Starting from the minimal complex matrix algebra 𝑴 2 (C), we show that two biquaternion products-... more Starting from the minimal complex matrix algebra 𝑴 2 (C), we show that two biquaternion products-𝑴 = 𝑹 𝑻 𝑮 and 𝑱 𝑫 = 𝑹(𝑼 𝑻 𝑮)-suffice to derive the complete conservation structure of a relativistic perfect fluid and its dilatation current, without postulating a Lagrangian and without invoking Noether's theorem as an external tool. The product 𝑴 = 𝑹 𝑻 𝑮 decomposes automatically into the action density, angular momentum density, and moment-of-energy density as algebraically forced channel outputs, and the closure condition 𝝏 𝑴 = 0 packages all three conservation laws into a single Maurer-Cartan flatness condition on a Lie-algebravalued current in the adjoint representation of the Lorentz group. The perfect-fluid reduction projects 𝑴 onto the adjoint orbit of the fluid velocity, yielding the relativistic Lagrangian fluid equations in comoving coordinates whose 𝜸-scaling encodes the dilatation structure of the flow. The companion product 𝑱 𝑫 = 𝑹(𝑼 𝑻 𝑮) constructs the dilatation current explicitly: under the perfect-fluid constraint it reduces to 𝑱 𝑫 = 𝜺𝑹, a pure Lorentz four-vector whose conservation is the relativistic virial theorem and whose sole surviving constraint in the thin-disk limit is the Euler homogeneity relation ∇ • r + 1 = 0-the differential signature of a renormalisation-group fixed point, derived without assuming scale invariance. The construction demonstrates that the Lagrangian, the Euler-Lagrange equations, and the Noether currents are outputs of the algebraic structure of 𝑴 2 (C) rather than its foundation, and identifies the biquaternion transposed product as the minimal algebraic operation from which the standard framework's results follow as necessary consequences.
This paper shows that the full algebraic content of relativistic physics arises from the minimal ... more This paper shows that the full algebraic content of relativistic physics arises from the minimal complex matrix algebra 𝑴 2 (C). Two interlocking 4-sets-the geometric minquat 𝑲 𝝁 and the spin-norm Pauliquat 𝝈 𝝁-satisfy the fundamental relation 𝑲 𝝁 = 𝒊𝝈 𝝁 , identifying time as the spin pseudoscalar and placing space-time vectors and spin vectors in a single algebraic structure. The Pauli-BQ bilinear 𝑨 𝑻 𝑩 generates the Minkowski metric, rotations, boosts, Maxwell electrodynamics, stress-energy, angular momentum, and relativistic dynamics. Gravity enters through a single rotor acting on the line element, yielding the PG coframe and metric without tensors or connections. The Dirac framework follows naturally as the 𝑴 4 (C) doubling of this structure, with 𝜷 𝝁 = 𝒊𝜸 𝝁 inherited from 𝑲 𝝁 = 𝒊𝝈 𝝁. Thus space-time, spin, electromagnetism, and gravity emerge as complementary sectors of the smallest nontrivial complex algebra. Contents 1 Unification Through Minimalism in 𝑀 2 (C) 2 Electromagnetism and Gravity: Norm Suppression vs. Norm Affection 3 Domains of Physics Where the Spin-Norm 4-Vector Appears Naturally 4 Unifying Consequences of the Identity 𝐾 𝜇 = i 𝜎 𝜇 5 Why the 𝑀 2 (C
This paper presents a compact and fully transparent derivation of the algebraic sequence that lin... more This paper presents a compact and fully transparent derivation of the algebraic sequence that links biquaternionic rapidity fields to the Painlevé-Gullstrand (PG) coframe and the corresponding spacetime metric 𝒈 𝝁𝝂. Building on the gravitational rotor 𝑸 𝒈 introduced in earlier work, we show explicitly how a general rapidity field generates the boosted BQ displacement 𝒅 / 𝑹 𝑮 , and how the PG coframe 𝜽 𝒂 can be read off directly as its linear coefficient structure, without any prior metric assumptions. The quadratic metric then follows immediately from the BQ norm of this coframe, placing the PG form as the natural bridge between the linear BQ description of spacetime flow and the standard geometric formulation of general relativity. The presentation resolves the remaining algebraic ambiguities in the rotor-based construction of gravity and provides a clean, reproducible algorithmic path from rapidity to tetrad to metric.
We develop the explicit first-order calculus of the gravitational rotor Q g ∈ Spin(1, 3) and show... more We develop the explicit first-order calculus of the gravitational rotor Q g ∈ Spin(1, 3) and show how general-relativistic metrics, gravitational flows, and MOND-type galactic dynamics arise directly from its adjoint action in the Dirac-biquaternion (BQ) algebra. Starting from rapidity fields in the rotor exponent, we derive the Schwarzschild, de Sitter, and Kerr forms in Painlevé-Gullstrand coordinates, demonstrating that tetrads, the spin connection, and the metric follow from Q g without invoking Christoffel symbols or second-order curvature equations. The Q g current conservation law D µ J µ = 0, together with the Bernoulli-Noether constraint, selects the Constant-Lagrangian with Hubble boundary (CL-H) flow as the unique stationary solution for axisymmetric rotating disks. This flow yields an azimuthal velocity profile v 2 ϕ (r) = 3 2 w 2 (R)-w 2 (r), where w(r) is the mixed Schwarzschild-Hubble river velocity, and produces a covariant acceleration law containing the Newtonian, MOND, and cosmological de Sitter regimes as limiting cases. The formalism therefore links the algebraic structure of the Dirac equation to relativistic metric construction and to observed galactic rotation curves within a single first-order gravitational framework.
We show that a single three-rapidity Painlevé-Gullstrand (PG) metric provides a numerically predi... more We show that a single three-rapidity Painlevé-Gullstrand (PG) metric provides a numerically predictive and observationally consistent description of the M 87 jet across more than seven orders of magnitude in radius. Imposing the observed asymptotic opening angle of θ jet ≃ 6 • fixes the ratio v ϕ /v esc = 0.105 and thereby fully constrains the PG jet geometry. This single condition determines the stagnation surface (w θ = 0 at ∼ 10-20 r g), the tightly wound inner helical motion, the parabolic collimation funnel, the Bondi-scale collimation break, the natural formation of a recollimation region at HST-1, and the asymptotic conical jet. The PG geometry also predicts intrinsic helical streamlines, nested cones corresponding to spine-sheath structure, and the large-scale double-helix morphology. All major dynamical and morphological features of the M 87 jet-from the EHT-scale launch region to the kiloparsec-scale spine-sheath and helical patterns-thus emerge from a single PG spacetime, without requiring additional free parameters. These results indicate that the PG metric provides a unified geometric backbone onto which the full dynamical physics of relativistic jet formation can be consistently added.
Astrophysical jets exhibit universal geometric features-narrow opening angles, long coherent heli... more Astrophysical jets exhibit universal geometric features-narrow opening angles, long coherent helices, and bipolar symmetry-across a wide range of systems. In this paper we show that these properties arise naturally from the stationary three-rapidity Painlevé-Gullstrand (PG) metric, treated purely as a kinematic flow generator. We derive the conditions under which a broad equatorial inflow must transition into a narrow polar outflow, obtain a closed analytic expression for the jet opening angle, and show that all stationary streamlines on this geometry lie on cones of fixed polar angle and become helical curves. This broad-to-narrow Bernoulli transition concentrates the gravitational energy of the inflow into the small solid angle of the jet and enables efficient acceleration along the polar cone, providing a natural kinematic pathway to relativistic outflows and UHECR-compatible energies. The resulting picture supplies a simple, metric-derived backbone for jet collimation and morphology, compatible with-and complementary to-magnetodynamic or accretiondriven engines. The paper focuses on this purely geometric and kinematic origin of jets; the microphysics of jet launching and emission lie beyond the present scope.
We show that Milgrom's empirical coincidence a 0 ≃ cH follows directly from Lorentz-FRW rapidity ... more We show that Milgrom's empirical coincidence a 0 ≃ cH follows directly from Lorentz-FRW rapidity kinematics, since the gravitational acceleration is a = c ψ while the cosmic expansion sets ψH = H, yielding a natural MOND scale a 0 = cH and clarifying its long-standing phenomenological status.
This paper develops the cosmological consequences of the Q g rotor framework, a first-order algeb... more This paper develops the cosmological consequences of the Q g rotor framework, a first-order algebraic formulation of gravity derived from the Dirac-biquaternion language. Building on previous work that established the gravitational rotor, mixed tensor, and current as linear bilinears of the spinor field, we derive the corresponding continuity and constitutive relations from the first-order Lagrangian and apply them to a homogeneous and isotropic universe. The resulting dynamics describes cosmic expansion as a precession of the gravitational rotor and leads to a linear version of the Friedmann law in which the expansion rate depends directly on the local spinor density and on an asymptotic rotor precession constant. This first-order structure reproduces the empirical expansion history of the universe without invoking a separate dark-energy component and restores the direct, local coupling between matter density and cosmic flow that is averaged out in the classical quadratic formulation. The framework provides a natural, quantitative explanation for the Hubble tension: variations in the locally measured Hubble rate arise from genuine differences in matter density rather than from systematic errors or new physics. The Q g cosmology thus offers a linear, covariant, and testable extension of Einstein-Cartan gravity that unifies local and cosmic dynamics within a single algebraic formalism.
This paper completes the development of the Q g rotor framework by formulating a firstorder Lagra... more This paper completes the development of the Q g rotor framework by formulating a firstorder Lagrangian that unifies gravitational geometry and matter flow within the biquaternion (BQ) algebra. Building on earlier work defining the rotor Qg , the mixed tensor Mµν , and the conserved current Jν=uµMµν , the theory now achieves full variational closure. The resulting Euler-Lagrange equations reproduce the continuity condition DνJν=0, the constitutive relation between J and M , and the constant-Lagrangian (CL-H) invariant that underlies galactic and cosmological dynamics. This establishes the Qg system as a linear, first-order generator of Einstein-Cartan gravity, expressed entirely in the Dirac/BQ algebra, where curvature, matter flow, and time evolution emerge from a single conserved rotor field.
This paper presents the Qg rotor framework as an algorithmic bridge between Dirac algebra and gra... more This paper presents the Qg rotor framework as an algorithmic bridge between Dirac algebra and gravitation. The gravitational field is expressed within the biquaternionic (Dirac) algebra, where the adjoint action of the rotor Qg on the Dirac basis generates the metric and connection directly, without introducing curvature as an external postulate. The mixed tensor M = EΦ and the Dirac current J = uM describe the coupling between matter flow and gravitational geometry, forming a complete linear set of algebraic field operators. From this structure an explicit translation algorithm is derived that reproduces the general-relativistic metric while remaining entirely within the covariant Dirac–BQ domain. Applying the constant-Lagrangian with Hubble boundary (CL-H) condition yields galactic rotation laws and MOND-type accelerations as covariant consequences of the same invariant. The result shows that gravitational and galactic dynamics can be formulated as a single, self-consistent algebraic flow within the Qg field, linking the Dirac formalism, general relativity, and MOND phenomenology.
This work develops a unified first-order description of gravitation, quantum time, and cosmic exp... more This work develops a unified first-order description of gravitation, quantum time, and cosmic expansion through the gravitational rapidity field 𝑄 !. Starting from the insight that the Dirac adjoint's fixed time axis prevents a covariant treatment of time in quantum theory, the metric is redefined as the adjoint action of a local rotor, 𝑔 "# = 𝑄 ! $ 𝜂 "# 𝑄 !. A single closure condition-the Constant-Lagrangian (CL) postulate-governs both the temporal transport of frequency in relativistic geodesy and the spatial inflow that structures galaxies. When the universal Hubble expansion is included as an isotropic outflow generator, the resulting CL-H balance yields a new analytic class of Einstein solutions in which the apparent dark-matter and dark-energy effects arise as curvature energy of the flow itself. The framework thus links quantum time, metric dynamics, and cosmological structure within one rapidity-based field equation.
This collection unites seven complementary studies that together outline a linear, spinor-driven ... more This collection unites seven complementary studies that together outline a linear, spinor-driven formulation of spacetime geometry. All works are authored by E. P. J. de Haas and available via viXra, Zenodo, and Academis. 2 Collection OverviewCollection Overview This seven-part series develops the Q g unification program-a first-order, spinor-driven synthesis of quantum mechanics and general relativity. Each paper expands one aspect of the same algebraic idea: that the spacetime metric is generated from within the Dirac algebra itself by the gravitational rotor field Q g (x) and its rapidity scalar ψ g (x).
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Drafts by Paul de haas