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