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Conjugate curvatures.
...How to grasp the idea of local curvature, positive or negative, in a three-dimensional space. Let us take a sphere. Plant a nail somewhere. Attach to it a string of length L and fix a pencil to the other end. We will then be able to trace a circle, which will be a parallel. Let us carry out the same operation with a plane and a horse saddle.
...On a plane the perimeter is 2 p L and the area of the disk p L2.
...On the sphere the perimeter and the area of the cap are smaller. On a horse saddle the perimeter and the area delimited by this closed curve are larger. Example: if one takes a sphere of radius R and a length L equal to a quarter of the equatorial perimeter, that is p R/2:
...The area of the disk is 3.875 times larger than the area of the spherical cap. Its perimeter is 1.57 times larger than the equator.
...By carrying out analogous measurements on a surface one can know whether the local curvature is positive or negative. Analogous situation in 3d. One then takes a point, a string of radius L and traces... a sphere. If the area of this sphere is smaller than the Euclidean area 4 p L2 one will conclude that the local curvature is positive If this area is smaller than the Euclidean area 4 p L2 one will conclude that the local curvature is negative. Same conclusion for the volume. Let us content ourselves with these qualitative ideas. In three and four dimensions one can define a length R, called scalar curvature, which is calculated from a curvature tensor.
...In the cosmological model that we present, it is decided to conjugate two sheets of universe such that the values of the local scalar curvatures at conjugate points are inverse:
R* = - R
...This is the purely geometrical way of considering things. It is then easy to give a didactic 2d image of it, modulo the usual reservations as to the real scope of such representations. This is the drawing of the following figure:
At the top, a blunt posicone. The local curvature is zero on the cone trunk and positive in the spherical cap.
At the bottom a "blunt negacone". The curvature is zero on the negacone trunk and negative in the horse saddle.
...The object and the geodesics have been projected onto two Euclidean representation plane-spaces. The first is that of an observer physically located in sheet F, who will thus be able to see the massive object, but not the test-particle traveling in sheet F*.
...The invisibility of an object located in one sheet by an observer located in the other is purely geometrical in nature. It is assumed that photons follow (particular) geodesics of each sheet. Photons j travel in sheet F (our sheet of universe) and photons j*, which may be called "ghost photons" travel in sheet F*, the "ghost sheet" (ghost universe). The fact that the two sheets form a disjoint, non-connected set forbids any photon of one sheet from passing onto the other.
...The "operation" of such a geometrical system is less complicated than it appears.
...Sheet F has its geometry, which is entirely described by a "metric" g, from which its system of geodesics is constructed. From this metric g one can construct a geometrical tensor S and identify it with a tensor T, which is "source of the field", origin of this curvature, by writing the Einstein equation:
S = c T
The geometry of the second sheet, such that its scalar curvature is inverse, corresponds to a metric g*, from which one can construct a geometrical tensor S*. The inversion of curvature follows simply from:
S* = - S =- c T
...Which will absolutely not mean that g* = - g. The equations are non-linear. The metric g* also generates geodesics.
...Let us consider a geodesic of sheet F and plot the curve corresponding to the conjugate points, in the other sheet. It is not a geodesic of the latter.
Conversely:
...At this stage, where are we? We have endowed the universe (assumed to be sheet F, our own space-time) with a twin brother. The matter present in our universe (the tensor T) determines its geometry, but it also determines that of the twin. We assume that our universe contains only positive masses and, more generally, particles possessing positive energies. We do not consider the possible presence of negative masses in our space-time sheet. The tensor T is therefore either positive, where there is energy-matter, or zero where a high vacuum reigns. The local curvature of F is therefore either zero or positive, but cannot be negative.
...On the other hand the curvature of sheet F* (one will then speak of induced curvature) is either zero or negative.
...If particles exist in this sheet, we assume that they also follow geodesics of it. Now what do we observe by looking at the figure above? The grayed object, this mass present in our universe, in sheet F, behaves as a repulsive object (see the curvature of the trajectory-geodesic) in sheet F*.
...We have constructed an exact mathematical solution corresponding to this couple of "conjugate metrics" (g, g*). [See in the site: paper Geometrical Physics B], . The solution g is identical to what we have called the exterior (outside the star) and interior (inside the star itself) Schwarzschild metrics. We propose to call the second metric "Anti-Schwarzschild". [See in the site: Geometrical Physics A, 7, the paper 2: Conjugated steady state metrics. Exact solutions. ]
With "ghost matter".
In this perspective of conjugate geometries, one can invert the situation and assume that a (positive) mass is present somewhere in sheet F*. It then creates a positive curvature there and the didactic 2d image of this geometry corresponds to the blunt cone, to a Schwarzschild solution, but in sheet F*.
...Same remark for the way in which observers of different sheets perceive the effect of this mass on a test-particle traveling in their universe.
...The examination of the above diagram allows us to bring out the laws of interaction between matter and ghost-matter, located in the second universe the ghost universe.
-
Two particles of matter attract each other
-
Two particles of ghost-matter attract each other.
-
Matter and ghost matter repel each other.
...One sees that this is different from the scheme suggested by Souriau, according to which the particles of the second species not only repelled those constituting our matter, but repelled each other.
...The second geometry corresponds to the presence of positive masses m*, in sheet F*. One will be able to define there a density of matter matter r* > 0 (or more precisely of ghost energy-matter, since the second sheet, the ghost universe, also contains "ghost radiation", ghost photons and ghost neutrinos). The energy of the ghost particles is positive, as is the pressure p*.
...From these quantities one can construct a ghost energy-matter tensor T* (the most general energy-matter tensor is a little more than that, but this schematic description suffices "for usual needs").
The field equation giving the geometry in sheet F* is:
S* = c T*
The one giving the geometry of F* is:
S = -c T*
...These are the two equations from earlier, inverted. In sheet F the geometry is induced, in the sense that it is created by matter M* located in F* and therefore invisible from sheet F.
...One can now consider that the two sheets are each populated by their own matter, of densities r and r*.
The conjugate metrics are then deduced from the system of equations:
S = c (T - T*)
S* =c (T* - T)
which indeed gives the inverted curvatures:
R* = - R
Such a universe can therefore present, in its sheets, any curvature, positive, negative or zero.
If the curvature is locally positive in sheet F it is because:
T > T*
or:
r > r
*
it is then negative in the adjacent, conjugate region of sheet F*.
If the curvature is locally negative in sheet F it is because:
T < T*
or:
r < r
*
it is then positive in the adjacent, conjugate region of sheet F*.
If the curvature is zero in a region of sheet F, it is also zero in sheet F*, which will mean either that:
T = T* = 0
(the two regions are empty)
or that:
T = T* (
r = r
*)
Newton and anti-Newton.
Newton's law is deduced from the Einstein equation by assuming:
-
That the local curvature is weak
-
That the velocities are weak compared with the speed of light
-
That the system is "quasi-stationary" (cosmologically speaking).
...It is deduced in the same way from the system of coupled field equations presented above. See:
J.P.Petit and P.Midy: Matter ghost matter astrophysics. 1: The geometrical framework. The matter era and Newtonian approximation. [see on the site: Geometrical Physics A , 4 , 1998, section 4. ]
One therefore has the following dynamics:
-
Two particles of matter m1 and m2 attract each other according to Newton's law.
-
Two particles of ghost matter, of "ghost-matter" m*1 and m*2 attract each other according to Newton's law.
-
Two particles m and m*, belonging to different sheets repel each other according to "anti-Newton".
About the tests of General Relativity.
...The model is not in contradiction with these. Indeed, as matter and ghost matter repel each other, where there is a concentration of matter (for example our galaxy), in the adjacent, conjugate portion of the ghost universe, a high vacuum reigns: matter has cleaned up there by repelling the ghost masses far away.
In the vicinity of the sun the tensor T* (ghost matter) is therefore negligible compared with T (matter) and the field equation becomes:
S @
c T
that is to say the Einstein equation, which then presents itself as an approximate form of a "twin" equation.
Conclusion of this geometry part.
**...**Everything is now in place to attempt to exploit a new idea, that of a universe in permanent interaction with a ghost universe, whose existence would reveal itself to us only through the action on the gravitational field.
...In the paper: "J.P.Petit and P.Midy: Geometrization of antimatter through coadjoint action of a group on its momentum space. 3: Twin group. Matter anti-matter duality in the ghost space. Reinterpretation of the CPT theorem. [See on the site: Geometrical Physics B, 1-3 ], 1998.", the twin structure is approached differently: through group theory (twin group). One will see in particular that the matter antimatter duality exists in both sheets. In the ghost universe there exists ghost matter and anti-ghost matter.
...We know the Newtonian approximation. We can therefore carry out numerical simulations. We can also attempt to construct analytical solutions. We will also have to, in order to be complete, produce a model of cosmological evolution, which will be called Twin Bang, since it affects both universes. .
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