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Galaxies 2018, 6(3), 83; https://doi.org/10.3390/galaxies6030083

Applications of a Particular Four-Dimensional Projective Geometry to Galactic Dynamics

Institut de Physique de Nice—UMR7010-UNS-CNRS, Université de Nice Sophia Antipolis, Site Sophia Antipolis, 1361 route des lucioles, 06560 Valbonne, France
Received: 25 June 2018 / Revised: 31 July 2018 / Accepted: 1 August 2018 / Published: 3 August 2018
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Abstract

Relativistic location systems that extend relativistic positioning systems show that pseudo-Riemannian space-time geometry is somehow encompassed in a particular four-dimensional projective geometry. The resulting geometric structure is then that of a generalized Cartan space (also called Cartan connection space) with projective connection. The result is that locally non-linear actions of projective groups via homographies systematically induce the existence of a particular space-time foliation independent of any space-time dynamics or solutions of Einstein’s equations for example. In this article, we present the consequences of these projective group actions and this foliation. In particular, it is shown that the particular geometric structure due to this foliation is similar from a certain point of view to that of a black hole but not necessarily based on the existence of singularities. We also present a modified Newton’s laws invariant with respect to the homographic transformations induced by this projective geometry. Consequences on galactic dynamics are discussed and fits of galactic rotational velocity curves based on these modifications which are independent of any Modified Newtonian Dynamics (MOND) or dark matter theories are presented. View Full-Text
Keywords: projective geometry; general relativity; relativistic localizing systems; relativistic positioning systems; rotational velocity curve; hyperbolic space; Newton’s law; black hole projective geometry; general relativity; relativistic localizing systems; relativistic positioning systems; rotational velocity curve; hyperbolic space; Newton’s law; black hole
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Rubin, J.L. Applications of a Particular Four-Dimensional Projective Geometry to Galactic Dynamics. Galaxies 2018, 6, 83.

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