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 Pof
the square. From a geometrical point of view, the points A , B , Cand Dmust
be identified to an antipode of Pon 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 Pis
zero too. Same thing for a mass located in (H,K) or (M,N) . See figure 4.
Fig. 4 : On a torus
a point Powns 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 Pand Q, one being the shortest. The figure 5 corresponds
to the spatially periodic description.
Fig. 5 :Two geodesics joining two distinct points Pand 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 Pto Q.
On figure 7 we have figured a
longer path.
Fig.7
: A longer path, from point Pto 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.
