But there are surfaces that are intrinsically singular, possessing singularities that are not due to a choice of coordinates. Example below: the conical singularity.
Casting, as formulated in 1917 by Schwarzschild in coordinates t, r, q, j (time, a radial distance, and two angles, equivalent to azimuth and polar angle: "spherical" coordinates), the Schwarzschild sphere is singular. For a certain value Rs of the "radial coordinate" r (supposed to be measured from a "geometrical center"), this metric plays the worst tricks. On this sphere, one of the terms has a zero denominator. In short, it is singular on this sphere. Was it an intrinsic singularity or an artifact induced by a bad choice of coordinates? This was the question we asked ourselves.
Note in passing that the "Schwarzschild geometry" is a four-dimensional hypersurface, which makes the matter even more complex.
Kruskal focused on this point. He constructed a coordinate change that, among other things, provides a constant speed of light along a radial trajectory. By doing so, he concentrates the singular aspect "at the center of the object", in a "central singularity". Psychologically, it seems like an improvement. The solution becomes "regular almost everywhere", a term mathematicians use to say that the solution is regular, free of pathology, except at a single point.
- You won't quibble, you won't argue with me over a simple point...
Unfortunately, Kruskal's formulation has a serious flaw: it does not recover 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 non-existence of infinity, for example).
We sought a new interpretation of the same Schwarzschild geometry by trying to eliminate all singularities, and we succeeded. Our answer is therefore:
- The singularity of the Schwarzschild solution is simply induced by a bad choice of coordinates.
Technically, everything rests on the change of variable:
r = Rs + Log ch r
which reads "r equals Rs plus the logarithm of the hyperbolic cosine of the variable r". Simple enough for a scientist, specialist or student. For anyone who can handle this formula, the quantity r can no longer become less than Rs, even when r takes all possible values from minus infinity to plus infinity.
Consider a surface obtained by rotating a parabola around a straight line, like this:
This figure is taken from the article. The surface is infinite, in fact, like the meridional parabola that generates it by rotating around the z-axis. If you insist on representing it with coordinates (r, z, j), you can expect problems when you ask "what does this surface look like for r < Rs?"
You will find an answer... imaginary, with square roots of negative quantities. Simply because you are then "outside the surface".
In mathematics, this surface is said to be "not simply connected", a fancy term that simply means surfaces where any closed curve cannot have its perimeter decrease, by sliding it along the surface, until it reaches zero.
This is possible on a sphere, which is "simply connected". But on this surface, it is clear that any closed curve that "goes around this kind of central well" cannot have its perimeter tend to zero, the limit being the perimeter of the "throat circle". The same applies to a torus, which is also "not simply connected".
We defined such a surface starting from its metric, which illustrates the point very well. Keeping the coordinate r, this surface seems singular. Using the above variable change, it is no longer singular. What does this coordinate r correspond to? It simply runs along the meridional parabola as shown in the figure, taking the value zero on the throat circle. Half of the surface corresponds to positive r, the other to negative r. In the coordinate system [r, j] of the points, there is no longer any singularity.
We decided to call this type of object a "toroidal bridge", by analogy with the torus.
But it can be easily shown, again starting from metrics, that one can pass to an object, a 3D hypersurface, which contains a "hyper-toroidal bridge". There is then not a throat circle, but a throat sphere. Just as for the surface above, a throat circle seemed to connect two 2D sheets, the throat sphere then connects two "half-spaces 3D". When you are in one of these 3D half-spaces and you dive into the throat sphere, you emerge in the other half-space.
Returning to the 2D surface shown above, the following drawing shows that by drawing "circles that one might think are concentric", their perimeter decreases, reaches a minimum, and then increases again.
In 3D, imagine a sphere completely surrounding the throat sphere. Then another, inside this one (one should say "beyond" following a given direction, towards this throat sphere). Imagine that the surface of this sphere could be smaller. However, when you reach the throat sphere, the area reaches a minimum, then starts increasing again... up to infinity, when you continue the process.
We have constructed the "metrics" of these 2D and 3D surfaces with a "toroidal passage" and a "hyper-toroidal passage", and in the latter case, we were struck by the similarity with the Schwarzschild metric, so we performed this coordinate change, revealing its "non-simply connected" nature, "the inside" of the object becoming simply "the other side of its throat sphere".
Thus, it was possible to eliminate all singularities.
At this stage, we have simply extended the black hole model to a "black hole-white hole" pair. However, for this "external observer", the traversal time of this hyper-toroidal bridge was still infinite. It seemed we had simply improved the black hole model by explaining what it led to.
We said that the choice of variables is entirely arbitrary in a geometric solution. But what applies to space also applies to time. So we looked for a temporal variable change invented by Eddington in 1924:
Again, we mention it for the scientist or student.
t is the old "cosmic time", the old "chronological variable" present in the initial Schwarzschild solution of 1917.
t' is this new "Eddington time". Rs is the "Schwarzschild radius (one should then speak of the Schwarzschild circumference, divided by 2π)."
c is the speed of light (here, constant).
Something that may seem strange: we mix time and space, but in this matter, we have all the rights. The choice of the time coordinate, the chronological marker (time-marker), is entirely arbitrary. We simply ask:
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that the metric be asymptotically Lorentzian, that is, that at infinity, space-time becomes Minkowski space-time, that of special relativity. In our case, it works (not for Kruskal).
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that this new time t' identify...