BQ Gravity Rotor: from Rapidity to Metric

Gravity enters the BQ-Pauli algebra as a local Lorentz rotor field. Rapidity potentials feed the rotor G(r, t) = e^χσ̂/2: at every point it boosts the coordinate line element, dR^G = G⁻¹dR G⁻¹, where a Lorentz transformation would use one global rapidity. The exact image has a flat quadratic: a local Lorentz transformation alone gives back Minkowski space, and so does every consistent slow-flow truncation of it. The gravitational content is in the choice of a Galilean coframe with a Lorentzian quadratic: the slow-flow coframe θ⁰ = c dt, θ^r = dr − v dt, squared with the Minkowski signature, gives curved Einsteinian metrics, exactly Schwarzschild with its horizon for v = √(2GM/r). The Einstein tensor then shows which riverbeds are vacuum. Choose the rapidities that feed the rotor.

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

From the rapidity field to g_μν in eight steps

The steps follow de Haas (2026), The Gravitational Rapidity Potential, Sections 2–5, at the BQ-Pauli level. Here C_r = cosh χ_r, S_r = sinh χ_r, C_φ = cosh χ_φ, S_φ = sinh χ_φ, s_θ = sin θ.

Comparison with the known metric

Check by explicit matrix multiplication

Curvature: the Einstein tensor of the prescribed metric

A field of Lorentz transformations

Global versus local. A Lorentz transformation relates two inertial frames with one constant rapidity. Here χ(r, t) is a spacetime field, so G(r, t) assigns a boost to every point. A constant χ gives back the global boost of the Lorentz pages.
Two exact projections. On the reference vector cT̂ the rotor gives the gravitational four-velocity U_g = G⁻¹(cT̂)G⁻¹ = c cosh χ T̂ − c sinh χ Î, with γ = cosh χ and v^FR = c tanh χ < c. On dR it gives the exact image dR^G.
Where the gravity enters. The exact quadratic (dR^G)^TdR^G = dR^TdR is flat, and so is every consistent truncation of dR^G. The metric comes from a Galilean coframe with a Lorentzian quadratic: (1) the slow-flow, Galilean limit of the rotor action, θ⁰ = c dt and θ^r = dr − v dt with v = cχ, and (2) the Minkowski-signature quadratic of that coframe. The coframe carries the physics; its Lorentzian quadratic gives the Einsteinian metric.
Sign of the flow. With G = e^χσ̂_r/2 and dR^G = G⁻¹dRG⁻¹, the matrices give θ^r = dr − cχ_rdt and g_tr = −v_r, as in the paper’s coframe and metric (eqs. 46, 51). For v_r > 0 this is the outgoing PG chart, and for the Hubble term a contracting one: the time reverse of infall plus expansion. The time-orientation toggle flips the rotor rapidity to −χ and gives the ingoing, expanding chart, g_tr = +v_r. Both charts describe the same geometry.

Two boosts

With a radial and an azimuthal rapidity the boosts are applied one after the other, radial first: dR^G = G_φ⁻¹(G_r⁻¹dR G_r⁻¹)G_φ⁻¹. Each step keeps dR^G a minquat, and the exact quadratic stays flat. The single product G⁻¹dRG⁻¹ with G = G_φG_r is not the same: G_φG_r is not a pure boost (it contains the Thomas–Wigner rotation), and that product leaves the minquat space at order χ_rχ_φ. The order of the two boosts and the Wigner rotation only affect terms of order χ_rχ_φ, which the slow-flow projection drops.

Rapidity fields, rotor, coframe and metric as in de Haas (2026), The Gravitational Rapidity Potential, and de Haas (2025), Constructing Spacetime from Rapidity. Part of the BQ algebra models.