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Conjugate Curvatures.
...How to grasp the concept of local curvature, positive or negative, in a three-dimensional space. Take a sphere. Drive a nail somewhere. Attach a string of length L to it, and fix a pencil to the other end. We can then trace a circle, which will be a parallel. Perform the same operation with a plane and a horse saddle.
...On a plane, the circumference is 2πL and the area of the disk is πL².
...On the sphere, both the circumference and the area of the cap are smaller. On a horse saddle, the circumference and the area enclosed by this closed curve are larger. Example: if we take a sphere of radius R and a length L equal to a quarter of the equatorial circumference, i.e., πR/2:
...The area of the disk is 3.875 times larger than the area of the spherical cap. Its circumference is 1.57 times larger than the equator.
...By performing similar measurements on a surface, one can determine whether the local curvature is positive or negative. The same applies in 3D. We then take a point, a string of length L, and draw... a sphere. If the area of this sphere is smaller than the Euclidean area 4πL², we conclude that the local curvature is positive. If this area is larger than the Euclidean area 4πL², we conclude that the local curvature is negative. The same conclusion holds for volume. Let us limit ourselves to these qualitative ideas. In three and four dimensions, one can define a quantity R, called the scalar curvature, which is computed from a curvature tensor.
...In the cosmological model we present, we decide to conjugate two sheets of the universe such that the values of the local scalar curvatures at conjugate points are opposite:
R* = -R
...This is the purely geometric way of viewing things. It is then easy to provide a didactic 2D image of this, subject to the usual caveats regarding the actual scope of such representations. This is the drawing in the figure below:
At the top, a blunted posicone. The local curvature is zero on the conical trunk and positive on the spherical cap.
At the bottom, a blunted negacone. The curvature is zero on the negaconical trunk and negative on the saddle.
...We have projected the object and its geodesics onto two Euclidean representation spaces. The first is that of an observer physically located in the sheet F, who can thus see the massive object, but not the witness particle moving through the sheet F*.
...The invisibility of an object located in one sheet to an observer in the other is purely geometric in nature. We assume that photons follow geodesics (special curves) within each sheet. Photons j travel through sheet F (our universe sheet), while photons j, which we may call "ghost photons," travel through sheet F*, the "ghost universe." The fact that the two sheets form a disjoint, disconnected set prevents any photon from one sheet from crossing over to the other.*
...The "operation" of such a geometric system is less complicated than it may seem.
...Sheet F has its own geometry, entirely described by a "metric" g, from which its system of geodesics is constructed. From this metric g, we can construct a geometric tensor S and identify it with a tensor T, which acts as the "source of the field," the origin of this curvature, by writing Einstein's equation:
S = c T
The geometry of the second sheet, such that its scalar curvature is inverted, corresponds to a metric g*, from which we can construct a geometric tensor S*. The inversion of curvature follows simply from:
S* = - S = -c T
...This does not mean that g* = -g. The equations are nonlinear. The metric g* also generates its own geodesics.
...Consider a geodesic in sheet F and draw the curve corresponding to the conjugate points in the other sheet. This is not a geodesic of that sheet.
Conversely:
...Where do we stand at this point? We have endowed our universe (assumed to be sheet F, our own spacetime) 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 our universe contains only positive masses and, more generally, particles with positive energy. We do not consider the possible presence of negative masses in our spacetime sheet. The tensor T is therefore either positive where energy-matter exists, or zero in a perfect vacuum. The local curvature of F is thus either zero or positive, but cannot be negative.
...On the other hand, the curvature of sheet F* (we will then speak of induced curvature) is either zero or negative.
...If there are particles in this sheet, we assume they also follow geodesics within it. What do we observe when looking at the figure above? The gray object, this mass present in our universe, in sheet F, behaves as a repulsive object (see the curvature of the geodesic trajectory) in sheet F*.
...We have constructed an exact mathematical solution corresponding to this pair of "conjugate metrics" (g, g*). [See on the website: paper Geometrical Physics B]. The solution g is identical to what we have called the exterior (outside the object) and interior (within the object itself) Schwarzschild metrics. We propose to call the second metric "Anti-Schwarzschild." [See on the website: Geometrical Physics A, 7, paper 2: Conjugated steady state metrics. Exact solutions.]
With "ghost matter".
From the perspective of conjugate geometries, we can reverse the situation and assume that a (positive) mass is present somewhere in sheet F*. It then creates a positive curvature, and the didactic 2D image of this geometry corresponds to a blunted cone, a Schwarzschild solution, but in sheet F*.
...The same remark applies to how observers from different sheets perceive the effect of this mass on a witness particle moving through their universe.
...Examining the above diagram allows us to deduce the laws of interaction between matter and ghost matter (ghost-matter), located in the second universe, the ghost universe.
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Two particles of matter attract each other.
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Two particles of ghost matter attract each other.
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Matter and ghost matter repel each other.
...We see that this differs from the scheme suggested by Souriau, in which particles of the second kind not only repelled those constituting our matter, but also repelled each other.
...The second geometry corresponds to the presence of positive masses m* in sheet F*. We can define a matter density ρ* > 0 there (or more precisely, ghost energy-matter, since the second sheet, the ghost universe, also contains "ghost radiation," ghost photons, and ghost neutrinos). The energy of ghost particles is positive, as is the pressure p*.
...From these quantities, we can construct a ghost energy-matter tensor T* (the most general energy-matter tensor is slightly more complex, but this schematic description suffices "for ordinary purposes").
The field equation giving the geometry in...