Abstract
Relativistic localizing 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.
Highlights
In this paper, we present consequences of a local projective geometry of spacetime
The local spacetime geometry is Euclidean and the changes of scale are due to the additional transformations that allow to pass from the Poincaré group to the Weyl group
The fits presented in this paper, based on the modified Newton’s laws (31) were obtained from very simplified data and data processing
Summary
We present consequences of a local projective geometry of spacetime. This geometry is strongly suggested based on only purely metrological characteristics of systems of relativistic localization of events in spacetime [1,2,3]. As a result, considering in particular the normal Riemann coordinates that always exist on a (pseudo-)Riemannian manifold, any change of Riemann normal coordinates attached to a given fixed point is no longer only linear but can be homographic The mathematical formalism was presented in a very little synthetic form and difficult to access except for some mathematicians in the field This projective geometry that relativistic localizing systems unveil is truly inherent in spacetime and somehow superimposes itself on the underlying pseudo-Riemannian geometry. We conclude in the last section in which we indicate in particular other motivations that led to the publication of this note
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