Matter ghost matter astrophysics.
2:
Conjugated steady state metrics. Exact
solutions. (p6)
- Dynamics.
From the external Schwarzschild
and Schwarzschild-like metrics we can compute the geodesics, outside the sphere
r = ro, which correspond, as usal, to plane trajectories [2].
(63) Fold F, matter trajectories
(64) Fold F, null geodesics, photon
path.
(65) Fold F*, matter trajectories :
(66)Fold F*; null geodesics, photon
path.
With :
j =
polar angle q = 1/r
b , l and h are path parameters.
Mbeing the total mass contained in the sphere r = ro .
(63) gives quasi-Keplerian trajectories (elliptic, circular, parabolic and
hyperbolic)
(64) gives hyperbola-like trajectories (positive lensing effect).
(65) gives hyperbola-like material test particle trajectories
(66) gives hyperbola-like null geodesic trajectories (negative lensing effect).
In that model, all the masses
and energies are positives. But a mass, if located in th other fold, acts
as a negative one, through an "anti-Newton law". See papers [2], [4] and [5].
What happens if the mass is located
in the other fold ? The geometries are simply exchanged. See figure 10.
Fig. 10 : Matter located
in F*. Didactic image.
The figure 10 is only a didactic
image, for we deal with 4d geometry. Some ghost matter, located in thbe fold
F*, repels a test particle, cruising in F, at the visinity. Conversely, a
test particle, located in F*, is attracted. We get the same conclusion, about
dynamics, that in the paper [6]. In this model the local curvature may be
positive, negative or null.
In both cases the presence of
a mass in a fold induces a conjugated geometry in the other fold. We will
call it an induced geometry . Depends on the sign of
(r - r*).
- (r - r*)
0 goes with a positive curvature
in F, negative in F*.
-
(r - r*)
< 0 goes with a negative curvature
in F, positive in F*. -
(r - r*)
= 0 (either because r
= r*
or r
= r*
= 0) goes with a null curvature
in both folds (Minkowski metrics in such steady or quasi steady conditions).
In general relativity, a mass bends
space, produces positive contribution to the curvature. If we consider a portion
of space which is empty, except the presence of a mass, space will be flat outside,
and curved inside. Here we get a third possibility. In a portion of space which
looks perfectly empty, if if there is some evidence of negative lensing effect,
it means that some mass is present in the other fold, which behaves like a negative
one. We call call this "negative mass" a apparent mass. All particles
own a positive intrinsic mass. Some have positive apparent masses. It
means that they are in the fold where the observer lies. Others, located in
the other fold, own a negative apparent masses. (with respect to this
observer). They behave like positive apparent masses for an observer located
in the other fold.
In this model, the interaction of particles of distinct folds is only gravitational.
They cannot collide. Photons, following null geodesics of a fold, cannot be
absorbed by particles located in the other fold.
In our fold, a zero mass neutrino
can pass through a star, like the sun, so that we may use an internal Schwarzschild
solution to compute its path. But presently we have no neutrino telescopes,
so that such a calculation shows little interest. But, if important masses
are located in the ghost fold, they will bend the trajectories of photons
cruising in ours and act like diverent lens, gives observational effects.
This will be described in a future paper.
