But there are surfaces that are intrinsically singular, possessing singularities that are not due to a choice of coordinates. Example below: the conical singularity.
Rough as cast, as formulated in 1917 by Schwarzschild in coordinates t , r , q , j (time, a radial distance and two angles, equivalent to azimuth and elevation: "spherical" coordinates) the Schwarzschild sphere is singular. For a certain value Rs of the "radial coordinate" r (supposedly measured from a "geometric center") this metric plays the worst tricks on us. On this sphere one of the terms has a zero denominator. In short, it is singular on this sphere. Was this an intrinsic singularity or an artifact induced by a bad choice of coordinates? That is the question we asked ourselves.
Let us note in passing that the "Schwarzschild geometry" is a four-dimensional hypersurface, which makes the thing even more awkward.
Kruskal focused on this point. He constructed a change of coordinates which, among other things, provides a constant value of the speed of light along a radial trajectory. In doing so he concentrates the singular aspect "at the center of the object", in a "central singularity". Psychologically, one has the impression of gaining from it. The solution becomes "regular almost everywhere", an expression that mathematicians use to say that the solution is regular, free of pathology, except at a single point.
- You are not going to quibble, to pick a quarrel with me, over a single point.....
Alas this formulation of Kruskal has a serious flaw: it does not give back the space of special relativity at infinity. Technically, it is not Lorentzian at infinity, "asymptotically Lorentzian".
This is an essential question in physics: do singularities exist? Does Nature tolerate singularities? The answer is formulated in terms of belief (as for the existence or nonexistence of infinity, moreover).
We have sought a new interpretation of this same Schwarzschild geometry by trying to eliminate all singularity and we have succeeded. Our answer is therefore:
- The singular character of the Schwarzschild solution is simply induced by a bad choice of coordinates.
Technically, everything rests in the change of variable:
r = Rs + Log ch r
which reads " r equals Rs plus logarithm of the hyperbolic cosine of the variable r ". Simple aside for the scientist, specialist or simple crammer. For anyone who knows how to handle this formula, the quantity r can then no longer become less than Rs, even when r takes all possible values from minus infinity to plus infinity.
Consider a surface that is obtained by rotating a parabola around a line, like this:
This figure is taken from the article. The surface is infinite, in fact, like the meridian parabola that generates it by rotating around the z axis, shown.. If one absolutely insists on representing it with coordinates (r,z,j) one can expect trouble when one asks "what is this surface like for r < Rs ?"
One will find an answer .. imaginary, with roots of negative quantities. Quite simply because one is then "off the surface".
This surface, in mathematics, is said to be "not simply connected", a barbarous term that simply designates surfaces where any closed curve cannot see its perimeter decrease, by sliding it along the surface, until it takes the value zero.
This is possible on a sphere, which is "simply connected". But on that surface one clearly sees that any closed curve that "goes once around this kind of central well" cannot see its perimeter tend toward zero, the limit being the perimeter of the "throat circle". Same thing for a torus, which is also "not simply connected".
We have defined such a surface starting from its metric, which illustrates the point very well. Keeping the coordinate r, this surface seems singular. Using the change of variable given above, it no longer is. What does this coordinate r correspond to? It simply "runs" along the meridian parabola as indicated in the figure, takes the value zero on the throat circle. Half of the surface corresponds to positive r, the other to negative r. In the coordinate system of the points [r , j ] there is no longer any singularity.
We have decided to call this type of object a "toroidal bridge", by analogy with the torus.
But one easily shows, still starting from metrics, that one can pass to an object, a 3d hypersurface, that includes a "hypertoroidal bridge". There is then no longer a throat circle, but a throat sphere. Just as for the surface above, a throat circle seemed to put two 2d sheets in communication, the throat sphere then puts two "3d half-spaces" in communication. When one is in one of these 3d half-spaces and one plunges into the throat sphere, one emerges in the other half-space.
Let us return to the 2d surface shown above. The drawing below shows that by tracing "circles that one believes to be concentric" one sees their perimeter diminish, pass through a minimum, then grow again.
In 3d one must imagine a sphere completely surrounding the throat sphere. Then another, inside that one (one should say "beyond" by following a given direction, toward this throat sphere). One imagines that the surface of this sphere could be smaller. But, when one reaches the throat sphere, the area passes through a minimum, then begins to grow again... to infinity, when one extends the operation.
We have constructed the "metrics" of these 2d and 3d surfaces including a "toroidal passage" and a "hypertoroidal passage" and, in the second case, we were struck by the resemblance to the Schwarzchild metric, where we therefore carried out this change of coordinates, bringing out its "not-simply-connected" character, "the interior" of the object simply becoming "the beyond of its throat sphere".
It was thus possible to eliminate all singularity.
At this stage we have simply extended the black hole model to a "black hole-white fountain" tandem. But, still for this "external observer", the crossing time of this hypertoroidal bridge was still infinite. We seemed to have simply improved the black hole model by explaining what it led to.
We have said that the choice of variables was, in a geometric solution, totally arbitrary. But what holds for space also holds for time. We therefore went looking for a change of temporal variable invented by Eddington in 1924:
There again, we mention it for the scientist or simple crammer.
t is the old "cosmic time", the old "chronological variable" present in the initial Schwarzschild solution, from 1917.
t' is this new "Eddington time". Rs is the "Schwarzschild radius (one should then speak of Schwarzschild perimeter, divided by 2p ).
c is the speed of light (here, constant).
Something that may seem bizarre: one mixes time and space but, in the matter, one has every right. The choice of the time coordinate, of the chronological marker coordinate (time-marker) is totally arbitrary. One simply asks:
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for the metric to be asymptotically Lorentzian, that is to say that at infinity, space-time becomes the Minkowski space-time, that of special relativity. In our case, it works (not with Kruskal).
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for this new time t' to identify itself, still at infinity, with the "proper time of an observer supposed to be motionless". This is also the case (not with Kruskal)
In doing so the free-fall time of a test particle, motionless at infinity and falling toward the Schwarzschild sphere, becomes infinite, relative to the time experienced by the "external observer", who is far away and motionless.
The particle, on the other hand, would emerge from this "aven" in an infinite time. As in the black hole one can therefore enter this kind of 3d aven, but not leave it, except in an infinite time.
The other side is a "resurgence". But, with such a choice of time (t') the particle emerges from this resurgence in an infinite time, whereas it can penetrate it in a finite time. It was getting stuck somewhere. The solution consisted in carrying out, which one has every right to do, a kind of double change of variable. For the bit of space-time that is supposed to be ours:
In the "twin universe":
The cosmic mechanism then works perfectly.
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No singularity.
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One can enter the "aven" but not leave it (black hole)
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One can leave the resurgence, but not penetrate it (white fountain).
Well, you will tell me, we are making progress....
Yes and no. The transit time of matter in this hypertoroidal passage is some tens of thousandths of a second, and this Moloch is capable of swallowing anything, ten solar masses, for example, in less time than it takes a bullet to cross a playing card.
The conclusion is that through this more rational reinterpretation of the geometric solution black holes could not exist. They are ... mathematical fictions. They could only exist by virtue of this "freezing of time". Now with this "Eddington time" which satisfies all the requirements of physics, the transit time becomes finite.
Conclusion: this Schwarzschild geometry would, according to us, be only an instantaneous image of an unsteady process of hyperspatial transfer. It is a bit as if one showed you a photo of an anvil that someone had thrown into the air and you concluded that anvils can float in the air.The Schwarzschild solution is also a solution of an equation that implies that the universe be perfectly empty, that the density of matter energy be zero at every point. It is then a bit as if one showed you a photo of a football field, taken when the players have gone to rest during the half-time break and you concluded that football is played on a.. empty field.
But then, what would happen?
We have shown that during the passage of the throat sphere the time coordinate was inverted. If one calls t' the (Eddington) time corresponding to our "space-time slope" and t'* the "time marker" of the twin universe, one has:
t'* = - t'
Let us note that Andrei Sakharov was the first to suggest in 1967 that two twin universes having opposite times were created at the moment of the "Big Bang".
It remained to know what the meaning of this "inversion of time" signified. Did it mean that when one plunged into the twin universe one grew younger? We have shown that it did not. One "brings one's proper time along" and if one came out a little further away through a symmetrical structure, one would not emerge younger than when one entered this twin universe. Impossible to "kill one's own father" as in the theme of the "Imprudent Traveller", by Barjavel.
Groups, again, allowed us to elucidate the "ontological" meaning of this inversion of the time coordinate. When a particle plunges into the twin universe, its gravitational action is still felt, but its contribution to the gravitational field then becomes negative. Its "gravific mass" is inverted.
Incidentally, this fully justifies the model developed in the site and in the book "On a perdu la moitié de l'univers" (Albin Michel). The masses that roam in the twin universe behave with respect to the masses present in ours as repulsive masses. The complete dynamics is:
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In our universe the masses attract according to Newton
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In the twin the masses attract according to Newton
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When masses located in two "adjacent" portions of space-time interact, they repel each other.
It is a simple consequence of this inversion of the temporal variable (but not of proper time).
Groups also show that the matter antimatter duality exists in both universes, as conjectured by Andrei Sakharov.
When a particle of matter succeeds in passing into the twin universe (we will see further on how) it remains matter, but "CPT symmetric". This is the meaning of the famous "CPT theorem" of physics (never proven. What Souriau calls a "physicist's theorem"). Classically physicists say "the CPT symmetric of matter is identical to matter". By CPT symmetry, understand:
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The particle, in this new frame, travels "back in time": T-symmetry.
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It is enantiomorphic, inverted right-left, "in a mirror": P-symmetry.
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All its "charges" are inverted, including its electric charge, if it possesses one. It is C-symmetry.
For us, the CPT symmetric of a particle is a particle of the twin (or passed into the twin). It is a twin particle. As there is T-symmetry, its mass is automatically inverted (result initially obtained by J.M.Souriau in 1974).
The C-symmetric of a particle is its antiparticle.
Feynman had found that the PT-symmetric of a particle behaved like an antiparticle. Correct, but it is a matter of .. the antimatter of the twin, with negative mass (since there is also T-symmetry). All this emerges from stories of groups This work makes the link with everything that had been published until then (in the site, see Geometrical Physics B under the heading "geometrization of antimatter"). One can illustrate in a speaking way this business of inversion of space. In the paper, one insists quite a bit on the notion of representation space. It is the space in which we mentally represent geometric objects. We used above an image where one saw Lanturlu putting his hand into this throat sphere, which seemed to emerge in another 3d universe. For drawing convenience the two figures have been separated. But there is one thing you will doubtless not have noticed at first glance: Lanturlu puts a left hand into this sphere and it is a right hand that emerges from it. This had not been done by chance.
Where is this second universe?
It is nested in ours, which is difficult to understand. But this will be simpler by returning to the 2d surface, to "flatland".