But
surfaces do exist which are intrinsically singular, possessing singularities
that are not due to a choice of coordinates. Here is an example: conic singularity.
Rough
from the foundry, as formulated by Schwarzschild in 1917, the coordinates t
, r , q , j (time,
a radial distance and two angles, equivalent to azimuth and site : "spherical"
coordinates) Schwarzschild's sphere is singular. For a certain value Rs of the
"radial coordinate" r (supposed to be measured from a "geometric centre") this
metric plays all sorts of games with us. One of the terms on the sphere has
a non-nil denominator. In short, it is singular on this sphere. Is it an intrinsic
singularity or an artefact introduced by a bad choice of coordinates ? That
is the question we asked ourselves.
Let
us note in passing the "Schwarzschild's geometry" is a four dimensional hypersurface,
which makes the thing even more untrustworthy.
Kruskal
focussed on this point. He constructed a coordinate change which, among other
things, supplied a constant value for the speed of light along a radial trajectory.
In doing so he concentrated the singular aspect "at the centre of the object"
in a "central singularity". Psychologically we feel we have gained something.
The solution becomes "regular almost everywhere", an expression that mathematicians
use to say that a solution is regular, exempt of pathology, except on a unique
point.
- You're
not going to be difficult just over one little point surely....
Alas,
Kruskal's formulation has a serious weakness : it does not give back the space
of infinite special relativity. Technically it isn't infinitely lorentzian,
"asymptomatically lorentzian".
It is
an essential question in physics : do singularities exist . Does Nature tolerate
singularity ? The answer is formulated in terms of belief (as for the existence
or inexistence of infinity in fact).
We sought
a new interpretation of this same Schwarzschild geometry by trying to eliminate
all singularity and we managed to do so. Our answer is therefore :
- The
singular character of the Schwarzschild solution is simply induced by a poor
choice of coordinates.
Technically,
everything rests on the change of variable :
r = Rs +
Log chr
which is read : r equals
Rs plus logarithm of the hyperbolic cosine of the variable
r . Simple except for scientists, specialists or higher maths students.
For those who are able to play with the formula, the value r can no longer become
inferior to Rs, even when r takes any possible value
from negative to positive infinity.
Consider
a surface obtained by turning a parabola around a straight line, like this :
This
figure was taken from the article. The surface is infinite, in fact, just like
the parabola-meridian that created it by turning around the axis z shown. If
we absolutely must represent it with its coordinates (r,z,j) we can expect trouble
when we ask "what is this surface like when r < Rs ?"
An answer
will be found ... imaginary, with roots of negative quantities. Simply because
we are then "extra-surface".
This
surface, in mathematics, is said to be "non-simply connected", a barbaric term
which simply designates surfaces where any closed curve is unable to see its
perimeter decrease, by sliding into the surface until it takes a value of zero.
This
is possible on a sphere, which is "simply connected". But on the surface in
question we can see quite clearly that a closed curve which "circles around
the sort of central well" cannot see its perimeter tend towards zero, the limit
being the perimeter of the "gorge circle". The same goes for a torus, which
is also "non-simply connected".
We defined
such a surface from its metric, which is a good illustration of our subject.
When keeping the coordinate r, the surface seems to be singular. By using the
change of variable given above, it is no longer so. What does the coordinate
r correspond to ? It simply "runs" along the parabola's
meridian as shown on the figure, taking the value zero on the circle of the
gorge. Half the surface corresponds to r positive,
the other to r negative. In the marking system of
points [r , j ] there
is no longer singularity.
We have
decided to call this type of object a "toric bridge" by analogy with the torus.
However
it is easily shown, still starting from metrics, that we can give an object
a 3d hypersurface which has a "hypertoric bridge". Then there is no longer any
gorge circle but a gorge sphere. The same goes for the surface above, a gorge
circle seems to connect two 2d layers so that the gorge sphere connects two
"3d half-spaces". If we are in one of these 3d half-spaces and we dive into
the gorge sphere, we emerge in the other half-space.
Let
us return to the 2d surface shown above. The following figure shows how when
we draw "circles that we believe to be concentric" we see their perimeter reduce,
reach a minimum, then increase once more.
In 3d
we have to imagine a sphere which completely englobes the gorge sphere Then
another inside this (we should say "beyond" when following a given direction,
towards the gorge sphere). We suppose that the surface of this sphere is less.
However when we reach the gorge sphere, the surface passes through a minimum
and then begins to increase ... to infinity if we continue the operation.
We have
built "metrics" of the 2d and 3d surfaces which include a "toric passage" and
a "hypertoric passage" and, in the second case, we have been struck by the resemblance
with the Schwarzschild metric, where we undertook the change of coordinates
and brought out its "non-simply connected" character, the "interior" of the
object becoming simply "beyond the gorge sphere".
It was thus
possible to eliminate all singularity.
At that
stage we had simply extended the black hole model to a tandem of "black hole-white
fountain". However, for the "exterior observer", the time required to cross
the hypertoric bridge is still infinite. We seem to have simply improved the
black hole model by explaining what it emerged as
We said
earlier that the choice of variables, in a geometric solution is totally arbitrary.
What goes for space also goes for time. We therefore sought a change of temporal
variable as invented by Eddington in 1924 :
Once again, we only mention
it for simple scientists and higher maths students.
t is the old "cosmic time",
the ancient "chronological variable" presented in Schwarzschild's initial solution
of 1917.
t' is the new "Eddington
time". Rs is the "Schwarzschild radius (in which case we should say the Schwarzschild
perimeter, divided by 2p ).
c is the speed of light
(constant here).
Something
which might seem odd : we mix time and space but, with matter, anything goes.
The choice of time-marker is completely arbitrary. We ask only that :
-
the metric be asymptomatically
Lorentzian, that is to say that at infinity, space-time becomes Minkowski space-time,
that of special relativity. In our case it works (not for Kruskal). -
the new time t' identifies
itself, still to infinity, with the "time of a supposedly immobile observer".
Which is also the case (not for Kruskal).By doing
so the time of free fall of a test-particle, infinitely immobile and falling
towards the Schwarzschild sphere, becomes infinite in relation to "the time
passed by the "exterior observer", who is far away and immobile.However
the particle would emerge from the swallowhole in an infinite time. As in the
black hole we can enter into this sort of 3d swallowhole but can't come out,
except in an infinite time.The
other side is a resurgence. But with such a choice of time (t') the particle
emerges from the resurgence in an infinite time whereas it is able to penetrate
in a finite time. That was a sticking point. The solution consists of effecting
a sort of double change of variable, which we have the perfect right to do,
for the bit of space-time that is supposed to be ours :
In the
"twin universe" :
The
cosmic mechanism thus functions perfectly.
-
No singularity.
-
We can enter the "swallowhole"
but we cannot come out (black hole). -
We can come out of the
resurgence but cannot go in (white fountain).Good,
you're saying, we're making progress...Yes
and no. The problem is that the transit time in the hypertoric passage is just
a few hundredths of a second. And this Moloch is capable of swallowing anything,
ten solar masses for example, in less time than it takes for a bullet to go
through a playing card.The
conclusion is that through this more rational representation of the geometric
solution, black holes could not exist. They are ... mathematical fiction. They
can only exist by virtue of this "time freezing". However with "Eddington time",
which satisfies all the demands of physics, transit time becomes finite.
Conclusion : this Schwarzschild
geometry is, in our opinion, just a snapshot, an instant image, of an unstationary
process of hyperspatial transfer. It is as if when shown a photo of an anvil
someone had thrown in the air you concluded that all anvils float in air. The
Schwarzschild solution is also the solution to an equation that implies that
the universe is perfectly empty, that the density of energy matter is nil at
every point. A bit like if someone showed you the photo of a football stadium
taken after the players had left the field for half-time and you concluded that
football was played on empty fields.
So what
would happen ?
We have
shown that during the passage through the gorge sphere, the time coordinate
was inverted. If we call t' the (Eddington) time corresponding to our "side
of space-time' and t'* the "time marker" of the twin universe we have :
t'* = -
t'
Note
that in 1967 Andrei Sakharov was the first to suggest that two universes with
inverted times were created at the moment of the "Big Bang".
It remains
to understand what is meant by "inversion of time". Does it mean that when we
dive into the twin universe we become younger ? We have shown that this is not
so. We "take our own time" and if we emerged a little further via a symmetrical
structure, we would not be any younger than when we entered the twin universe.
Impossible then to "kill one's own father" as in Barjavel's "Imprudent Traveller"
("Le Voyageur Imprudent").
Groups,
once again, have allowed us to elucidate the "ontological" sense of the time
coordinate inversion. When a particle plunges into the twin universe its gravitational
action makes itself felt, but its contribution to the gravitational field then
becomes negative. Its "gravific mass" inverses.
By the
way, this totally justifies the model developed on the site and in the book
"We have lost half the universe" ("On a perdu la moitié de l'univers")
(Albin Michel). The masses that wander in twin universes behave as repulsive
masses in relation to the masses present in our universe.
-
According to Newton,
Masses attract each other in our universe. -
According to Newton,
masses attract each other in the twin universe. -
When masses situated
in two "adjacent" portions of space-time interact, they repel each other.
This is a simple consequence
of the inversion of the temporal variable (but not of time).
The
groups also show how the duality matter-antimatter exists in both universes,
as imagined by Andrei Sakharov.
When
a particle of matter manages to pass into the twin universe (we'll see how later)
it remains as matter, but "CPT symmetric". This is the meaning of the famous
"CPT theorem" of physics (never proved. What Souriau calls a "physician's theorem").
Classically physicians say " the CPT symmetric of matter is identical to the
matter". CPT symmetry means :
-
The particle, in its
new lair, moves in "backward time" : T-symmetry. -
It is enantiomorphic,
inverted right and left, in "mirror" : P-symmetry. -
All its "charges" are
inverted, including its electric charge, if it has one. This is C-symmetry.For
us, a particle's CPT symmetric is a particle of the twin (or that has entered
the twin). It is a gemellary particle. As it has T-symmetry, its mass is automatically
inverted (a result initially obtained by J. M. Souriau in 1974).
The C-symmetric of a particle
is its antiparticle.
Feynman
found that the PT-symmetric of particle behaved like an antiparticle. Exact,
but it is question of ... the antimatter of the twin, with negative mass (because
it also has T-symmetry). All that appears from the history of groups. This work
makes the link between everything published up to now (on the site see Geometrical
Physics B in the section 'geometrising antimatter'). A good illustration of
this business of inversion of space can be obtained. In the paper we frequently
insist on the idea of a representation space. It is the space in which we mentally
represent geometric objects. Earlier we used an image in which Lanturlu thrusts
his hand in the gorge sphere and it seems to emerge in another 3d universe.
For the requirements of illustration the drawing is separated into two figures.
But there is something that you probably didn't notice at first : Lanturlu puts
his left hand in the sphere but it is his right hand that
emerges. This is not an accident.
Where is
the second universe ?
It is
fitted into ours, which is a little difficult to understand. It will be simpler
if we return to the 2d surface, the "flatland".
