Here, we have slightly "degreased" the figure to make it more readable. A surface is a 2D object, here "embedded" in a 3D Euclidean space, otherwise in R3. Above, we can "see" it. It turns out that this surface is embeddable in this R3 space "isometrically". That is, if we stick a piece of tape on it, this tape will actually lie on a geodesic connecting two points A and B of the surface. Moreover, the length measured along this geodesic arc is also correct. It is isometric, etymologically "same length". At the bottom is a 2D representation space, which gives a representation that is not isometric. The length of the arc A'B' is not equal to that of the arc AB. Make the following object, using a sheet of paper, a pencil, and a pair of scissors:
This drawing is not isometric. First, the curve indicated is obviously not a geodesic of the plane. Second, the length of the arc AB is not "the real length", the one we would measure on "the real surface", which "is not pierced". This pierced sheet of paper is only a convenient representation, nothing more. Similarly, this technique of drawing one part on the front side of the sheet and the other on the back, with the whole curve only appearing in transparency.
In the following drawing, the geodesics of this surface are shown, calculated by computer (which is in the article).
The parts of the curves that are dashed correspond to the branches that are "on the other side" (as if we were looking at 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 can change the variable r into the variable r. Then the geodesics can perfectly be represented in a plane of "polar coordinates" (r, j). The geodesics (here a non-radial geodesic) have the following appearance:
This representation is isometric. Let three points A, B, C belong to the surface, located on the same geodesic. A', B', and C' are the corresponding points in this representation [r, j]. Points A and B are located on the same half-disk and the geodesic arc connecting 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 the arc A'B' is equal to that of the arc AB, measured on the surface.
The arc BC crosses the throat sphere. Same thing.
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 on itself.
Geodesics are our only way of understanding a surface or, more generally, a non-planar, non-Euclidean space. They are reliable references (even though we have distorted views of them through our 2D or 3D representation systems (in perspective). These geodesics, we know they "exist", they are intrinsic. Those of a sphere, for example, are great circles. Regarding space-time, they are populated by an infinite number of space-time geodesics. These geodesics exist intrinsically and, to understand (etymologically: to enclose, to embrace), we search, like the blind, to "feel" these geodesics. But the lines of time and space coordinates have no intrinsic reality, just as the two sets of meridians and parallels are not an integral part of a sphere. They are not "provided with". The Schwarzschild geometry, a solution to Einstein's field equation, is a 4D hypersurface. On it, theorists have imposed families of curves "at constant t", "at constant r", etc.
Carve into your head that these gestures remain completely arbitrary. But even theoretical cosmologists often lose sight of this point, which mathematicians-geometers occasionally have to remind them of. Therefore, it was perfectly legitimate to change space and time coordinates.
At this point, you might say: but then, what allows us to say that one choice of coordinates is better than another? Where is the reasonable and the unreasonable? It is a matter of taste. Choosing space and time coordinates is to impose a physical view on 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 towards the North Star, an object 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. On the other hand, the inhabitant of one of the poles has an unpleasant surprise: their domain, a point, becomes a straight line...
There are thirty-six thousand ways to project a sphere onto a plane. Imagine this one:
Imagine we were to make geographical maps in this model and sell them. Immediate success among the inhabitants of the two poles: these projections are then, in these regions, almost isometric. Very convenient for getting an idea of distances in these corners. If the Earth had been habitable at its poles and relatively inhospitable elsewhere, the maps would probably have been created this way. It should be noted that then 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. Moreover, on this strange map, part of the territory would have to be represented in solid lines and the other in dashed lines, since it is beyond this parallel where the object strangely seems to "fold back". Unless we provide maps in the shape of disks, with the rest of the terrain represented on the other side, on the back of the sheet.
Let's try to "think all this in 3D". We have represented Lanturlu inserting his left arm into the throat sphere and we have separated the two drawings, which seems to suggest that this second 3D space could be "elsewhere". To be more accurate, we should have overlapped the two drawings in perspective, representing the hand (right) emerging "in dashed lines".
I tried to do it, although it was not extremely easy. One could also have used two different colors, for example red for what would be in the first 3D side of our non-simply connected 3D space and green for what is in the other side. A red Lanturlu would then see, for example, his red left hand, which he had plunged into the throat sphere, emerge as a green "right" hand.
Too bad Raymond Devos is not interested in math. Although...
Obviously, "inside" the throat sphere, there is nothing. This appearance of an interior, of a volumetric content, is due only to the choice of this 3D representation space. Similarly, inside the hole made in the sheet of paper, there was no paper either. It was just an accident related to the choice of this planar representation space. Someone who would persist in using this planar representation without removing the cut-out disk from the paper and who would continue to...