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The two folds are separated. We assume that particles follow the geodesics of each fold. Call "normal particles" the particles of normal matter, which travel in the fold F. Call "normal photons" the ones which travel in the fold F, along their special "null-geodesics".
Call "ghost matter" the matter which follows the geodesics of the fold F*.
Call "ghost photons" the photons which travel along their (special, null-geodesics) paths, in fold F*.
Light emitted by matter, in fold F, cannot be received by ghost matter, for photons cannot cross from fold F to fold F*.
"Ghost light", emitted by "ghost atoms" in the fold F*, cannot be received by matter, located in the fold F, for ghost photons cannot cross from the fold F* to the fold F.
As a conclusion, the objects located in F* are optically invisible from the fold F, and vice versa. We assume that these two worlds only communicate through gravitation.
The invisibility of the other fold's objects is based on pure geometric grounds.
Introducing a field equations system.
Classical general relativity was ruled by the Einstein field tensor equation :
(129)
S = c T
The tensor T can be considered as the input of the problem, the question being :
- What geometry goes with a given energy-matter field ?
A geometry is (locally) fully contained in a mathematical object called a metric g (which is a tensor), from which we can build the "geometric tensor S", and solve the field equation.
From the metric tensor g we can also build the geodesic system of the hypersurface-solution and "read it".
Here we have two interacting hypersurfaces, each owning its metric. Call g the metric of the hypersurface F (fold F) and g* the metric of the hypersurface F* (fold F*).
The hypothesis of conjugated curvatures gives :
S* = - S ****
S being the geometrical tensor built from the metric g and S* the geometrical tensor built from the metric g*.
(but this does not imply g* = - g).
The opposite curvatures hypothesis is justified in the paper :
** J.P.Petit & P.Midy : Geometrization of matter and anti-matter through coadjoint action of a group on its momentum space. 4 : The Twin group. Geometrical description of Dirac's anti-matter. Geometrical interpretations of anti-matter after Feynman and so-called CPT-theorem. Geometrical Physics B, 4, march 1998.**
on group theory arguments.
Induced geometry.****
The figure (128) corresponds to an induced geometry effect. Matter is present in the fold F, inside the (circular) border. It corresponds to grey area. In 3D this matter would fill a sphere with constant density.
The fold F* is totally empty. Inside the circular border, facing the grey disk, which belongs to F, we keep the surface white. It means that this negative curvature is due to the presence of a mass in the other fold. This is an induced geometry.
In (128) the mass is in F. We can describe it by a tensor T (local energy-matter content). The geometries correspond to equations :
**S = *c T
S = - c T i.e :
S* = - S
From this system one computes the geodesics of the two folds (see Geometrical Physics A, 5).
Important point :
Consider a geodesic of the fold F and the curve composed by its conjugated points M* on fold F*. They do not form a geodesic of F*
(131)
Conversely, consider a geodesic of fold F* and its image, point to point (conjugated point) in fold F. This is definitely not a geodesic of fold F.
(132)
We have given our Universe (supposed to be the fold F) a twin brother (supposed to be the fold F*). We have assumed that our Universe contains positive mass, which produces positive curvature in this fold F (or zero curvature in the regions where no energy-matter is present).
We have assumed that the system produced an induced geometry in the twin fold F*, with negative or zero curvature (conjugated curvature).
The two geometries are supposed to obey the field equations system.
(133) S = c T
(134) S* = - c T
where T is supposed to describe the energy-matter content of fold F.
From the projected geodesics (fig. 128) we see that a mass located in the fold F attracts a test-particle cruising in that fold, but repels a test-particle cruising in the twin fold F*, along a geodesic of this one, as if it repelled the many that could be located in that twin fold F* (supposed to follow geodesics of this fold).
Ghost photons follow (null) geodesics of the fold F*. As we can see, the presence of a mass M in the fold F produces a negative gravitational lensing effect in fold F*.
We have built the exact mathematical solution of the above field equations system. See :
J.P.Petit & P.Midy : Astrophysics of ghost matter. 2 : Conjugated steady state metrics. Exact solutions. Geometrical Physics A, 5, march 1998.
In fold F the solution corresponds to the classical so-called Schwarzschild solution. We suggest that we call the conjugated metric solution, describing the fold F*'s geometry : "nega-Schwarzschild".