twin universe cosmology Matter ghost matter astrophysics. 2: Conjugated steady state metrics. Exact solutions. (p6)
4) Dynamics.
From the external Schwarzschild and Schwarzschild-like metrics we can compute the geodesics, outside the sphere r = ro, which correspond, as usual, 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.
M being 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 positive. But a mass, if located in the 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 the fold F*, repels a test particle, cruising in F, at the vicinity. 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 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 this "negative mass" an 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 mass. (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 divergent lens, giving observational effects. This will be described in a future paper.
