...Let
us now try to imagine (the figures are taken from the article) a group of four
children's marbles, still in a 3d representation space, which form a tetrahedron
(a very orientable object) and which fall into a sphere-shaped gorge sphere,
according to "geodesic radials".
They
will "bounce off" the gorge sphere (according to the imagery of our
choice of representation space. In fact the geodesics are continuous in the
3d hypersurface).
Iremember,
when Iwas younger, you often found chrome balls at the ends of stairway banisters.
If you live somewhere that has this sort of thing you can try the experience
for yourself by throwing small steel marbles at it.
After
the rebound the four marbles will form an inverted tetrahedron :
Let us increase the size of the tetrahedron so that we can see the inversion
more clearly. In the initial configuration it presents itself in the following
way :
We are
"orienting" its faces. For example we give the route ADB a direction
etc., in such a way as to compare the "movement" to that of a corkscrew
with its point towards the exterior (arrows). The four faces are thus oriented.
Let us now compare this tetrahedron with that formed by the marbles that "rebounded"
from the gorge sphere :
The
orientation of the faces has been inverted. If my drawing had been more precise
the two objects would have been placed on either side of a mirror, one being
the enantiomorphic image of the other.
It's
the same thing for Schwarzschild : objects reappear "on the other side",
and if we could "see them in transparency" they would appear enantiomorphic.
But we can't "see them in transparency". For us to "see"
the photons must be able to set up communication between two "adjacent"
regions on either of the two "sides of space-time", which are therefore
P-symmetric.
In passing,
what of "non-radial" trajectories ? Calculations of geodesics give
planar trajectories which "rebound" on the Schwarzschild sphere. See
the following figure.
The
question of variable time, briefly touched on above, remains. As Isaid, we
have the complete right to choose any variable we like. The choice is completely
arbitrary as the object, the space-time hypersurface, is an "invariant
coordinate", it exists independently of the choice of coordinates used
to mark the points shown above, which are "event-points", points of
a spatio-temporal object, a 4d hypersurface.
So,
what is time then, what is space if all that is arbitrary ?
There
is one time that we cannot touch, the only intrinsic scalar of the hypersurface
: it is its proper time. Its proper time is "length" in spatio-temporal
hyperspace. Let us suppose that objects can move along geodesics (4d). Let us
take a couple of points (A, B) on a geodesic. The length Ds which separates
the two points, divided by c, a constant, the speed of light in a region far
from the gorge sphere as it happens, and the period of its proper time, is the
lapse of own time Dt separating the two "events", and this whatever
spatio-temporal coordinates are chosen.
Ds
is the only quantity to have an intrinsic physical sense.
Imagine
that you are moving over the terrestrial globe along a geodesic (a big circle),
going from point A to point B. If you say :
- I
went from a point of longitude jA
and latitude qA to a point
of longitude jB and latitude
qB
what is meant by the quantities
(jB - jA)
et (jB - jA)
? They will be dependent on the points you chose for your poles, on your choice
of marker points. However if you said :
- I've
travelled 2,347 kilometres.
The
measure would mean something whatever marker coordinate system you had chosen.
We saw with the sphere that we could use coordinates that show up one or several
singularities. A pole is a place where longitude is no longer defined. We also
saw how, with a simple change of coordinates, we could make an "undesirable
region of a surface (or r<Rs) disappear and where we would find a 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, Iagree, 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, Poincar�, 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 Poincar� 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 Poincar� 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
Iconstructed 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 Inoticed 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 Poincar�'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 Ibegan 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 Iare 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 ...
