astrophysical twin universe and cosmology
Matter ghost matter astrophysics.
5: Results of 2D numerical simulations. VLS.
About a possible scheme for galaxy formation.
.(p2)
Another method, also mentioned, introduces a distance truncation at the antipode of each point. Note that our square is an Euclidean, flat torus, with zero curvature everywhere. See figure 3.
Fig. 3 :** The "Euclidean torus".** We have indicated the center P of the square. From a geometric 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 represent the geodesics of the Euclidean torus. The lower-left image of figure 3 is incorrect, because 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 also zero. Same for a mass located in (H, K) or (M, N). See figure 4.
Fig. 4 :** On a torus, a point P has three antipodal points :**
(A, B, C, D) (M, N) (H, K)
The corresponding geodesic path lengths are fundamentally different :
(1)
Note that, on a torus (whatever its curvature), we have an infinite number of geodesics joining two given points P and Q, one being the shortest. 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 indicated the shortest path. The non-Euclidean torus representation is just a topological description, because this torus has 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 indicated a longer path.
Fig.7 : A longer path, from point P to point Q.
We see that things are not as simple as they seem at first glance.
If we place the mass points on a S2 sphere, a unique geodesic connects two given points. See figure 8.
Fig. 8 : Two points on a sphere, connected by a single geodesic.
When calculating 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 diametrically opposite positions. 