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The two folds are separated. We assume that particles follow
the geodesic of each folds. 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 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 Fto 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's 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 andS* 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 Feynmann and so-called CPT-theorem. Geometrical Physics
B, 4, march 1998.
on group's 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 tensorT (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 Fand 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 definitively 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 produce positive curvature in this
fold F ( or zero curvature in the regions where no energy-matter is present).
We have assumed that 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's
content of fold F.
From the projected geodesics ( fig. 128 ) we see that a mass located in the
fold Fattracts a test-particle cruising in that fold, but repels a test-particle
cruising in the twin fold F*, alongs 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 follows (null) geodesics of the fold F*. As we can see, the
presence of a mass Min the fold Fproduces a negative gravitational lensing
effect in fold F*.
We have built the exact mathematical solution of the above field equations
sustem. See :
J.P.Petit & P.Midy : Matter ghost matter astrophysics. 2 : Conjugated
steady state metrics. Exact solutions. Geometrical Physics A , 5 , march 1998.
In Fold Fthe 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".
