PQ4

...Ας προσπαθήσουμε τώρα να φανταστούμε (τα σχήματα λαμβάνονται από το άρθρο) μια ομάδα τεσσάρων παιδικών μαρμάρων, που βρίσκονται ακόμα σε έναν χώρο αναπαράστασης 3D, οι οποίες σχηματίζουν ένα τετράεδρο (ένα πολύ προσανατολήσιμο αντικείμενο) και που πέφτουν σε μια σφαιρική «χαράδρα» σφαίρα, σύμφωνα με τις «γεωδαιτικές ακτίνες».

Θα «αναπηδήσουν» από τη σφαίρα-χαράδρα (σύμφωνα με την εικονογραφία του χώρου αναπαράστασης που επιλέξαμε. Στην πραγματικότητα οι γεωδαιτικές είναι συνεχείς στην 3D υπερεπιφάνεια).

Θυμάμαι, όταν ήμουν νεότερος, συχνά έβρισκετε χρωμιωμένες σφαίρες στα άκρα των κιγκλιδωμάτων των σκαλών. Αν ζείτε κάπου που έχει κάτι τέτοιο, μπορείτε να δοκιμάσετε την εμπειρία μόνοι σας ρίχνοντας μικρά ατσάλινα μάρμαρα πάνω τους.

Μετά την αναπήδηση, τα τέσσερα μάρμαρα θα σχηματίσουν ένα ανεστραμμένο τετράεδρο:

Ας αυξήσουμε το μέγεθος του τετραέδρου ώστε να δούμε πιο καθαρά την αναστροφή. Στην αρχική διαμόρφωση, αυτό παρουσιάζεται ως εξής:

«Προσανατολίζουμε» τις έδρες του. Για παράδειγμα, δίνουμε κατεύθυνση στη διαδρομή ADB κ.λπ., έτσι ώστε να συγκρίνουμε την «κίνηση» με αυτή ενός ανοιχτηριού με την αιχή του προς το εξωτερικό (βέλη). Οι τέσσερις έδρες thus προσανατολίζονται. Ας συγκρίνουμε τώρα αυτό το τετράεδρο με εκείνο που σχηματίζεται από τα μάρμαρα που «αναπήδησαν» από τη σφαίρα-χαράδρα:

Ο προσανατολισμός των εδρών έχει αναστραφεί. Αν το σχέδιό μου ήταν πιο ακριβές, τα δύο αντικείμενα θα είχαν τοποθετηθεί εκατέρωθεν ενός καθρέφτη, το ένα being η εναντιομορφική εικόνα του άλλου.

Το ίδιο ισχύει για τον Schwarzschild: τα αντικείμενα επανέρχονται «στην άλλη πλευρά», και αν μπορούσαμε να «τα δούμε διαφανή», θα εμφανίζονταν εναντιομορφικά. Αλλά δεν μπορούμε να «τα δούμε διαφανή». Για να «δούμε», τα φωτόνια πρέπει να μπορούν να δημιουργήσουν επικοινωνία μεταξύ δύο «γείτονων» περιοχών στις δύο «πλευρές του χωροχρόνου», οι οποίες είναι therefore P-συμμετρικές.

Παρεμπιπτόντως, τι γίνεται με τις «μη-ακτινικές» τροχιές; Οι υπολογισμοί των γεωδαιτικών δίνουν επίπεδες τροχιές που «αναπηδούν» στη σφαίρα Schwarzschild. Δείτε το ακόλουθο σχήμα.

Το ζήτημα του μεταβλητού χρόνου, που θίχτηκε συνοπτικά παραπάνω, παραμένει. Όπως είπα, έχουμε κάθε δικαίωμα να επιλέξουμε οποιαδήποτε μεταβλητή θέλουμε. Η επιλογή είναι εντελώς αυθαίρετη καθώς το αντικείμενο, η υπερεπιφάνεια του χωροχρόνου, είναι μια «ανInvariant συντεταγμένη», υπάρχει ανεξάρτητα από την επιλογή των συντεταγμένων που χρησιμοποιούνται για να σημειωθούν τα σημεία που φαίνονται παραπάνω, τα οποία είναι «σημεία-γεγονότα», σημεία ενός χωρο-χρονικού αντικειμένου, μιας 4D υπερεπιφάνειας.

Άρα, τι είναι ο χρόνος τότε, τι είναι ο χώρος αν όλα αυτά είναι αυθαίρετα;

Υπάρχει ένας χρόνος που δεν μπορούμε να αγγίξουμε, ο μόνος εγγενής βαθμωτός της υπερεπιφάνειας: είναι ο ιδιος χρόνος του. Ο ιδιος χρόνος είναι «μήκος» στον χωρο-χρονικό υπερχώρο. Ας υποθέσουμε ότι τα αντικείμενα μπορούν να κινούνται κατά μήκος γεωδαιτικών (4D). Ας πάρουμε ένα ζεύγος σημείων (A, B) σε μια γεωδαιτική. Το μήκος Ds που χωρίζει τα δύο σημεία, διαιρεμένο με το c, μια σταθερά, την ταχύτητα του φωτός σε μια περιοχή μακριά από τη σφαίρα-χαράδρα όπως συμβαίνει, και η περίοδος του ιδιου χρόνου, είναι το διάστημα ιδιου χρόνου Dt που χωρίζει τα δύο «γεγονότα», και αυτό ανεξάρτητα από τις χωρο-χρονικές συντεταγμένες που επιλέγονται.

Ds
is the only quantity to have an intrinsic physical sense.

Φανταστείτε ότι κινείστε πάνω στο terrestrial globe κατά μήκος μιας γεωδαιτικής (μεγάλος κύκλος), πηγαίνοντας από το σημείο A στο σημείο B. Αν λέτε:

- Πήγα από ένα σημείο μήκους jA
και πλάτους qA σε ένα σημείο
μήκους jB και πλάτους
qB

τι εννοείται με τις ποσότητες
(jB - jA)
et (jB - jA)
? Θα εξαρτώνται από τα σημεία που επιλέξατε για τους πόλους σας, από την επιλογή των σημείων αναφοράς. Ωστόσο, αν λέγατε:

- Έχω διανύσει 2,347 χιλιόμετρα.

Η μέτρηση θα σήμαινε κάτι ανεξάρτητα από το σύστημα συντεταγμένων αναφοράς που είχατε επιλέξει.

Είδαμε με τη σφαίρα ότι μπορούσαμε να χρησιμοποιήσουμε συντεταγμένες που εμφανίζουν μία ή περισσότερες singularities. Ένας πόλος είναι ένα μέρος όπου το μήκος δεν ορίζεται πλέον. Είδαμε επίσης πώς, με μια απλή αλλαγή συντεταγμένων, μπορούσαμε να κάνουμε μια «ανεπιθύμητη περιοχή μιας επιφάνειας (ή r<Rs) να disappear και όπου θα βρίσκαμε ένα purely imaginary element of length Ds.
Indeed, it is the fact that in its initial formulation the Schwarzschild metric brings a purely imaginary element of length (proper time) that we supposed that we were "off hypersurface". There is no absolute coordinate
system. But we can decide to make the choice of a coordinate in space that at least makes singularities disappear, which is what we have done. Nor is there
any "absolute cosmic time". With Midy, in our latest paper, we showed
that the "initial singularity", considered as the "instant of
the creation of our universe", is a result of a particular choice of time
marker variable and that a different choice would keep all observables, beginning
with the redshift, but would make the original singularity disappear like the
sin of the same name. The question "what was there before the Big Bang
?" no longer makes sense. Troubling, I agree, but the question results
from a spatio-temporal paradigm. It is equivalent to "what is there at
the centre of a black hole ?" It is therefore perfectly licit to change
the temporal coordinate by using "Eddington time" (the change of variable
was shown above), inasmuch as it allows this local geometric structure to be
joined to Minkowski space-time, that of relativist space (in the sense of special
relativity) and flat, without curves, empty. But the idea is to be able to describe
the whole of space-time with just one metric. Once again the guiding thread
is to be found in group theory and in examining the 'isometry group" of
the Schwarzschild metric.

The

isometry group contains every geometric transformation that leaves
the metric invariant (therefore an invariant hypersurface). The object sphere's
isometry group is the group of rotation in space plus symmetries (in relation
to a plane or an axis through its centre, in relation to a point that is this
centre). We call this group O3 (an abbreviation of "orthogonal group of
size 3"). (See Introduction
to Geometrical Physics B. All this is in there.) However if we remove the symmetries
in relation to an axis, a plane or a point, it becomes SO3 ("special orthogonal
group of size 3").

Schwarzschild

geometry has symmetries. Until now we have been used to giving it S03 symmetry
(rotations in space). But in fact it has an isometry group O3, and so contains
P-symmetry (symmetry in relation to a point). Let us go back to the tetrahedron
we used earlier. Its symmetry in relation to a point is enantiomorphic, a first
class p-symmetric.

In the

section 'groups' on the site we showed how the group "secreted space"
or, more precisely secreted geometric objects. Souriau calls them group "species".
So it is not the sphere that engenders the SO3 group, but the opposite. The
spheres are species of this group. Species in the taxonomic sense of the term
(Taxonomy : science of the classification of species). We said earlier that
sometimes physicians do mathematics without realising it and vice-versa. Relativist
physics and the progress made in groups date from the beginning of the century
: Klein, Poincaré, Lorentz, Cartan, etc., followed on from the work of the brilliant
Norwegian Sophus Lie. Everything began to hang together. Was it the work of
physicists that stimulated the work of mathematicians or vice-versa ? No doubt
they stimulated each other. Special Relativity has its own space-time, that
of Minkowski (defined by its "metric"). Its "isometry group"
is the Poincaré group, itself built around the Lorentz group (see Introduction
to Geometrical Physics B). Souriau, in his book "Structure of System Dynamics",
Dunod 1974, pages 197 to 200, was the first to show that the Poincaré group
"secreted retrochronal objects" and that this went hand in hand with
the inversion of their mass. We can see the mechanism therefore : physicists
put their finger on a physical phenomenon, such as the invariability of the
speed of light : the Michelson and Morley experiment. Mathematicians reinterpret
this in terms of groups. But among the groups there are elements that seem to
refer to new objects : negative masses.

This makes physicists knit
their brows. If a negative mass meets a positive mass the result would be ...
nil, nothing. Not to be confused with matter-antimatter annihilation (which
has a positive mass in fact) which produces the equivalent in energy-matter
in the form of photons. As negative masses m* = -m have a negative energy E*
= m*c2 = -mc2 , the evaluation gives ... zero. During a quarter of a century
these negative masses, discovered by Souriau, remained "a purely mathematical
curiosity" (which Souriau himself believed in fact).

In 1998

I constructed a geometric, gemellary context (see the papers of Geometrical
Physics B) in which these negative masses could frolic in complete liberty,
in their natural habitat : a world where the coordinate time is inverted.

This

text is a vulgarisation of the work (from the article "Questionable black
hole") and is based on group theory. First of all I noticed that the Schwarzschild
metric was non SO3XR
(3d rotations plus temporal translations, which express the fact that the object
is invariant in time, stationary), but 03XE1
(including, among other things, P-symmetry et T-symmetry). This is the guide
for an extension of the geometric context, which goes hand in hand with Eddington's
vision of 1924. Symmetries are then exploited with a "PT-symmetric"
model : where space and time coordinates are inverted in the gemellary universe,
an idea originally proposed by Andrei Sakharov in 1967.

Does

all that seem complicated to you ? Let the Higher Math student have a look at
the Minkowski metric, that of Special Relativity :

ds2
= c2 dt2 - dx2 -dy2 -dz2

Change

t ...to

• t

x ...to

• x

y ...to

• z

z ...to
-z

Invariance. The isometry
group (the one which leaves this metric invariant) is greater (it is Poincaré's
group "with its four components"). The transformation is only a part
of the whole but you can see that the Minkowski metric is invariant by PT symmetry.

The metric of Special Relativity goes with a relativist space

(t , x ,
y , z )

But

it can also describe a universe in which space and time coordinates will be
inverted (PT-symmetric to ours). They are not tachyons. Nothing like them. In
the second universe speeds remain subluminic.

In short,

the Schwarzschild metric, revisited with Eddington's idea, became PT symmetric.
The time coordinate should therefore invert "naturally" in passing
through the gorge sphere. Does that mean that the time experienced by an eventual
spaceship passenger going into the twin universe would be inverted? That time
is just a coordinate. On Earth when you cross the Equator your latitude becomes
negative but you don't start walking backwards...

We then

included this geometry in a larger context, ten dimensions, this number, according
to a theorem of Wiener and Graustein, corresponds to the minimum number of dimensions
required to receive a space of n dimensions, with n higher than 2.

These

six additional dimensions have already been introduced in the articles presented
in Geometrical Physics B. They refer to quantic aspects. The conclusion :

• The duality matter-antimatter
  exists on both sides of the universe.

• When a particle of matter
  passes through the hypertoric bridge corresponding to Schwarzschild geometry,
  its contribution to the gravitational field is inverted. The field equation
  system proposed from 1994 in Nuovo Cimento (reproduced in Geometrical Physics
  ) is thus confirmed, as are the developments we have presented in a vulgarised
  manner in "We have lost half the Universe" (Albin Michel).

• When a particle of matter
  crosses one of these "hyperspheric tunnels" matter remains (but CPT
  symmetric). The same goes for a particle of antimatter.

  However,
  in that case, transit time is FINITE. Therefore black holes cannot exist. When
  Schwarzschild geometry was tinkered with using a bad choice of variables and
  a bad choice of "geometric context" it led to this "time freezing",
  that we consider to be a mathematical artifice.

  But
  if black holes do not exist, what happens to a neutron star whose mass exceeds
  the fateful critical value (two solar masses : which will send the pressure
  at its centre shooting towards infinity) ?

  The
  following figure shows the pressure value (in "logarithmic" coordinates)
  according to the distance from the centre of the neutron star (supposed to be
  of constant density), for different values of exterior radius (of mass therefore)
  obtained by using the classic Tolmann Oppenheimer Volkov model. The critical
  curve corresponds to a value of two solar masses.

We

can see that as long as the star's mass remains largely inferior to the critical
value, the increase in pressure towards the centre remains moderate. But as
soon as the mass approaches the critical value the pressure bolts to become
infinite at the centre (critical curve).

The

rest of the paper presents what is a project for a model and not a
model. In our opinion, the sudden rise in pressure must have an influence on
the "physical constants", including the local value of the speed of
light, which should also tend towards infinity. We think that this should provoke
the opening of a hypertoric passage at the centre of the star. As a guide we
calculated the pressure, still with the TOV model, for masses above critical
mass, 2 solar masses, which brings about the increase in pressure towards infinity
(criticity of a physical nature) but below 2.5 solar masses, which corresponds
to classic "geometric criticity" : when the Schwarzschild radius reaches
the exterior radius of the star. As the TOV model is based on a stationary solution
it has obviously no value as a model. However let us note the extremely rapid
extension of the sphere (p = infinity) from the centre of the star outwards
with the addition of moderate masses.

The pressure curve seems to go towards the right like a "whiplash".

(Note that we have used
the word "infinity" whereas a little earlier we were putting the legitimacy
of the word into doubt. Let's say that the phenomenon will occur when the pressure
goes beyond a value limit. But this would no doubt require that we bring "quantic
contributions" to the model). Pierre Midy and I began to study the question.
In our opinion there are two possible scenarios.

**Soft

version** : a neutron star receives an influx of matter from a companion
star (stellar wind) which brings it to two solar masses, a mass which will send
the pressure at its core towards infinity. A hyperspatial bridge then opens
in its centre and from which excess matter is evacuated. It disperses on arriving
in the twin universe, as its mass has been inverted, repelled by the neutron
star, which makes itself felt and which behaves towards the transferred mass
as a repulsive object. Evacuation via the hypertoric passage takes place at
relativist speed and the size of the structure (the surface of the gorge sphere)
depends on the flow rate required. If the influx is continuous, the hypertoric
bridge will behave like an "overflow" that functions continually and
ensures a leak flow. The following figures evoke the two regions of the star
in sub-criticity :

and with a "leak flow"
:

Hard version : The fusion of two neutron stars. The process
will be a lot more violent. The hypertoric bridge will form and grow very rapidly,
at relativistic speed, by swallowing a large part of the mass. All this will
take place with the emission of gravitational waves and "gamma leap".
We think that only a part of the mass would be transferred. In effect, once
matter crosses to the other side its mass is inverted and it contributes negatively
to the gravitational field. In doing so it reduces the original gravitational
pressure on the neutron star. However only a correctly developed unstationary
solution, and referring to a object which is not spherically symmetrical (an
unrealistic idea for neutron stars) but axisymmetrical, will begin to bring
answers.

We spoke

earlier of this aspect of things and a specialist could say :

• Neutron stars cannot
  have spherical symmetry. Black holes do not stem from the Schwarzschild metric
  but that of Kerr, which is different (it possesses a different isometry group).

  Currently
  Midy and I are reworking all this using the Kerr metric, which doesn't seem
  to present any particular technical difficulty. The gorge surface instead of
  being spherical, simply becomes elliptic.

  Let
  us return to the project for a hyperspatial transfer model. The "hard"
  phenomenon" could transfer the majority of mass to the twin. Once "gravitational
  tension" had reduced sufficiently, the hyperspatial bridge would automatically
  close. The phenomenon would probably be extremely brief, of the order of a few
  hundredths of a second. A residual mass would remain in our universe, in the
  "neighbourhood", while continuing to be repelled by matter (the neutron
  star) which had been almost completely transferred to the twin. The residual
  matter remaining on our side of space-time, would form a ring of gas, like a
  smoke ring, which would rapidly cool through radiation if there was no source
  of energy in the vicinity, a hot star for instance. The minimum temperature
  reached by the object could not be inferior to that of the cosmic oven in which
  it bathes : 3°K. That is the key-observable. The following figure is a 2d representation
  of the phenomenon.

If this

model holds up, we should find rings of cold or relatively cold gas that seem
to be organised around an invisible object. Dynamically these objects orbit
around a repulsive object, fundamentally invisible : the neutron star transferred
to the twin. Are some of the recently discovered "proplyds" objects
of this type ? Observation will tell us ? The difficulty comes from the fact
that the objects were only discovered because they showed up against a more
luminous background (like the proplyds which appear against the Orion nebula).
They are then heated by radiation from relatively nearby stars.

The

"good toroidal nebula" will be far from any radiation source, dark
therefore. But maybe a phenomenon of polarisation of background light could
allow its detection. Polarisation cartography is an important area in observational
astronomy. However, the phenomenon could also take place in the twin universe
which would then send us matter and just as violently.

In the

Geometrical Physics A papers we developed arguments in which the stellar phenomenon
would not take place in a twin that was hotter than ours. In such a case gemellary
matter would group in large conglomerates radiating in infrared and structured
like enormous spheroidal proto-stars, but whose cooling time would exceed the
age of the universe. The conglomerates would function like proto-stars that
were never lit up. Repelling our matter, they would be responsible for VLS,
the very large structures of our own matter, incomplete, arranged around immense
empty bubbles whose characteristic diameter is of the order of hundreds of millions
of light years and whose existence, outside this explanation with a gemellary
model (numeric simulation) remains fairly inexplicable.

A final

remark. We do not find antimatter on our side of the universe. We also notice
a violation of the parity principle and some believe that the two are linked.
In 1967 A. Sakharov suggested that the violation of the parity principle could
be inverted in the twin. If that is the case when there is a link with the subsistence
of one of the two species, the immense conglomerates would then be made of gemellary
antimatter, PT-symmetric with ours (of negative mass because it evolves in a
universe with an inverted time coordinate).

Let

us end by giving a series of drawings which are an attempt at a 2d description
(a simple educational model) of the hyperspatial transfer phenomenon. In the
papers reproduced on the site we have shown (it ensues from the structure of
coupled star equation systems) that the scalar curves of the two universes are
inverted in two adjacent regions :

R* = - R

The

2d educational model of a mass situated in our universe is, geometrically speaking,
that of a "blunt posicon". The twin will then have the appearance
of a "blunt negacon" ("conjoined geometries"). The twin's
geometry, where there is only emptiness, is therefore an "induced geometry".

Rough
educational image of "conjoined geometries" in the two universes.

Matter

is in the blunted part of the posicon (greyed area). When criticity is reached,
a "conical point (infinite curve density) appears in the greyed area (equivalent
to an increase in pressure to infinity). A conical point is a point in which
"curve density" is infinite.

The

drawings show the continuation of the process. The gorge is created in the following
figure.

The

following figure (which is supposed to represent a total transfer of matter
to the twin universe) represents 'half time".

In our

opinion it is at this instant that reference is made to Schwarzschild geometry.
The gorge circle is flooded on both surfaces. The scalar curve is nil everywhere
(the reason for solutions with secondary nil members). A simple remark : the
geodesics are inscribed on the folds without difficulty. Try with a roll of
sticky tape.

The

following figure shows the moment immediately before the closure of the hypertoric
point, when it constricts, in the gemellary leaf, according to a conic point.

After

separation the mass (greyed) has entered the twin, which produces an "induced
negative curve" in our universe.

September 1999. To be
continued ...