In formers papers ([22], [23] and [24]) we have developped an attempt to describe particles in a ten dimensional space :
(148)

with six additional dimensions : an extension of the five dimensional Kaluza-Klein space (see reference [25], chapter 5 "La Relativité à 5 dimensions", page 413, where de inversion x5 ® - x5, the inversion of the Kaluza coordinate, is identified to the charge conjugation). This work was based on a group, suggesting that the associated couple (space-time plus twin space time) corresponds to a CPT symmetry, the C-symmetry corresponding to :
(148)

(inversion of the six additional Kaluza like dimension, extension of Souriau's work [25]. This showed the duality matter-antimatter holds in the twin fold ([23] and [24]) and provides a new geometric interpretation of the so-called CPT theorem [24].

    Schwarschild space-time can be imbedded in a ten dimension space, which suggests that these additonal dimensions could correspond to quantum features. The subsequent symmetry corresponds to the group :
(149)

    It's a two component group, which is the isometry group of the metric, considering the Schwarzschild geometry as imbedded in a ten-dimensional space.
Introducing :
(150)

we get group whose dimension is 4.

    The value b = - 1 corresponds to C-symmetry. It means that in each space-time fold any geodesic has a "mirror image" z ® - z , which corresponds to an antimatter particle following the same path. Matter antimatter duality holds in boths half folds.

    b = m = -1 corresponds to CPT symmetry. When matter, which belongs fold F, is poured into a "black hole" and raises out of the associated "white fountain", although its proper time increment Ds is not changed (it cannot be), this particle, cruising in a the CPT-symmetrical fold F*, becomes CPT-symmetric. It is still a particle of matter. The transfer (including transient, fast hypothetical hyperspatial transfer, evoked above) does not transform matter into antimatter, and vice-versa, but the "appearent mass" m* = - m (see reference [15] and equation (110)) is changed.

    In "orthochron" fold F, matter and antimatter own positive mass and energy, as shown in references [23] and [24]. But, when they are transfered towards the twin fold F*, which owns an opposite time marker t* = - t, they behave like negative mass particles with respect to particles of the first one, see section 14.

Conclusion.

    Starting from the so called black hole model, considered as a physical interpretation of Schwarzschild geometry, we have reconsidered the problem of the fate of a neutron star when this lats overcomes its limit of stability. We have first presented a new geometric tool : hypertoric geometry, through 2d and 3d examples (section 2). We have shown that pathlogies associated to metrics, arising from their line element, expressed in a given coordinate system, can be cured by a more suitable choice, phrased in terms of "local topology". For example we have showed that in the two given examples, the 2d surface and 3d hypersurface, whose isometry groups were O2 ans O3, these geometric structures were not simply connected.

    We have extended the method to Schwarzschild geometry and showed that the singular features could be fully eliminated, considering not simply connected space time. We have given the Schwarzschild geometry a different physical significance, this last being considered as a bridge linking two universes, ours and a twin one.

    We have showed that the "freeze of time", keystone of the blach hole model, was a simple consequence of a peculiar time marker choice. Using another one, inspired by Eddington's work (1924) we have built a completely different model, with radial frame dragging (similar to the azimutal frame-dragging of the Kerr metric). We have showed that the Schwarzschild solution can be interpreted as a "space bridge" between two universes, two space-times, this link working as a one-way tunnel. We have showed that the transit time of a test particle was finite and short, which made the classical black hole model questionable.

    Extending the isometry group of the Schwarzschild metric we have showed that the two Universe were enantiomorphic (P-symmetric) and owned opposite time markers (t* = -t). Using groups' tool : the coadjoint action of a group on its momentum space, we have given the physical significance of this "time inversion", through Schwarzschild sphere, considered as a throat surface. When a positive mass particle passes through the space bridge, its contribution to the gravitational field is inversed : m* = -m (as shown by J.M.Souriau in 1974 the inversion of the time marker is equivalent to the inversion of the mass and energy).

    As the question of the fate of a destabilized neutron star became a still open problem, we have presented a project of an alternative model : the hyperspatial transfer of a part of the neutron star, through a space bridge, matter flowing towards the twin universe at relativistic velocity.

    By the way we have recalled some well-known defects of the Kruskall model, particularly the fact that it is not asymptotically Lorentzian at infinite.

    We have presented some attempts to imbed subsets of Schwarzschild geodesics, with peculiar parameters (zero velocity at the infinite, radial paths in plane q = p/2). We have suggested to consider Schwarzschild geometry as an hypersurface, imbeded in a ten dimensional space. Linking the present work to former ones, based on group theory, we have extended the model to a CPT symmetric one. Matter antimatter duality holds in both folds When matter is transfered towards twin Univers, it undergoes a CPT-symmetry and its mass (its contribution to the gravitional field) is reversed. But it is still matter. Similarly, antimatter flowing in space bridge remains antimatter, with opposite mass, for the inversion of the time marker, as shown by Souriau, implies the inversion of the mass.

Références.

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