- More about embedding and geodesics.
Not all surfaces can be embedded in R³. For example, consider the metric (134)
where Rs > 0 and r > 0
is defined on R modulo 2 
Expressed using these peculiar coordinates [r, ], this line element is regular almost everywhere (except at the point r = 0). Elsewhere, no problem arises. Its isometric group is O₂. The orbits of the group are circles r = constant. One might imagine that this surface could be embedded in R³, where it would then appear axisymmetric around some z-axis.
Geodesics with ( = constant ) exist. One might think they are "meridional lines" of the surface, and that the equation z( ) of such a meridian could be constructed as we did at the beginning of the paper. Along geodesics with ( = constant ): (135)
If this surface can be embedded in R³, along those geodesics: (136)
which gives: (137)
Conclusion: this surface cannot be embedded in R³.
This metric (135) suggests a repulsive effect.
Not all surfaces, as defined by their metric, can be embedded. Nevertheless, these surfaces "exist," even if we cannot grasp them in our hands. Consider the following 3D hypersurface, defined by: (138)
with Rs > 0 and r > 0
is defined on R modulo 2
We cannot embed such a hypersurface. But it exists and possesses "planar geodesics" ( 
= /2).
We can compute the geodesic system of these 2D and 3D hypersurfaces. We can represent them in a plane (r, ). They are real. (139)
Their shape is identical to that of the two previous surfaces, as defined by their line element (134). These two geometric objects are simply connected.
Fig. 25: Geodesics corresponding to line elements (134) and (138).
(Note that it resembles a repulsive action.)
There is something puzzling. Given a line element, we can compute the geodesic system. For example, the one corresponding to the classical representation of Schwarzschild geometry is: (140)
We can calculate the curves r( ) corresponding to this differential equation. They are real, even for values r < Rs!
Fig. 26: Complete geodesic line corresponding to the Schwarzschild line element.
We now understand why physicists were puzzled upon observing this strange result. But there is a mathematical fact: a line element can produce a real geodesic system, with some portions corresponding to an imaginary length element ds.
What about physics? We identify ds with a proper time increment. Earlier, we decided that imaginary ds does not correspond to a physical path, which forced us to reconsider the "local topology" of the hypersurface, changing the "local spheroidal topology" into a "local hypertoroidal topology."
In previous works, people retained the assumption of "local spheroidal topology," making the physical interpretation of the "interior" of the Schwarzschild sphere problematic. In reference [1], in section 6.8, we read:
(Inside the Schwarzschild sphere) It would thus appear natural to reinterpret r as a time marker and t as a radial marker (...) ... which would imply that ds² < 0 along this worldline.
- Kruskal's analytical extension.
In the classical coordinate system [x°, r, , ], the radial velocity of light is: (141)
so that it tends to zero as r approaches Rs. Kruskal's argument is as follows (reference [1], section 6.8):
This is an undesirable feature of Schwarzschild coordinates that we can eliminate as follows: we seek a transformation for r and t into new variables u and v such that the line element takes the form: (6.187)
... we arrive at a transformation appropriate for the interior of the Schwarzschild radius: (6.204)
While, outside this sphere: (6.201)
The fundamental requirement is that f be regular on the Schwarzschild sphere r = Rs. Still from [1]:
Thus u serves as a global radial marker, and v serves as a global time marker.
Moreover, from (6.187), null geodesics (ds = 0) yield a "constant speed of light": (142)
From (6.201), we see that as r tends to infinity, f tends to zero, so Adler, Schiffer, and Bazin state [1]:
They do not, however, correspond to spherical coordinates for flat space at asymptotic distance, as Schwarzschild coordinates do.
The Kruskal metric is also a nonsingular solution of the Einstein equations in these regions and is equivalent to the Schwarzschild solution, but has no singularity at the boundary (the Schwarzschild sphere). It is an analytical extension of the manifold.
Kruskal focuses on the problem at this boundary, which becomes nonsingular, with the singularity concentrated at the "geometric center" where f tends to infinity. Still using reference [1], we reproduce the passage devoted to inward-directed photon radial trajectories:
In terms of u and v, the trajectory is simple; in terms of r and t, however, we see that it begins at some finite r > Rs and finite x°, moves inward toward r = Rs as x° tends to infinity, and crosses the line x° = infinity into the interior of the Schwarzschild sphere. After that, r continues to decrease along the trajectory, but x° decreases. ... This treatment also clarifies that x° is not a reasonable time marker inside the Schwarzschild sphere.
We see that "nothing is perfect." With his particular choice of coordinates, Kruskal manages the crossing of the Schwarzschild sphere, confining the singular feature of the geometric solution to a "central singularity." But the metric is no longer Lorentzian at infinity.
This demonstrates how the choice of coordinates alters the interpretation of the solution. Our approach introduces a change in the "local topology" (hypertoroidal bridge), but eliminates all singularities.
- Back to embedding.
The Wiener-Graustein theorem states that any n-dimensional surface, with n > 2, can be embedded in a space whose minimal dimension is (143)
For 4D hypersurfaces, this corresponds to a 10-dimensional space. We know that Schwarzschild geometry geodesics lie in planes. = π/2 corresponds to one of them. Thus, we can focus on a subset of geodesics with ( = π/2). These geodesics depend on two parameters l and h. We know that geodesics with (l = 1) correspond to particles whose velocity is zero at infinity. Furthermore, choose the subset of geodesics with ( = constant). Then: (144)
Introduce an additional coordinate z and write: (145)
ds² = dr² + dz²
(146)
A differential equation whose solution is: (147)
We can represent these geodesics in a 3D space [z, r, ]. They are meridional lines of an axisymmetric surface.
Fig. 27: The meridian of the surface in which an isometric embedding of the Schwarzschild geodesics ( = constant) is performed.
In 3D space, this surface resembles Figure 28 (a half-cut).
Fig. 28: The embedding surface.
If we draw the "radial" geodesics on it, we obtain Figure 29.
Fig. 29: Representation of "radial" geodesics. Bottom: their projection onto a plane [r, ].
This is a very partial embedding, limited to the set of "radial" geodesics. Figure 29 evokes a pleat and suggests enantiomorphy. Indeed, consider a set of three points following radial geodesics. We obtain:
Fig. 30-a: Three mass-points falling toward the throat along "radial" paths.
and:
Fig. 30-b: The same, after crossing the throat.
The triangle has been inverted.
On the plane projection [r, ], the orientation of the triangle is reversed. Now imagine four test particles following radial trajectories, falling toward the Schwarzschild sphere, forming a tetrahedron. See Figure 31.
Fig. 31: Four particles falling onto the Schwarzschild sphere along "radial" geodesics in a Euclidean 3D space.
Fig. 32: After "bouncing" off the Schwarzschild sphere, the particles travel in the twin space. The tetrahedron is inverted (enantiomorphy).
Returning to the previous representation, the normal vector is also reversed:
Fig. 33: A particular geodesic ( = constant) in its representation within the set of (l = 1) geodesics, in a space (r, , z).