الكون المزدوج الفيزياء الفلكية والكونيات

twin universe astrophysics and cosmology

Matter ghost matter astrophysics.
5 : Results of numerical 2d simulations. VLS.
About a possible schema for galaxies' formation.

.(p2)

Another method, also mentioned, introduces a distance truncature at the antipode of each points. Notice that our square is an euclidean, flat torus, with null curvature everywhere. See figure 3.

Fig. 3 :** The "euclidean torus".** We have figured the center P of the square. From a geometrical point of view, the points A , B , C and D must be identified to an antipode of P on the torus. On our square, straight lines figure geodesics of the euclidean torus. The lower-left image of figure 3 is not correct for we simply cannot draw a "flat torus". The gravitational action of a mass located at the antipodal point (a,B,C,D) on the point P is zero too. Same thing for a mass located in (H,K) or (M,N) . See figure 4.

Fig. 4 :** On a torus a point P owns three antipodal points :**

(A,B,C,D) (M,N) (H,K)

The corresponding geodesic pathes lengths are basically different :
(1)

Notice that, on a torus (whatever it owns curvature or not) we have an infinite number of geodesics joining two given point P and Q, one being the shortest. The figure 5 corresponds to the spatially periodic description.

Fig. 5 :** Two geodesics joining two distinct points P and Q.** Spatially periodic description.

On figure 6 we have figured the shortest path. The non-euclidean torus representation is just a topologic description, for this torus owns local positive and negative curvature. A geodesic of such a torus is obviously not a geodesic of our "flat torus".

**Fig. **6 : The shortest path from P to Q.

On figure 7 we have figured a longer path.

Fig.7 : A longer path, from point P to point Q.

We see that the things are no so simple they look at a first glance.

If we figure the mass points on a S2 sphere, a unique geodesic links two given points. See figure 8

Fig. 8 : Two points on a sphere, linked by a single geodesic.

When computing the corresponding gravitational interaction we must consider two lengths :
(3)

d = a R d' = R ( 2ap - a )

If the two points attract each other, they tend to meet. Conversely, if they repel each other, they tend to occupy diametrally opposed positions.