twin universe cosmology Matter, ghost matter, astrophysics. 2: Conjugated steady state metrics. Exact solutions. (p4)
3) Conjugated scalar curvatures.
From the general field equations system (1) + (2), we get :
(58)
R* = - R
In two conjugated points M and M*, belonging respectively to the folds F and F*, the scalar curvatures R and R* are opposite. We will call such geometries conjugated geometries. We can try to illustrate this concept by means of a didactic image. Consider figure 1: at the top, a "smoothed posicone"; at the bottom, a "smoothed negacone", facing each other. A smoothed posicone is constructed from a truncated cone, connected along a circle to a portion of a sphere (surface with constant angular curvature density).
Fig. 1: Didactic image of conjugated geometries (R = –R). The mass M is in the fold F. The fold F is empty.
Illustrated: a pair of conjugated points (M, M*).
The horse saddle is the equivalent, for negative curvature, of a portion of a sphere (surface with constant angular curvature density). A sphere contains a total curvature equal to 4π. A portion of a sphere contains an amount of angular curvature q given by:
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A cone is a surface containing a concentrated angular curvature point S, corresponding to a positive angular curvature q > 0. We construct it according to figure 2.
Fig. 2: Construction of a "posicone".
Definition of the angular curvature contained at the apex of the cone: If one draws a triangle formed by three geodesics, two cases arise. If it does not contain the apex, the sum of the angles is the Euclidean sum π. If it contains the apex, this sum equals π plus the corresponding punctual curvature q. See figure 3.
Fig. 3: Punctual positive angular curvature
located at the apex of a (posi)cone.
Similarly, we can construct a "negacone", as follows:
Fig. 4: Construction of a "negacone" with punctual negative curvature, located at S.
We can assemble a set of small posicones, corresponding to elementary curvatures dqi, and glue them together. See figure 5.
Fig. 5: Set of elementary posicones.
The angular curvature is an additive quantity. If the number of elements tends to infinity and the dqi tend to zero, the overall object tends to a bounded regular surface. On any portion of this surface, we can measure the angular curvature (the sum of the angles dqi). We can also define a local angular curvature density as follows:
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Thus, this set of assembled elementary posicones tends to a regular surface with a tangent plane. If C(M) is constant and positive on a surface, then it is a sphere or a portion of a sphere. The integral of the angular curvature density over the surface of the sphere gives its total curvature equal to 4π. If C(M) is zero, the surface is locally flat (plane, wall of a cone, cylinder, for example).