Here, we have slightly "trimmed down" the figure to make it more readable. A surface is a 2D object, here "immersed" in a 3D Euclidean space, otherwise known as R3. Above, we can "see" it. It turns out that this particular surface can be immersed in this R3 space "isometrically." This means that if we stick a strip of tape onto it, the tape actually lies along a geodesic connecting two points, A and B, on the surface. Furthermore, the length measured along this geodesic arc is also correct. It is isometric, etymologically meaning "same length." Below is a 2D representation space, which provides a non-isometric representation. The length of arc A'B' is not equal to that of arc AB. Create the following object using a sheet of paper, a pencil, and a pair of scissors:
This drawing is not isometric. Firstly, the indicated curve is clearly not a geodesic of the plane. Secondly, the length of arc AB is not the "true length," the one that would be measured on the "true surface," which is "not holed." This holed sheet of paper is merely a convenient representation, nothing more. Similarly, this technique of drawing one stroke on the front of the sheet and another on the back, with the entire curve appearing only in transparency, is just a representation.
In the following drawing, the geodesics of this surface, calculated by computer (as mentioned in the article), are depicted.
The dashed parts of the curves correspond to branches that are "on the other side" (as if we were viewing the surface "from above").
Now, a question: Can I construct a planar and isometric representation of these geodesics? The answer is yes. We have seen that we could change the variable r into the variable r. Thus, the geodesics can be perfectly depicted in a plane of "polar coordinates" (r, j). The geodesics (here, a non-radial geodesic) have the following appearance:
This representation is isometric. Let A, B, and C be three points belonging to the surface, situated on the same geodesic. A', B', and C' are the corresponding points in this [r, j] representation. Points A and B are located on the same half-sheet, and the geodesic arc joining them does not cross the throat circle. Measured in this plane, along the image of this geodesic (which is obviously not a geodesic of this plane), the length of arc A'B' is equal to that of arc AB, measured on the surface.
The arc BC
crosses the throat sphere. The same applies.
But this isometry does not apply to all geodesics of the surface. There is one, unique in its kind: the throat circle, reduced here to a point. It is the only geodesic of this surface that closes in on itself.
Geodesics are our only way of understanding a surface or, more generally, a non-planar, non-Euclidean space. They are reliable reference points (even if we have distorted visions of them through our 2D or 3D representation systems (in perspective)). We know that these geodesics "exist," that they are intrinsic. Those of a sphere, for example, are great circles. Regarding spacetime, it is populated by an infinity of spacetime geodesics. These geodesics exist intrinsically, and to understand (etymologically: to enclose, to take in one's arms) them, we seek, like blind people, to "feel" these geodesics. But the time and space coordinate lines have no intrinsic reality, just as the two sets of meridians and parallels are not an integral part of a sphere. They are not "delivered with it." Schwarzschild geometry, a solution to Einstein's field equation, is a 4D hypersurface. Theorists have superimposed families of curves "at constant t," "at constant r," etc., onto it.
Keep in mind that these choices remain totally arbitrary. But even specialists in theoretical cosmology often lose sight of this point, which mathematician-geometers must remind them of from time to time. It was therefore perfectly legitimate to change space and time coordinates.
At this stage, you might ask: what allows us to say if one choice of coordinates is better than another? Where do reasonableness and unreasonableness lie? It is a matter of taste. Choosing space and time coordinates means superimposing a physical vision onto a mathematical object. In the case of the Earth, we gave it poles because it rotates. The North Pole is simply the point on the "Earth" surface whose normal points toward the Pole Star, a celestial body that remains fixed in the starry vault.
Regarding isometry and non-isometry, cartography illustrates the problems resulting from attempts to represent a sphere on a plane. The Mercator projection (projection of the Earth's sphere onto a cylinder tangent along the equator) is very pleasant for people living near the equator. However, an inhabitant of one of the poles gets a nasty surprise: their domain, which is punctual, transforms into a line...
There are thirty-six thousand ways to project a sphere onto a plane. Let us imagine this one:
Imagine that we make geographical maps based on this model and sell them. Immediate success among the inhabitants of the two poles: these projections are then, in these regions, quasi-isometric. Very convenient for getting an idea of distances in those corners. If the Earth had been habitable at its poles and relatively inhospitable elsewhere, maps would probably have been created this way. Note that in this case, the boundary circle of the planar projection no longer corresponds to the equator, but to a parallel (here belonging to the northern hemisphere). Near this region, the map will be far from isometric. Furthermore, on this strange map, part of the territory must be depicted with solid lines and the other part with dotted lines, since it lies beyond this parallel where the object, strangely, seems to "fold back." Unless we provide maps shaped like disks where the continuation of the terrain would appear on the other side, on the back of the sheet.
Let us try to "think all this in 3D." We represented Lanturlu plunging his left arm into the throat sphere and separated the two drawings, which seems to evoke the fact that this second 3D space could be "elsewhere." To be more accurate, we should have superimposed the two drawings in perspective, representing the (right) hand emerging "in dotted lines."
I tried to do this, although it was not extremely easy. We could also have used two different colors, for example, red for what would be in the first 3D slope of our non-simply connected 3D space and green for what is in the other slope. A red Lanturlu would then, for example, see the red left hand he plunged into the throat sphere re-emerge as a green "right" hand.
It is a shame that Raymond Devos is not interested in math. Although...
Obviously, there is nothing "inside" the throat sphere. This appearance of an interior, of volumetric content, is due solely to the choice of this 3D representation space. Just as there was no paper inside the hole made in the sheet of paper either. It was merely an accident linked to the choice of this planar representation space. Someone persisting in using this planar representation without removing the disk cut out of the paper and continuing to ask "what is inside?" would be "off the mark" (or rather... inside it). This "plate" "does not exist."
Let us return to 3D. When Lanturlu plunges his arm into the throat sphere, it also has no interior. This appearance of an interior is due solely to this choice of representation space. One could consider that Lanturlu and his emerging hand were drawn on a sheet of paper possessing three dimensions, from which we had removed... a sphere (the 3D equivalent of the disk in the sheet of paper). Mathematically, a disk is a "b2 ball" and a "volume sphere" is a "b3 ball." A "ball" is called a contractile cell (see the Topologicon, in the "CD-Lanturlu"), that is to say, an object that can contract to a point while traversing itself. These 2D and 3D examples are here to illustrate the approach followed in the article: the Schwarzschild sphere has no "interior," nor a "center." When one crosses it (hypertoric passage), one finds oneself in "another slope of spacetime."
What is the justification for this new interpretation of "Schwarzschild geometry"?
Answer: the elimination of singularities. Kruskal, with his "analytic continuation," went to great lengths to penetrate "inside" this damned sphere at all costs. He only succeeded in compressing the singularity (initially held by the Schwarzschild sphere) into a point located "at the center of this object." We settled for this sleight of hand. But we think that without singularities, it is still better.
Nature protests, when viewed from the wrong angle, by secreting singularities. That is how we see things. It is an a priori viewpoint on "reality." We believe that singularities do not exist in nature. We also believe that infinity does not exist either. But, as Kipling would say, that is another story. I had heated discussions with Souriau on this question a year ago.
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What proves to me that infinity exists? ....
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But without infinity, there are no more mathematics! ....
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Have you ever encountered infinity? Have you ever seen it, held it in your hand?
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It is a .... convenience.
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We generate infinitely large numbers by assuming that we can add 1 to a number, indefinitely. So we use a sequential infinity to generate a numerical infinity. Your reasoning is circular.
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Well, let's say it is a convenience. Man has invented two important things throughout his history: infinity and toilets....
I do not believe either that the infinitely small exists, neither physically nor mathematically. But that will be the subject of other papers. Let us leave these questions aside for the moment. A simple digression.
Incidentally, see "Scale Invariant Cosmology" by P. Midy and J.P. Petit, International Journal of Modern Physics D, June 1999, pages 271-280, where we made "the initial singularity of the universe" disappear. It is a more mathematically constructed reformulation of the idea I had published in 88-89 in Modern Physics Letters A (papers on the site).
I hope that when you read these lines, we will have had the time, with the help of Messrs. Lecot or Boland, to include this new article on the site. But in any case, even if this work were reproduced there, do not venture into it. As Archimedes wrote, I believe, at the entrance to a science-sanctuary: "Let no one enter here who is not a geometer." It is tensors and the like, a domain in which Midy delights, but it is also as indigestible as English pudding.
We see, therefore, through this discourse, that the physical vision we have of phenomena stems from the way we decided to represent them. By modifying spatial coordinates, we changed the "local topology," a word that would deserve to be mathematically defined, says Souriau. In fact, this sentence is a gentle euphemism: he simply started screaming when I pronounced this word. His cat Pioum and I had all the trouble in the world calming him down. Souriau is the Tintin of mathematics. He readily practices mathematical indignation carried to its peak. But these fits of indignation should not be confused with anger in the trivial sense of the term. I would rather be "Monsieur Jourdain" in this domain. Physicists often do mathematics without knowing it (and vice versa, for that matter).
Admitting, provisionally, the use of "undefined" words, everything happens as if we had considered until now that the "local topology" of Schwarzschild geometry was "hyperspherical" (that the Schwarzschild sphere "contains" a "b3 ball"). We made it "hypertoric." This is why I proposed the (uncontrolled) designation of "hypertoric geometries."
We mentioned space inversion earlier. This is negotiated with groups. Can this be understood otherwise? We saw that Lanturlu, plunging his left arm into the throat sphere, saw emerge ... a right hand. In fact, each atom of his hand followed a "radial" geodesic, directed perpendicularly to the surface.
Let us note in passing, we had forgotten to mention, that this mode of representation is not isometric. As with the holed paper trick earlier. If we measured the length traveled by a witness atom belonging to Lanturlu's hand in the two half-spaces, this would not coincide with the true length that can be calculated with a string.
Let us take up a drawing given earlier.
Here, we have depicted a geodesic arc AB that crosses the throat circle, and its image in the planar representation space, below. The non-isometric nature of the representation then appears even more clearly. The lengths of arcs AB and A'B' are very different.
It is obviously quite difficult to imagine that one can thread a string through the throat sphere of a hypertoric passage. By tightening the string, one would obtain a geodesic (shortest path line). Moreover, if one measured the length of this string in the 3D representation space (Lanturlu threading his arm) and ventured to measure the length of said string in this space, one would find a shorter length A'B'. The true length, measured in the 3D hypersurface, would be longer, as depicted above in 2D. The 3D representation featuring Lanturlu is therefore non-isometric, like the planar representation in the figure above.
Ultimately, these subtle concepts, stemming from group theory, become less opaque thanks to a few drawings, provided one "sees in space." What I am trying to teach you is to see in a curved 3D space.
Let us return to this matter of enantiomorphism, the inversion of objects when they cross the throat structure, in 2D or 3D. Consider, in 2D, "radial" geodesics. The term has become improper since a radius is in principle a line starting from a point. These are actually geodesics at constant j. See a figure, earlier, where this azimuthal coordinate was indicated. But, for conciseness, we will continue to use the word "radial" in quotation marks. Note that this word "radial" is still a consequence of the choice of a representation space. Imagine that a letter R (which is not identical to its mirror image, its enantiomorphic image) slides, like a poorly stuck decal, along our toric passage, each of its points moving along a geodesic. This letter will end up "on the other side." What is interesting is to look at the result of the operation in the planar projection, in the representation space.
We represented a sort of ribbon, whose edges consist of two geodesics. What do we observe? In the representation space, the letter R has inverted, becoming a Russian "ia," an upside-down R, enantiomorphic. We are beginning to understand now why Lanturlu's hand, upon re-emerging in "our 3D representation space," seems to have inverted, becoming enantiomorphic.