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Potential problems arising from a choice of coordinates.
...We will discuss the risks involved in imposing a coordinate system onto a geometric solution, expressing this solution within that particular coordinate system: one must ensure that the system is appropriate. When we look at the solution above, assuming this geometry solves a field equation, the use of a (r, q) coordinate system presupposes a "locally spherical" topology, in two dimensions, of course. That is, within any circle "centered on this hypothetical geometric center," one could always inscribe a smaller circle, down to the point where the circle becomes a single point. Mathematically, we would say that every circle of radius r defines a "contractible cell."
...In three dimensions, locally, the universe would be "nested like Russian dolls." Inside any sphere, one could always inscribe a sphere with smaller surface area. In three dimensions, this corresponds to a locally spherical topology.
Can it be otherwise?
Yes, if the surface's topology is "locally toroidal." In two dimensions, this looks like this:
...Note: The object in the figure above is a 2D surface in the sense that two parameters are needed to locate a point on it. In this sense, a curve is a "surface of one dimension." When a geometer refers to a circle, he uses the expression "sphere S1," meaning "sphere of a single dimension": only one parameter, the abscissa, is needed to locate a point on a curve, a one-dimensional object. The sphere S2, the "ordinary" sphere, and the circle, the sphere S1, share something in common: they are all "closed" objects (a concept borrowed from topology).
...The number of quantities required to define a point's position in space precisely defines the dimension of that space. Thus, spacetime (x, y, z, t) is considered a four-dimensional hypersurface because four quantities are needed to define a point, called an "event."
End of this note on the concept of dimension.
...One must keep one crucial point in mind. The geometer constructing a particular solution to a field equation is blind—he cannot see the geometric object he obtains. He can only explore it through its geodesics, describing them within a specific coordinate system. The polar coordinates used earlier corresponded to the intersection of the surface with a family of coaxial cylinders:
and with a family of planes passing through the common axis of those cylinders.
In three dimensions, it would correspond to the intersection of space with a family of concentric spheres.
...But what happens if we cut the surface exhibiting this tubular bridge-like structure with a family of concentric cylinders? As long as the cylinders intersect the surface, everything is fine. But when the cylinder's circumference becomes smaller than that of the "throat circle," the sections become... imaginary curves. Let p be the circumference of the throat circle. Associate it with a length Rg such that p = 2πRg.
...It is clear that any cylinder in the family with r < Rg does not intersect the surface. When the geometer examines the shape of the surface's geodesics for r < Rg, he will find imaginary geometric objects.
...When we seek the intersection of a line, for example x = x₀, with a circle, we find two real values for y when the line actually cuts the circle. Otherwise, these values are purely imaginary.
...If a person, exploring a surface in the dark, unable to perceive its shape, and unaware that its topology is locally toroidal, could be extremely bewildered.
The surface can be described using two families of curves:
...Each curve is defined by a single parameter. A point M, at the intersection of these two curves, is uniquely determined by two quantities (a, b), the two values of the curves passing through M.
...The first family consists of circles that are not geodesics of the surface (except for the throat circle), while the second family consists of geodesics with hyperbolic shape, orthogonal to these circles. The hyperbolic curves evoke plunging trajectories that allow passage from one sheet to another.
...Of course, a similar situation can occur in a 3D space with a locally hypertoroidal topology. The circles would be replaced by a family of spheres, among which one would find a throat sphere with minimal area. The lines orthogonal to this family of spheres form plunging trajectories that allow passage through this hypertoroidal tunnel and re-emergence in another 3D sheet (or layer).
...This remark is not idle. We will return to it when we examine the black hole model. Indeed, in this model, when one penetrates "inside the horizon sphere," the mass of a particle becomes... purely imaginary (and many other things as well). We are then justified in asking whether we are still within the spacetime hypersurface. Is the particular choice of coordinates (t, r, θ, φ), which implies a locally hyperspherical topology (existence of a radial coordinate r capable of taking values smaller than the Schwarzschild horizon radius), still relevant?
A well-known astrophysicist wrote a few years ago:
- We now know much more about the interior of black holes.
But if black holes exist, do they have an interior, or do they instead correspond to a locally hypertoroidal topology?
...We see how much can be induced by a choice of coordinate system. The geometric solution exists. It has geodesics. But we can only "read" this through projecting it into our mental representation space: a Euclidean spacetime, which is not even relativistic. Choosing a coordinate system is equivalent to choosing a reading system, a projection system.
...Like Plato's characters, we can only observe shadows on a "Euclidean screen." But we must still choose the right lens for the "projection system."
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