twin universe cosmology Twin Universes cosmology (p 3)
4) Inverse gravitational lensing
...The problem of gravitational lensing must be reconsidered. As suggested in the previous paper [1], in the present model the confinement of galaxies is mainly due to the action of the surrounding antipodal matter, located in the twin fold, in order to account for the strong missing mass effect. Numerical simulations have provided a description of a galaxy surrounded by halos of antipodal matter [1]. See figure 7.
Figure 7: Concentration of mass confined by the action of the surrounding antipodal matter, from 2D numerical simulations.
...As a confirmation of this confinement effect, if we remove the antipodal matter from the system, the central object dissipates immediately. Although this figure focuses on the surrounding halo, all the surrounding antipodal matter contributes to this confinement effect, so that we can schematically represent the galaxies as nested in some sort of holes of the antipodal matter, as suggested in figure 8. The intensity of the confinement effect obviously depends on the density r* of the antipodal matter distribution, which should be at least ten times larger than r.
Figure 8: Galaxies nesting in a wide antipodal matter cloud (the galaxy and the antipodal matter repel each other)
...Classically, matter "attracts" photons and produces gravitational lensing. The trajectory of photons, bent by the presence of a positive point mass, can be calculated from the Schwarzschild solution:
(3)
...Notice that m is an arbitrary constant of integration. For weak fields and slowly moving bodies, we can relate the goo term of the metric to the gravitational potential Y through:
(4)
The gravitational potential, due to a mass M is:
(5)
whatever this mass M would be positive or negative. If M is negative, it repels the test particle. Then
(6)
whence:
(7)
If M is positive, the characteristic Schwarzschild length is (8)
...As pointed out above, m is nothing but an arbitrary constant of integration in the Schwarzschild solution. If we take m < 0, then the associated mass M becomes negative. We can define a characteristic length, positive (the Schwarzschild radius Rs) from:
(9)
The trajectory, in polar coordinates, corresponds to:
(10)
See reference [10], page 203. For the photon, following the null geodesics, we get:
(11)
j is the polar angle for this plane trajectory. u = 1/r
A positive mass (M > 0; m > 0) produces a positive gravitational lensing:
Figure 9: Classical (positive) gravitational lensing
...For a test particle, located in one fold, a mass located in the adjacent fold behaves like a repulsive negative mass (M < 0; m < 0) and then produces a negative lensing effect:
Figure 10: Negative lensing effect due to a "negative" mass ** ** Notice that these hyperbolic paths are familiar to specialists of plasma physics (e-e or p-p scatterings)
