23)Description in a ten-dimensional space.
In former
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 "Relativity in 5 dimensions", page 413, where the inversion
x5 � - x5, the inversion of the Kaluza coordinate,
is identified with 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 that 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].
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 rises out of the associated
"white fountain", although its proper time increment Ds
is not changed (it cannot be), this particle, cruising in a 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 "apparent
mass" m* = - m (see reference [15] and equation (110)) is changed.
In the "orthochronous"
fold F, matter and antimatter own positive mass and energy, as shown in references
[23] and [24]. But, when they are transferred 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 the latter 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 pathologies associated with 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 shown 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 shown that the singular features
could be fully eliminated, considering not simply connected space time. We have
given the Schwarzschild geometry a different physical significance, the latter
being considered as a bridge linking two universes, ours and a twin one.
We have
shown 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 shown 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 shown 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 shown 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 the 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 Kruskal model, particularly
the fact that it is not asymptotically Lorentzian at infinity.
We have
presented some attempts to embed subsets of Schwarzschild geodesics, with peculiar
parameters (zero velocity at infinity, radial paths in plane q
= p/2). We have suggested to consider Schwarzschild geometry as a 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 the 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
the 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 ; reprinted in Lorentz-Einstein-Minkowski
"Das Relativitätsprinzip", 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] Missner 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 Mathematics
and black holes. 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
Physic 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) transl. Preprint R2-4267, JINR,
Dubna
[31] D.Novikov, ZhETF Pis'ma
3:223 ( 1966 ) ; JETP Lett. 3:142 (1966), transl. Astr. Zh. 43:911 (1966) Sov.
Astr. 10:731 (1967 )
[32] J.P.Petit : "Enantiomorphic universes with opposite proper times", Compte Rendu de
l'Académie des Sciences de Paris, may 1977, t.285 pp. 1217-1221
[33] J.P.Petit : "Universes
in interaction with their image in the mirror of time". 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).
