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 Mand
M*, which belong to the folds Fans F*, the scalar curvatures Rand R* are
opposite. We will call such geometries conjugated ones. We
can try to illustrate this concept through a didactic image. Consider the
figure1. Up is a smoothed "posicone", down a "smoothed negacone", face to
face. A smoothed posicone is built with a truncated cone, linked along a circle
to a portion of sphere (constant curvature density surface).
Fig. 1 :
Didactic image of conjugated geometries ( R * = - R)
The mass Mis in the fold
F . The fold F*
is empty.
Shown : a couple of conjufated points (M,M*).
The horse saddle is the equivalent
of a portion of sphere, for negative curvature (constant angular curvature
density surface). A sphere contains a total curvature equal to 4p.
A portion on a sphere contains an amount of angular curvature
q which
is
(59)
A cone is a surface with contains
an (angular) concentrated curvature point S , corresponding to a positive
angular curvature q >
0. We build it according
to the figure 2.
Fig. 2 : Building a
"posicone".
Definition of the angular curvature
contained in the summit of the cone : If one draws a triangle with three geodesics,
we have two cases. If it does not contain the summit, the sum of the angles
is the euclidean sum p.
If it contains the summit this sum is p
plus the corresponding punctual curvature q
. See figure 3 .
Fig. 3)
: Punctual positive angular curvature
located at the summit of
a (posi)cone.
Similarly we can build a "negacone", as follows :
.
Fig.4 : Building
a "negacone"
with punctual negative curvature,
located in S.
We can build a set of small posicones,
corresponding to elementary curvature dqi
and glue these objects together.
See figure 5.
Fig. 5
: set of elementary posicones.
The angular curvature is an additive
quantity. If the number of elements tends to infinite and the dqi
tend to zero, the global object tends to a bounded regular surface. On any
portion of the surface we can measure the angular curvature (the sum of the
angles dqi
). We can also define
a local angular curvature density as follows :
(60)
Then this set of joined elementary posicones tends to a regular surface, with
tangent plan. If C(M) is constant and positive on a surface, its a sphere,
or a portion of a sphere. The integrated angular curvature density, over the
surface of the sphere, gives its total curvature 4p.
If C(M) is zero, the surface is locally flat (plan, wall of a cone, cylinder,
for example). .
