23)** Description in a ten-dimensional space.**
In previous papers ([22], [23] and [24]) we have developed an attempt to describe particles in a ten dimensional space :
(148)
( x , y , z, t , z 1, z 2, z 3, z 4, z 5, z 6 ) = ( x , y , z, t , z** ) = ( r , t , z ** )
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 the 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)
z** ** ® -z** **
(inversion of the six additional Kaluza-like dimensions, extension of Souriau's work [25]). This showed the matter-antimatter duality holds in the twin fold ([23] and [24]) and provides a new geometric interpretation of the so-called CPT theorem [24].
Schwarzschild space-time can be embedded in a ten dimension space, which suggests that these additional 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 embedded in a ten-dimensional space.
Introducing :
(150)
we get a 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 both half folds.
b = m = -1 corresponds to CPT symmetry. When matter, which belongs to fold F, is poured into a "black hole" and comes out of the associated "white fountain", although its proper time increment Ds is not changed (it cannot be), this particle, traveling in the CPT-symmetrical fold F*, becomes CPT-symmetric. It is still a particle of matter. The transfer (including the hypothetical fast hyperspatial transfer, mentioned above) does not transform matter into antimatter, and vice versa, but the "apparent mass" m* = - m (see reference [15] and equation (110)) is changed.
In "orthochron" fold F, matter and antimatter have positive mass and energy, as shown in references [23] and [24]. But, when they are transferred towards the twin fold F*, which has 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 it exceeds 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 pathologies 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 and 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 black 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 azimuthal 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 universes 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 inverted : 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 Kruskal model, particularly the fact that it is not asymptotically Lorentzian at infinite.
We have presented some attempts to embed 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, embedded 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 transferred towards twin universe, it undergoes a CPT-symmetry and its mass (its contribution to the gravitational 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.
References.
[1] R. Adler, M. Bazin and M. Schiffer : Introduction to General Relativity, Mc Graw Hill Book Cie 1975
[2] Schwarzschild K. : Über das Gravitational eines Massenpunktes nach der Einsteineschen Theory, Sitzber. Preuss. Akad. Wiss. Berlin, 1916, p.189-196
[3] Birkhoff G ; "Relativity and Modern Physics", Cambridge, Mass. 1923
[4] Einstein A : "Spielen Gravitationsfelder im Aufbau des materiallen Elementarteilchen eine wessentliche Rolle. Sitzber. Preuss. Akad. Wiss. Berlin ; reprintend in Lorentz-Einstein-Minkowski "Das Relativitâtsprinzip", Leipzig 1922. English translation : "The Principle of Relativity" , London 1922, reprinted by Dover, New York.
[5] Finlay-Freundlich E. On the empirical Foundations of the General Theory of Relativity, in A. Beer Ed. "Vistas in Astronomy", vol.1 Pergamon Press, London 1955.
[6] Misner C.W, Thorne S. and Wheeler J.A. "Gravitation", San Francisco Ed. 1973
[7] Vishveshwara C.V. Generalization of the Schwarzschild surface to arbitrary Static and Stationary metrics, J. of Appl. Physics 9, 1968.
[8] Weinberg S : Gravitation and Cosmology, New York, 1972.
[9] N. SRAVROULAKIS Mathématiques et trous noirs. Gazette des mathématiciens n°31, juil.86 pp.119-132
[10] Eddington A.S. : A comparison of Withehead's and Einstein's formulæ Nature 113 : 192 (1924)
[11] P. Midy and J.P. Petit : Scale Invariant Cosmology. International Journal of Physics D, June 1999, pp.271-280
[10bis] J.P. Petit : Dynamic groups of Physics. 1998. Geometrical Physics B,1. (website http://www.jp-petit.com)
[12] J.M. Souriau : Structure of dynamical systems, Birkhauser Ed, 1998 and Editions Dunod (french) 1974.
[13] J.P. Petit : The missing mass problem. Il Nuovo Cimento B Vol. 109 July 1994 and Geometrical Physics A,2 (website http://www.jp-petit.com : Geometrical Physics A,1)
[14] J.P. Petit : Twin Universe cosmology. Astronomy and Space Science 1995, 226 pp. 273-307 and Geometrical Physics A,2 (website http://www.jp-petit.com Geometrical Physics A,2 )
[15] J.P. Petit and P. Midy : Matter ghost matter astrophysics 1 : the geometrical framework. The matter era and Newtonian approximation. Geometrical Physics A,4 (website http://www.jp-petit.com).
[16] J.P. Petit and P. Midy : Matter ghost matter astrophysics 2 : Conjugated steady state metrics. Exact solutions. Geometrical Physics A,5 (website).
[17] J.P. Petit and P. Midy : Matter ghost matter astrophysics 3 : The radiative era : The problem of the "origin" of the Universe. The problem of the homogeneity of the early Universe.. Geometrical Physics A,6. 1998 (website http://www.jp-petit.com).
[18] J.P. Petit : Cosmological model with variable velocity of light. Modern Phys Letters A3, 1988, pp. 1527
[29]** **J.P. Petit, Mod. Phys. Lett. A3 ( 1988) 1733
[20]** **J.P. Petit, Mod. Phys. Lett. A4 ( 1989) 2201
[21] J.P. Petit and P. Midy : Repulsive dark matter Geometrical Physics A,3. 1998 (website http://www.jp-petit.com)
[22] J.P. Petit and P. Midy : Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 1 : Charges as additional components of the momentum of a group acting on a ten dimensional space. Geometrical definition of antimatter. Geometrical Physics B,2. 1998 (website http://www.jp-petit.com).
[23] J.P. Petit and P. Midy : Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 2 : Geometrical definition of Dirac's antimatter. Geometrical Physics B,3. 1998 (website http://www.jp-petit.com)
[24] J.P. Petit and P. Midy : Geometrization of matter and antimatter through coadjoint action of a group on its momentum space. 3 : A first geometric description of antimatter after Feynmann. So-called CPT theorem. Geometrical Physics B,4. 1998 (website http://www.jp-petit.com)
[25] J.M. Souriau : Géométrie et Relativité (in french only), Hermann Ed. 1964
[26] A. Sakharov : "CP violation and baryonic asymmetry of the Universe". ZhETF Pis'ma 5 : 32-35 ( 1967) ; Translation JETP Lett. 5 : 24-27 (1967)
[27] A. Sakharov : "A multisheet Cosmological model" Preprint Institute of Applied Mathematics, Moscow 1970
[28] A. Sakharov : "Cosmological model of the Universe with a time vector inversion". ZhETF 79 : 689-693 ( 1980 ) ; translation in Sov. Phys. JETP 52 : 349-351 (1980)
[29] A. Sakharov : "Topological structure of elementary particles and CPT asymmetry" in "Problems in theoretical physics", dedicated to the memory of I.E. Tamm, Nauka, Moscow 1972, pp. 243-247
[30] A.D. Sakharov , ZhETF Pis'ma 5 : 32 ( 1967 ) ; JETP Lett. 5.24 (1967) trad. Preprint R2-4267, JINR, Dubna
[31] D. Novikov, ZhETF Pis'ma 3:223 ( 1966 ) ; JETP Lett. 3:142 (1966), trad Astr. Zh. 43:911 (1966) Sov. Astr. 10:731 (1967 )
[32] J.P. Petit : "Univers énantiomorphes à temps propres opposés", Compte Rendu de l'Académie des Sciences de Paris, May 1977, t.285 pp. 1217-1221
[33] J.P. Petit : "Univers en interaction avec leur image dans le miroir du temps". Compte Rendu de l'Académie des Sciences de Paris, June, 7, 1977, t. 284, série A, pp. 1413-1416
[40] J.P. Petit Le Topologicon, Ed. Belin, France, 1983 (available on cd-rom. Ask the author).
