- Suggestions for hyperspatial transfer models.
Soft scenario:
Assume a neutron star, close to criticality, is located in the vicinity of a companion star. The latter sends it matter (stellar wind). When critical conditions are reached, a small hypertoroidal bridge would form at the center of the star, which would rapidly evacuate the excess matter towards the twin space. This transferred matter behaves as if its mass had been inverted (for it moves in a reversed time-marker fold F*, see section 14). The neutron star repels it, and it is rapidly sprayed into space, in the twin fold. This process would ensure the stability of the neutron star, for the bridge would close when the density and pressure at its center are reduced sufficiently. This could be accompanied by gravitational waves and gamma-ray emissions (gamma-ray flashes).
Hard scenario:
Pairs of neutron stars exist. It has been shown that their rotational motion is continuously slowed down due to energy loss via gravitational wave emission, so that they should eventually merge. The abrupt merging of two neutron stars would turn into a catastrophe (including the mathematical meaning of the term). Constructing a complete non-steady-state solution of the system (115) plus (116) would provide a description of such a process. The following is conjectural.
Note that the full transfer of matter would lead to a configuration corresponding to:
(126)
S = - c T*
(127)
S* = c T*
But, since the process is a priori reversible, the transferred neutron star would be critical. A possibility is an almost complete transfer of matter into the twin space. After the process ends, the hypertoroidal bridge would close, and a new equilibrium would be reached, corresponding to:
(128)
S = - c (T - T*)
(129)
S* = c ( T* - T )
The sizes of the bold letters are intended to indicate the relative importance of the tensor terms. The small T is supposed to represent the remnant matter remaining in our fold.
What might it look like?
This remnant matter would be kept at a distance by the transferred neutron star (self-attractive, but repelling the remnant matter due to the inversion of its mass), now located in the twin space. As explained in references [13], [14], [15], and [21]:
- Matter attracts matter, via Newton's law (in the Newtonian approximation).
- Twin matter (transferred matter) attracts twin matter, via Newton's law.
- Matter and twin matter repel each other, via an "anti-Newton's law."
In our fold, the remnant matter would cool via radiative processes. If no energy source exists in its vicinity, its temperature would tend toward that of the cosmic background (3 K). It would form a sort of hollow shell of cold gas surrounding an (invisible) repelling object. See Figure 17.
Fig. 17: Schema of hyperspatial transfer of the majority of the matter of a neutron star.
If this idea is correct, such cold objects would be observable in our galaxy. Perhaps some proplyds (recently discovered), if they are made of cold gas, could correspond to such remnant shells. Of course, if they are located in the vicinity of hot stars, their temperature could not be so low. Some people think proplyds are young stars, or young planetary systems undergoing a formation process. This is just a suggestion.
- Criticality in a neutron star.
Spherically symmetrical neutron stars (a somewhat unrealistic model) are classically described by an internal Schwarzschild geometry, corresponding to the well-known metric:
(130)
The stability condition is:
(131)
We have two characteristic lengths. Left: the Schwarzschild radius. Right: the characteristic radius associated with the internal solution. rn is supposed to be the radius of a (constant density) neutron star. As it approaches criticality, this corresponds to Figure 18.
Fig. 18: A neutron star tending towards criticality.
Chapter 14 of reference [1], "The role of relativity in stellar structure and gravitational collapse," presents the TOV equation (Tolman-Oppenheimer-Volkoff model) in section 14.1. It is shown that if:
(132)
the pressure becomes infinite at the center of the (spherically symmetric) neutron star. This critical radius is:
which is slightly smaller (and corresponds to a smaller critical mass: two solar masses, instead of 2.5).
It shows that this rise in central pressure is the first symptom of criticality.
...Figure 19 shows the evolution of the pressure value inside a neutron star for different values of the external radius, up to criticality, via the TOV model. When the neutron star's mass becomes critical (for a value close to 2 solar masses), the pressure rises to infinity.
Fig. 19: Pressure inside a neutron star (TOV model) for different values of external radius.
The following curves are still based on the (steady-state) TOV equation, so it cannot be considered a correct model. Anyway, it seems to indicate how fast the (p = infinite) sphere could grow inside the neutron star when the radius increases slightly.
Fig. 20: Internal pressure computed using the steady-state TOV equation.
Although basically incorrect, this seems to show how fast the singularity (p = infinite) could grow with a slight mass increase.
- A didactic model of hypertoroidal transfer.
In reference [16], we presented a coupled metrics solution (g, g*), describing the geometries of the two folds when a constant density sphere is present in one fold (ours), with vacuum outside, and when the adjacent portion of twin space is empty. It was shown that the local scalar curvatures were conjugated through:
(133)
R* = - R
A (crude) model of a mass surrounded by void is a blunt cone (assuming that particles follow the geodesics of this surface. See the website). Its blunt part is a portion of a sphere, whose curvature density is constant. The rest is a portion of a cone, an Euclidean surface, whose local curvature density is zero.
Fig. 21a: Classical blunt cone (blunt "posicone").
Fig. 21b: Blunt posicone with conjugated "twin geometry": a "blunt negacone" (R* = - R)
The conjugated space was then depicted as a blunt "negacone," built around a horse saddle, whose constant curvature density is negative, surrounded by a portion of "negacone," an Euclidean surface.
Fig. 22: The two folds get connected through a conical point (infinite curvature density)
Pressure is an energy density per unit of volume. If we represent this pressure by local curvature density, when critical conditions are reached (infinite pressure at the center of the star), a conical point (which is a point of infinite curvature density) appears, and the two folds get connected.
Fig. 23: Appearance of a throat circle.
Then the small passage grows in size, which goes along with a modification of the geometrical design.
Fig. 24-a: which gets enlarged.
Fig. 24-b: The second fold becomes flat.
Fig. 24-c: The second fold becomes a "posicone".
Fig. 24-d: Symmetrical configuration: two truncated posicones linked along a circle.
Image of Schwarzschild geometry: The symmetrical "diabolo"
In a symmetrical process, corresponding to the full transfer of matter (positive curvature) towards the twin space, the halfway point would correspond to two truncated cones, connected along a circle. This would correspond to the "Schwarzschild solution."
**
Fig. 24-e: The first fold becomes flat.**
Fig. 24-f: The first fold F becomes a "negacone".
We can complete the series and illustrate a process of "curvature exchange" between two surfaces.
Fig. 24-g: The curvature transfer continues.
Fig. 24-h: The curvature transfer continues.
Fig. 24-i: The curvature transfer continues.
Fig. 24-j: Punctual contact, just before separation.
Fig. 24-k: End of the curvature transfer.