- Suggestions for hyperspatial transfer models.
Soft scenario:
Assume a neutron star, close to criticality, is located near a companion star. This latter sends it matter (stellar wind). When the 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 (as it moves in a reversed time marker fold F*, see section 14). The neutron star repels it, and it is rapidly ejected into space, in the twin fold. This process would ensure the stability of the neutron star, as the bridge would close when the density and pressure at the center become sufficiently low. This phenomenon could be accompanied by gravitational wave and gamma ray emissions (gamma flashes).
Hard scenario:
Pairs of neutron stars exist. It has been shown that their rotation is continuously slowed down due to energy loss by gravitational wave emission, so that they should merge. The abrupt merging of two neutron stars would turn into a catastrophe (in the mathematical sense of the term). Building a complete non-steady solution of the system (115) plus (116) would allow describing such a process. The following is conjectural.
Note that the total 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 to the twin space. Once the process is complete, 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 supposed to indicate the relative importance of the tensor terms. The small T represents the residual matter remaining in our fold.
What would it look like?
This residual matter would be kept at a distance by the transferred neutron star (self-attractive, but repelling the residual 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, according to Newton's law (in the Newtonian approximation).
- Twin matter (transferred matter) attracts twin matter, according to Newton's law.
- Matter and twin matter repel each other, according to an "anti-Newton's law".
In our fold, the residual matter would cool by radiative processes. If no energy source exists nearby, its temperature would tend towards the cosmic background temperature (3°K). It would form a kind of hollow shell of cold gas surrounding an (invisible) repulsive object. See figure 17
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Fig.17: Schematic of the hyperspatial transfer of the majority of the matter of a neutron star.
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If this idea is valid, such cold objects would be observable in our galaxy. Perhaps some proplyds (recently discovered), if they are composed of cold gas, could correspond to such residual shells. Of course, if they are located near hot stars, their temperature would not be so low. Some people think proplyds are young stars or young planetary systems in the process of formation. This is just a suggestion.
- Criticity in a neutron star.
Spherically symmetric 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. On the left: the Schwarzschild radius. On the right: the characteristic radius associated with the internal solution. rn is supposed to be the radius of a (constant density) neutron star. When it approaches criticality, this corresponds to figure 18.
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Fig.18: A neutron star approaching criticality.
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Chapter 14 of reference [1] "The role of relativity in stellar structure and gravitational collapse" presents, in section 14.1, the TOV equation (Tolman-Oppeinheimer-Volkov model). 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 increase in central pressure is the first symptom of criticality.
...Figure 19 shows the evolution of the pressure inside a neutron star, for different values of the external radius, up to criticality, according to the TOV model. When the critical mass of the neutron star becomes critical (for a value close to two solar masses), the pressure increases to infinity.
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Fig. 19: Pressure inside a neutron star (TOV model) for different values of external radius.
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The following curves are still based on the (steady-state) TOV equation, so they cannot be considered as a correct model. However, they seem to indicate how fast the (p = infinite) sphere could grow inside the neutron star when the radius increases slightly.
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Fig.20: Internal pressure calculated according to the steady-state TOV equation.
Although fundamentally incorrect, this figure seems to show how fast the singularity (p = infinite) could grow with a slight increase in mass.
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- 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 vacuum 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 represented as a blunt "negacone", built around a horse saddle, whose constant curvature density is negative, surrounded by a portion of "negacone", an Euclidean surface.
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Fig. 22: The two folds are connected by a conical point (infinite curvature density)
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Pressure is an energy density per unit of volume. If we represent this pressure by the local curvature density, when the critical conditions are reached (infinite pressure at the center of the star), a conical point (a point of infinite curvature density) appears, and the two folds are connected.
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Fig.23: Appearance of a throat circle.
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Then the small passage grows in size, which leads to a modification of the geometric configuration.
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Fig.24-a: which expands.
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 connected along a circle
Image of Schwarzschild geometry: the symmetrical "diabolo"
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In a symmetrical process corresponding to the total transfer of matter (positive curvature) to the twin space, the midpoint 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".
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We can complete the series and illustrate a process of "curvature exchange" between two surfaces.
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Fig.24-g: The curvature transfer continues.
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Fig.24-h: The curvature transfer continues.
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Fig.24-i: The curvature transfer continues.
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Fig.24-j: Punctual contact, just before separation.
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Fig.24-k: End of the curvature transfer.
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