A: HTML Presentation\PQ4.htm Let's now (and these figures are then extracted from the article) design, still within a 3D representation space, a set of four small balls forming a tetrahedron (an object highly orientable) that falls into a throat-shaped sphere, along "radial geodesics".
They will "bounce" off this throat-shaped sphere (according to this imagery derived from our choice of representation space. In fact, on the 3D hypersurface, geodesics are continuous).
I remember, when I was younger, there were often chrome balls at the ends of stair railings. If you live in a dwelling that has such an object, you could perform the experiment, using your four hands, by launching small steel balls onto it.
After bouncing, the four balls will form an inverted tetrahedron:
We will enlarge the tetrahedron to better see this inversion. In its initial configuration, it looks like this:
We "orient" its faces. For example, we give a direction ADB, etc., such that by assimilating this "movement" to that of a corkscrew, the tip of the corkscrew points outward (arrows). The four faces are thus oriented. Let's now compare this tetrahedron with the one formed by the balls that "bounced" off the throat-shaped sphere:
The orientation of the faces has been reversed. If my drawing had been more precise, the two objects could have been placed on either side of a mirror, one being the enantiomorph of the other.
For Schwarzschild, it's the same thing: objects reappear "on the other side", and if we could "see them in transparency", they would appear enantiomorphous. But we cannot see them "in transparency". To "see" them, photons would have to communicate between two "adjacent" regions of these two "sides of spacetime", which are therefore P-symmetric.
By the way, what about "non-radial" trajectories? Calculating geodesics gives plane trajectories that "bounce" off the Schwarzschild sphere. See the following drawing.
There remains this issue of the time variable, briefly mentioned above. As I told you, in terms of variable choice, we have all the rights. This choice remains completely arbitrary, since the object, the spacetime hypersurface, is "coordinate invariant", existing independently of the choice of coordinates used to locate the points on it, which are "event points", points of a spatio-temporal object, a 4D hypersurface.
But then, what is time, what is space, if everything is so arbitrary?
There is a time that we cannot touch, which is the only intrinsic scalar of the hypersurface: it is the proper time. Proper time is the "length" within this spacetime hypersurface. We assume that objects can only move along geodesics (4D). On a geodesic, we take a pair of points (A, B). The length Ds separating these two event points, divided by c, a constant, in this case the speed of light in a region far from the throat-shaped sphere, is the proper time Dt separating these two "events", and this is regardless of the chosen spacetime coordinate system. This quantity Dt is the only one that has an intrinsic physical meaning.
Imagine you are traveling along the Earth's sphere, along a geodesic (a great circle), from point A to point B. If you say:
- I went from a point of longitude jA and latitude qA to a point of longitude jB and latitude qB
What would the quantities (jB - jA) and (qB - qA) mean? They would depend on the points where you decided to place your poles, on your choice of coordinate system. However, if you say:
- I traveled 2347 kilometers along this geodesic.
This measurement will have meaning regardless of the coordinate system chosen.
We saw with the sphere that we could install coordinates that make one or more singularities appear. A pole is a place where the longitude j is no longer defined. We also saw that by a simple change of coordinates, we could make an "undesirable region of a surface" (where r < Rs) disappear, where we would find an element of length Ds purely imaginary. Indeed, it was the fact that, in its initial formulation, the Schwarzschild metric led to a purely imaginary element of length (proper time) that made us suppose we were then "outside the hypersurface". There is no absolute coordinate system. But we can decide to choose a spatial coordinate that at least has the merit of eliminating the singularities, which we have done. There is also no "absolute cosmic time". With Midy, in our last paper, cited above, we showed that the "initial singularity", considered as "the moment of the creation of our universe", resulted from a particular choice of the time variable, and that another choice, not only preserved all the observables, starting with the redshift, but also eliminated this initial singularity, like the original sin of the same name. The question "what was there before the Big Bang?" then loses its meaning. Disconcerting, I agree, but the question comes from a spacetime paradigm. It is equivalent to: "what is at the center of a black hole?" It is therefore perfectly legitimate to change the time coordinate, using "Eddington time" (the change of variable was indicated above), as long as it allows connecting this local geometric structure with Minkowski spacetime, that of a relativistic (in the sense of special relativity) and flat, non-curved, empty space. But the idea is to be able to describe the entire spacetime with a single metric. The guiding thread is once again in group theory and in the examination of the "isometry group" of the Schwarzschild metric.
The isometry group contains all the geometric transformations that leave the metric invariant (and thus the hypersurface invariant). The isometry group of the sphere is the group of rotations in space plus symmetries (with respect to a plane or an axis passing through its center, or with respect to a point which is the center). This group is called O3 (abbreviation of "three-dimensional orthogonal group". See the introduction of Geometrical Physics B). It contains all of that. But if we remove the symmetries with respect to an axis, a plane, or a point, it becomes SO3 (the "special orthogonal group of dimension three").
The geometry of Schwarzschild has symmetries. Until now, it was customary to attribute it the SO3 symmetry (rotations in space). But in fact, it has the O3 isometry group, so it contains the P-symmetry (symmetry with respect to a point). Take the tetrahedron from earlier. Its symmetric with respect to a point is enantiomorphous, P-symmetric of the first.
In the group section of the site, we showed how the group "secretes" space or more precisely secretes geometric objects. Souriau calls them "species" of the group. Thus it is not the sphere that generates the SO3 group, but the opposite. The spheres are the species of this group. Species in the taxonomic sense of the term (Larousse. Taxonomy: science of the classification of species). We have said that it happens to physicists to do mathematics without knowing it and vice versa. The relativistic physics...