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

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.

Physical Justifications and Possible Astrophysical Manifestations of the Projective Theory of Relativity

“…But then, the question is to know what other manifestations of the projective geometry of spacetime could appear, and in particular in astrophysics. Two results have previously been obtained on this subject [22]: A foliation of spacetime with a structure similar to those of black holes, and fits of galactic rotational velocity curves. These results are presented briefly in Sections 2-4 but their difficult interpretations are discussed in Section 5 in greater depth and in a new way both from a mathematical point of view and from the point of view of physical implications and meanings.…”

“…As indicated in previous section, the spacetime manifold we denote by M is an emergent geometric structure arising somehow "naked", i.e., neither time orientation nor any Lorentzian metric are really defined on the emergent spacetime manifold M from localization; appart the fundamental, projective ground structure. Nevertheless, we can start with a few following geometric assumptions about the geometric structure of M [22]: (1) The spacetime manifold M is time-orientable and simply connected, (2) we provide M with an Euclidean metric ds 2 such that (M, ds 2 ) is the four-dimensional hyperbolic space H 4 ⊂ PR 41 , (3) to each event e ∈ M, a system of four Riemann normal coordinates (u i p ) ≡ p ∈ M can be attached such that u α e = 0 (i = 1, . .…”

“…A second possible manifestation of projective geometry could be a modification of the Newton's law of gravitation. Indeed, it can be shown that due to invariance with respect to the group G + τ , we can deduce [22] a modified force of gravitation #" F such that…”

“…where M(r) is the mass enclosed in a sphere of radius r . Fits of rotational velocity curves have been obtained for around 10 galaxies [22] recorded in the SPARC astronomical database and 18 galaxies from the THINGS database. In the first "SPARC case", no mass models were used to deduce the masses M(r) for each galaxy, and in the second "THINGS case", Formula (5) was given without theoretical or physical justifications and from a pure phenomenological viewpoint just because of the quality of the fits obtained.…”

“…Fits of the rotational velocity curve[22] of galaxy LSBC F583-01 (SPARC database). Green solid curve: (α, β) = (0.0000, 20.8025) .…”