Contemporary Mathematics Volume 599, 2013 http://dx.doi.org/10.1090/conm/599/11945 On Dark Matter, Spiral Galaxies, and the Axioms of General Relativity Hubert L. Bray Abstract. We define geometric axioms for the metric and the connection of a spacetime where the gravitational influence of the connection may be interpreted as dark matter. We show how these axioms lead to the Einstein- Klein-Gordon equations with a cosmological constant, where the scalar field of the Klein-Gordon equation represents the deviation of the connection from the standard Levi-Civita connection on the tangent bundle and is interpreted as dark matter. This form of dark matter, while not quantum mechanical, gives virtually identical predictions to some other scalar field dark matter models, including boson stars, which others have shown to be compatible with the ΛCDM model on the cosmological scale. In addition, we quantify the already known fact that this scalar field dark matter, unlike the WIMP model, is automatically cold in a homogeneous, isotropic universe, suﬃciently long after the Big Bang. With these motivations in mind, we show how this scalar field dark mat- ter, which naturally forms dark matter density waves due to its wave nature, may cause the observed barred spiral pattern density waves in many disk galaxies and triaxial shapes with plausible brightness profiles in many ellip- tical galaxies. If correct, this would provide a unified explanation for spirals and bars in spiral galaxies and for the brightness profiles of elliptical galaxies. We compare the results of preliminary computer simulations with photos of actual galaxies. 1. Overview There are three ideas, each of independent interest, which, if correct, would create a new connection between differential geometry and astronomy, very much in the tradition of general relativity’s previous successes at describing the large-scale structure of the universe. We begin by discussing these ideas in general terms. Idea 1: Natural geometric axioms motivate studying the Einstein-Klein-Gordon equations with a cosmological constant. The Klein-Gordon equation is a wave type of equation for a scalar field which we propose as a model for dark matter. While this geometric motivation is new, modeling dark matter with a scalar field satisfying the Klein-Gordon equation is not (see [18] for a survey of scalar field dark matter and boson stars). Hence, ideas 2 and 3 apply to these other works as well. 2010 Mathematics Subject Classification. Primary 83C05, 83Cxx. c 2013 Hubert L. Bray 1

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