The missing mass problem
(p5)
.
The interpretation of the solution
From figure 2, we see that the potential
Y tends to a constant at infinity. In the classical Eddington solution, the potential has a logarithmic growth. Figure 3 shows the association of a matter cluster, located in the region
s
, surrounded by a smooth hollow located in the region
s*.
In both regions, matter attracts matter. However, the negative sign, from the field equation and the Poisson equation, causes matter and the "antipodal matter" to repel each other. This helps to confine the cluster. For a given thermal velocity, the necessary amount of matter to balance the pressure force is reduced. The smooth halo acts like a corset.
A field equation provides a macroscopic description of the universe. It does not take into account the corpuscular nature of matter. The model implies that particles and antipodal particles coexist in very distant, antipodal regions of space. In fact, their natures are identical. The physical meaning of the field equation is as follows: the particles and antipodal particles interact through gravitational effect, but not through electromagnetic effect. We assume that the antipodal particles, clusters, rings, and other antipodal structures are not observable with a telescope or a radio telescope. The observation of antipodal structures would presumably require some sort of gravitational telescope.
From equation (22), clusters can be located in the antipodal region. Therefore, associated large halos, surrounding wide rarefied regions, should also exist in the observable universe. Indeed, they do, as it corresponds, in our view, to the observed large-scale structure of the universe: galaxies seem to be arranged around large rarefied bubbles. According to our model, large clouds of antipodal matter should exist in the corresponding antipodal regions.
The universe was assumed to have a S3
x
R1 topology. The reader probably has some difficulties in understanding this strange three-dimensional geometry. In fact, the sphere S3 is simply represented as the double cover of a projective space P3. In such a configuration, each point
s
of the sphere is associated with its antipode A(s). The situation is analogous to that of a sphere S2 covering a projective space P2, which can be represented in our space R3 by the well-known Boy surface.
Figure 9: A pair of antipodal points on a sphere S2 and the Boy surface, image of the projective space P2
On figure 10, we have represented the equator of a sphere and its location on the Boy surface.
Figure 10: The vicinity of the equator of a 2-sphere and its location on the Boy surface
.
Figure 11 shows how the equator of a S2 sphere can be sewn onto itself along a Möbius strip with three half-turns. Locally, the surface can be assimilated to a fibered manifold whose fiber has two values, +1 and -1.
Figure 11: Enantiomorphic image corresponding to the cover of a Möbius strip.
In a 3-sphere S3, if one follows a geodesic, the antipodal point is at the halfway. If the 3-sphere is immersed in a four-dimensional space, it is possible to make any point coincide with its antipode. These pairs of points are associated through the antipodal differential involutive mapping A, but they are not identified.
As shown in figure 12, we can continuously go from a "gruyère" structure to a cluster structure. This peculiar feature had already been illustrated previously using 2D numerical simulations. When a region of space is placed "in front" of the antipodal region, as suggested in figure 12, the clusters nest in the "holes".
Figure 12: Two-dimensional image of the global large-scale structure of the universe
Figure 13: The interaction between two antipodal regions
This effect could act at the level of the galactic structure, as suggested in figure 14, each galaxy nesting in a "hole" of the conjugated antipodal region.