ_r ≈ √(2GM/r) − Hr in slow flow. The galaxy-scale preset takes M ≈ 10¹¹ M_⊙, R ≈ 2 kpc and a local H_z chosen so that r_c ≈ 20R.

How much of the acceleration is along the flow

From the rotor to the acceleration in eight steps, at the Dirac level

Weyl representation, following de Haas (2026), A Spin(1,3) Rapidity Theory for Galactic Rotation Curves, §2.2–2.5, lifted. Orthonormal legs β_r, β_θ, β_φ = β_I, β_J, β_K; α_x = β₀β_x; the BQ-Pauli blocks are U_g = G(cT̂)G^† and its time adjoint.

Velocity and acceleration in the disk

Check by explicit 4×4 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, lifted to the Dirac level; the BQ-Pauli version is the kinematic route. Every identity is checked exactly in bq_dirac_kinematic_route.py. Part of the BQ algebra at the Dirac level.