- Geodesics in a [r, j] representation.
Introducing (6) into (14), with dr = thr dr, we obtain: (17)
which gives the [r, j] representation of geodesics. When r tends to zero, dj/dr tends to a finite value, so that the tangent of the pitch angle: (18)
tends to zero at the origin. The image of the Schwarzschild throat circle, in this representation, is a conical point. ** ** **
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Fig. 4: The geodesic shown in figure 3, in a (r, j) coordinate system.
The crossing of the throat circle corresponds to point O
This is an isometric representation of the geodesics. Note that we can also represent the surface in a [z, r, j] system, but this is no longer an isometric one. We then obtain the associated meridian: (19)
When r tends to zero, z(r) is linear. When it tends to infinity, the function tends to a parabola.
Fig. 5: Meridian of the surface, in a [r, j] non-isometric representation of the surface. ****
The image of the Schwarzschild throat circle, in this representation, is a conical point. ** **
- **Extension to a spherically symmetric 3D hypersurface. **
This can be extended to a 3D hypersurface, described by the line element: (20)
This metric refers to a 3D hypersurface, here expressed in a [r, q, j] coordinate system. The variable r is not a "radial distance", corresponding to "spherical coordinates". We find similar pathologies in this new line element, which can be eliminated by introducing the same coordinate change (6).
[ r, q, j ] ® [ r, q, j ]
The line element then becomes: (21)
Its signature becomes (+, +, +) and its determinant: (22)
no longer vanishes.
The geodesics of this hypersurface are located in planes. q = p/2 is one of them. In their [r, j] representation, they coincide with those of figure 2. The isometry group is O3 and the corresponding orbits are spheres. Among these, one has a minimum area (the throat sphere of such a 3D toroidal bridge). The great circles of the sphere-orbits are not geodesic curves, except for the special ones located on the throat sphere whose perimeter is 2pRs. The geodesics of this particular sphere are the only closed ones. We can call this particular geometry a hypertoroidal geometry. This 3D surface is not simply connected. It has a single 3D fold, which can be considered as a set of two bounded half 3D folds, glued together along their spherical boundary (the throat sphere). At large distances from this "hypertoroidal bridge", the metric tends to the Euclidean one, hereafter written in spherical coordinates: (23)
ds² = dr² + r² ( dq² + sin²q dj² )
- **Schwarzschild geometry. **
Classically, it is considered that its isometry group is SO3 × R, where R refers to one-dimensional translations. It is then said that this metric is time-independent and spherically symmetric, considering that R corresponds to time translations.
Expressed in a [x°, r, q, j] coordinate system, where x° is the time marker, the line element is (24)
Classically, x° = ct is written, which is supposed to define the cosmic time t of "an external observer". When r >> Rs, (21) tends to the Minkowski metric. Classically, r is assimilated to a radial coordinate. (21) shows a singularity in the term grr and a change of signature when r = Rs.
Once again, we can regularize this metric using the coordinate change (6), switching to a [t, r, q, j] system. The line element then becomes: (25)
The orbits of the O3 isometry group are spheres. Among these, one, the throat sphere (Schwarzschild sphere), has a minimum area. The hypersurface is not simply connected. It forms a single space-time fold, which can be considered as a set of two half 4D space-time folds (twin folds), the first corresponding to r > 0 and the second to r < 0, so that the throat sphere corresponds to r = 0. We may compute the geodesics located in the q = p/2 plane. Following "spherical coordinates":
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Fig. 6: Spherical coordinates.
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the element is dr² = r² ( dq² + sin²q dj² )
The circles j = constant are geodesics of the sphere, but obviously, they do not represent all the geodesics of the surface. Only those passing through two antipodal points (poles).
The circles q = constant are not geodesics, except the one corresponding to q = p/2 (equator).
In a [r ≥ Rs, j] coordinate system, these (non-zero length) geodesics correspond to: (26)
The choice of the set of constants [l, h] determines the geodesic. Among them, we find hyperbola-like geodesics, which do not intersect the throat sphere r = Rs. See figure 7.
Figure 7: Schwarzschild geometry.
[r, j] representation of a hyperbola-like plane geodesic which does not cut the throat sphere r = Rs
We also find quasi-elliptic geodesics. See figure 8
**Fig. 8: Schwarzschild geometry.
[r, j] representation of quasi-elliptic geodesics. **
Now let us focus on geodesics which intersect the r = Rs throat sphere. In a [r, j] representation, call a the angle between the tangent to the geodesic and the radial vector. (27)
The first Lagrange equation gives: (28)
For r ≥ Rs values, the parameter l is strictly positive. Another Lagrange equation is: (29)
and gives a monotonic evolution of the angle j with respect to the proper time s. In this (q = p/2) plane, the rotation depends on the sign of h.
According to this new interpretation of the Schwarzschild geometry (considered as a non-simply connected hypersurface), we may represent the geodesic in a [r, j] representation as shown in figure 9.
Figure 9a: [r, j] representation of a geodesic which intersects the throat sphere **** (Schwarzschild sphere) corresponding to h ≥ Rs
One portion of the geodesic has been drawn as a dotted line, as it is supposed to belong to the second half-3D fold, connected to the first along the throat sphere, the Schwarzschild one. It suggests a break. But this last is due to this peculiar [r, j] system of representation, which is more familiar to our (limited) human geometric intuition. In a 3D representation space, we get figure 9b. Particles seem to "bounce" on the Schwarzschild sphere.
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Fig. 9b: In the 3D Euclidean representation space, particles seem to bounce on the Schwarzschild sphere.
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From this point of view, "there is nothing inside the Schwarzschild sphere", because, in this "inside", we are simply "outside the hypersurface". Recall that the throat sphere, the Schwarzschild sphere corresponds to the r = 0 value. The first half-fold corresponds to (r > 0) and the second one to (r < 0).
In a [r, j] representation, the appearance of the geodesic becomes quite different. Let us compute the tangent of the angle b, between the geodesic and the radial vector (see figure 6). (30)
When r tends to ±0, thr ≈ r, hence: (31)
In the [r, j] representation, the geodesics that go from one half-fold to the other are tangent to the radial vector. There is no longer an angular discontinuity at the origin, this last being the image of the throat circle (r = 0). To get a complete description of these geodesics, we must return to the line element expressed in the [t = x°/c, r, q, j] coordinate system (24), using the Lagrange equations system, with: (32)
Among these equations, we find: (33)
For a given h value, the evolution of j is monotonic with respect to the proper time s.
Fig. 10: [r, j] representation of a geodesic which passes ** from a half-fold (r > 0) to the other one (r < 0). **
As before, the portion of the geodesic which belongs to the second 3D half-fold has been drawn as a dotted line.
We cannot give an embedding of the 4D hypersurface, as we did at the beginning of the paper for a 2D surface. Moreover, we are dealing with 4D geodesics, not 3D ones. [r, q, j] and [r, q, j] spaces are nothing but representation spaces, which are supposed to make things a bit clearer. The real geodesics are inscribed in a 4D space. Anyway, the [r, q, j] representation suggests a 3D "hypertoroidal bridge", while the [r, q, j] representation suggests a 3D "hypercone". In this second (3D) representation of such a 2D surface, the geodesics go from one half-fold to the other, passing through the (r = 0) point. This is similar to a 2D cone. See figure 11
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Fig. 11: Geodesic of a cone. Right: a surface having a conical point.
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