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Some comments about the
metrics.
All the elements of the
group are built from the elements of the complete Lorentz group, which obey
:
(412)
with
(413)
This last matrix is linked
to the metric :
(414)
...So that the two folds have same signature.
If they are described as Minkowski space times, their metrics are identical.
But their arrows of time are opposite.
If one wants to describe the two folds, the two universes, one has to choose
his own arrow of time and space orientation.
...It
is clear that the duality matter-anti-matter holds in both folds. If we call
the second fold "twin fols" (A.Sakharov) or "shadow fold" (Green, Schwarz
and Salam) or " ghost fold" (the author's choice) the arrow of time in this
second fold is opposite (T-symmetry), as predicted by A.Sakharov, and space
structures are enantiomorphic (P-symmetry).
...In
the second fold the matter is CPT-symmetric with respect to ours. Whence,
in that fold, a proton owns a negative charge and an electron a positive charge.
...Conversely,
an anti-electron of that fold, PT-symmetric with respect to ours, owns a negative
charge, whence an antiproton of the second fold has a positive charge.
...To
sum up, the second fold is CPT symmetric with respect to ours. As suggested
by Andréi Sakharov, we can expect that the violation of the parity
principle could be reversed in that fold. ..If
the absence of anti-matter, in our fold, is a direct consequence of the violation
of the parity principle, it is possible that such dissymmetry would be reversed
in the other fold.
Interacting folds.
...All
our work in astrophysics and cosmology ( see Geometrical Physics A ) comes
from a system of two coupled field equations :
(10)
S = c
( T - T* )
(11)
S * = c
( T* - T )
...The
two minus signs were introduced as an a priori hypothesis. At the end of this
work, based on group theory, the explanation arises. The two folds musthave opposite arrows of time and must be enantiomorphic in order
to fit constrainsts coming from the group structure.
...So
that the other matter, located in the other fold, for an orbserver located
in the first, behaves as if it own a negative mass, which comes from the coadjoint
action and the T-symmetry.
Conclusion :
...The
part of the site, called Geometrical Physics B, devoted to group theory, fits
the first one, devoted to astrophysics and theoretical cosmology. Group theory
bring the starting point of the research.
...Geometrization
of elementary particles requires a multiple extension of the complete Poincaré's
group. Antimatter is geometrized. CPT-symmetrical of a matter particle cannot
be longer identified to normal matter, due to its negative mass and energy,
like PT-symmetrical of a metter particle cannot be identified to Dirac's anti-matter,
for the same reason. Existence of negative energy species (CPT and PT-symmetrical
) requires a two-folds geometry, in which the duality matter-antimatter holds.
Matter of this ghost fold is simply CPT-symmetrical and anti-matter PT-symmetrical
of a normal matter particles.
