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Geometries induced by ghost matter.****
In the preceding section we have studied the conjugated geometries due to the presence of a constant density, positive mass M, located in fold F. Now let us suppose that a (constant density r* > 0) positive mass M*>0 is present in the fold F*. We assume that in this portion of the universe the conjugated region of F is empty.
Then T* describes the content of energy-matter of the non-empty portion of fold F*. The corresponding field equation system is :
S = - c T*
S* =** *c T
The geometries simply commute :
(135)
Looking at figure (135) we see that a mass M*, located in fold F*, attracts ghost masses, which follow geodesics of this twin fold, and repel normal masses, following geodesics of fold F.
Looking at figure (135) we see that fold F gains an induced geometry, due to the presence of a ghost mass M* in its fold F*.
The interaction laws.
From (128) and (135) we can deduce the interaction laws :
-
Matter attracts matter
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Ghost matter attracts ghost matter.
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Matter and ghost matter repel each other.
See also :
J.P.Petit & P.Midy : Astrophysics of ghost matter-matter. 1. The geometrical framework. The matter era and the newtonian approximation. Geometrical Physics A , 4 , march 1998.
In this paper we show, in addition, that the interaction forces is newtonian.
We see this is different from the schema proposed by J.M.Souriau, where two particles of the second species repel each other.
In our schema, we see that all masses m and m* are positive. But induced geometry phenomenon makes it possible to have local negative curvature, somewhere, which was forbidden in classical general relativity.
To sum up we can write the field equation system :
(136) **S = *c (T - T)
(137) S* =** *c (T - T) ** ** which gives inverse scalar Riemann curvatures :
(138)
R = - R* ****
If the local curvature is positive in fold F, it means that :
(139) T > T*
or :
r > r *
Then the conjugated curvature is negative in the adjacent portion of F*.
Conversely, if the local curvature is negative in fold F, it means that
(140) T < T*
or: r < r *
Then it is positive in fold F*.
If the local curvature is zero, in fold F, it means that the curvature is null too in the adjacent portion of the twin fold F*.
In addition , either T = T* = 0 or : r = r * = 0 T = T* ( r = r *)
About classical general relativity tests.
Matter and ghost matter repel each other. A galaxy is a concentration of matter. Then the adjacent portion of the twin space F* is extremely rarefied, for masses m* have been pushed away. In the vicinity of the Sun the ghost matter density ( r* or T* ) can be neglected. Then the field equations system reduces to :
(141)
(141 bis )
(141) is the Einstein equation, from which we build all the local classical tests of general relativity. The Einstein equations become the limit case when the ghost matter density tends to zero.