The Clifford Conjecture and Cold Dark Matter

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This paper presents a proof of W.K. Clifford's 1876 conjecture* that ``...[the property of spacetime] being curved or distorted is constantly being passed from one portion of space to another after the manner of a wave...'' The modern statement of this as a theorem is: (Clifford Gravitational Potential Equations) Every affinely connected 4D mass-spacetime (MST) manifold with Minkowski signature tangent space, has local (vector) curvature field potentials, densities and associated MST 4-currents with evolution constrained by the full Bianchi identities, inhomogeneous wave equations and current conservation laws. These relations are valid over the whole manifold for the trace of the curvature potential matrix. This allows us to show the structure of the Riemann curvature of a 4D MST is naturally interpreted in terms of mass densities and currents and gravito-dynamical fields. This leads to a metric-free fundamental law of dynamical gravitation, linear in two field quantities. An application of the present theory to a 4D MST manifold is given for a compact Friedmann-Lemaitre-Robertson-Walker(FLRW)-model of galactic condensation, involving the relative contraction of a spacetime manifold about a galactic center. This provides a prediction of the distribution of the cold dark matter (CDM) continuum of the galaxy model. The dynamical gravitational field lines have the appearance of swirling axial magnetic field lines with polar jets developing. The theory gives an extension of Newton's static gravitation theory to a dynamical one and a solution to at least part of the galactic missing mass problem. *W.K. Clifford, Camb. Phil. Soc. Proc. 11, 157 (1876)

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