BQ-Dirac Gravity Rotor: from Rapidity to Metric

Gravity enters the BQ-Dirac algebra as a local Lorentz rotor field: the global boost Λ = diag(U⁻¹, U) of the Lorentz models, with its constant rapidity replaced by a rapidity field χ(r, t). Rapidity potentials feed the BQ-Pauli rotor G = e^χσ/2, and its Dirac double Q_g = diag(G⁻¹, G) = e^−χα/2 boosts the Dirac line element by one similarity. As at the Pauli level, the exact image dR_g is flat, and the metric comes from its Galilean limit θ with a Lorentzian quadratic. At the Dirac level that quadratic is simply the square θθ = ds²𝟙, and g_μν is the anticommutator of the β-matrices θ_μ, where the exact dR_g,μ only give back the flat g⁰_μν. Choose the rapidities that feed the rotor.

dR_g = Q_g dR Q_g⁻¹, Q_g = diag(G⁻¹, G) = e^−χα_I/2, θθ = ds² 𝟙, {θ_μ, θ_ν} = 2g_μν𝟙

Rapidities feeding the rotor

Evaluation radius r/r_s

Units: c = 1 and r_s = 2GM/c² = 1, so ψ_N = √(r_s/r). The equatorial plane θ = π/2 is used for the numbers.

The riverbed

The Dirac gravity rotor Q_g: the Pauli rotor, doubled

From the rapidity field to g_μν at the Dirac level

Here γ_r = cosh χ_r and γ_φ = cosh χ_φ; in steps 2–4 v = c tanh χ is the exact boost speed, from step 5 on v = cχ is the river velocity; s_θ = sin θ, η = diag(−1, 1, 1, 1). Three line elements: the flat dR, its exact image dR_g (flat), and the Galilean θ (curved).

Comparison with the known metric

Check by explicit 4×4 matrix multiplication

What the Dirac level adds

One similarity for both blocks. The BQ-Pauli image G⁻¹dRG⁻¹ and its time adjoint G dR^TG are the two blocks of Q_gdRQ_g⁻¹, exactly as for the global boost. And the nested rotor is one rotor, Q_g = Q_g,φQ_g,r.
The square instead of the adjoint. The metric comes from the plain square θθ = ds²𝟙, and the metric tensor from the anticommutators of the β-matrices θ_μ = θ^a_μβ_a; the same relation on the exact dR_g,μ gives the flat g⁰_μν.
Ready for spinors. The θ_μ are the β-matrices of a Dirac equation in this field, and Q_g acts on a spinor from one side, Ψ → Q_gΨ. That is for a later theme.
Same geometry. The metric, the riverbed and the comparison are those of the BQ-Pauli page, so its Einstein tensor applies unchanged: see the BQ gravity rotor, curvature section.

Sources and status

Rapidity fields, coframe and metric as in de Haas (2026), The Gravitational Rapidity Potential, at the BQ-Pauli level; the Dirac rotor Q_g, the boosted basis and the metric as the scalar part of the curved β-products as in de Haas (2025), BQ Dirac Gravity, Sec. 4 and eqs. 173, 191–192. Naming: the paper’s dR_g of §4.1 (the Galilean, PG line element) is θ here; dR_g here is the exact image, the paper’s A̸^G of eq. 124. The paper’s /G_μ = Q_gβ_μQ_g⁻¹ = θ^a_μβ̂_a (eqs. 193, 226) joins the two parts that the matrices separate: Q_gβ_μQ_g⁻¹ alone only gives η, the coframe θ^a_μ gives g_μν. Reported, not changed: eqs. 146–150 give Q_gβ₀Q_g⁻¹ = cosh ψ β₀ + sinh ψ β_r for Q_g = exp(−½ψβ₀β_r); with the BQ-Weyl matrices of this site that rotor gives the minus sign, and the plus sign belongs to rotor rapidity −ψ (the time-orientation toggle). The metric is not affected.

Every identity is checked exactly, in both time orientations, in bq_dirac_gravity_rotor.py. Part of the BQ algebra at the Dirac level.