6
The analogy
with a glass lens is relatively good. A positive mass M causes rays to converge. A mass M* causes them to diverge:
Analogy with optics:
When observing a dotted wallpaper through a divergent lens,
one can perceive a greater number of objects, each with a smaller apparent diameter. However, their brightness is reduced (their "apparent magnitude"):
Cosmologically speaking, conglomerates of ghost matter, acting as divergent lenses, should reduce the magnitude of high-redshift galaxies while increasing their number.
To evaluate this effect, one would need to know the diameter of the ghost matter conglomerates, which is difficult to determine. If they form, we cannot a priori predict what they might become. Do they coalesce into hypergiant galaxies?
If located at the centers of the "large empty bubbles," they are on average a hundred million light-years apart. However, their influence on the distant background strongly depends on their diameter f.
See:
J.P. Petit, P. Midy, and F. Landsheer: Ghost matter astrophysics.
5: Results of numerical 2D simulations. VLS. About a possible schema for
galaxy formation. [
On this site: Geometrical Physics A,
8
, 1998, section 3, expression (23) and figure 18.
]
In any case, if such objects exist, they must create the appearance of a large number of dwarf galaxies at high redshift. And this is precisely what we observe (P.J.E. Peebles: Principles of Physical Cosmology, Princeton Series in Physics, 1993). The classical interpretation is that dwarf galaxies form first, then merge and grow into larger objects through galactic fusion and cannibalism (merging).
Our model offers an alternative explanation for this dwarfing effect observed in high-redshift galaxies.
Toward a theory of galaxy formation.
This is a new scenario whose full implications deserve exploration.
The main difficulty—still unresolved—is to simultaneously treat both phenomena: one cannot separate the cosmological expansion from the formation of various structures. Currently, we are unable to handle both at once.
Nevertheless, let us sketch a hypothetical scenario. The clumps of ghost matter might form first, immediately exerting intense counter-pressure on ordinary matter, which would consequently heat up. See the cited paper above [
On this site: Geometrical
Physics A,
8
, 1998, section 4, diagrams 19, 20, and 21.
]
In astrophysics, whenever an object condenses or collapses, its temperature increases. This is true, for example, for proto-stars. This corresponds to a conversion of gravitational (potential) energy into kinetic energy (thermal agitation). Pressure is proportional to the product of density and temperature (p = n k T). As pressure increases, it resists further collapse. A proto-star, before ignition, is a spheroidal mass of gas at several thousand degrees, roughly the size of the solar system, radiating in the infrared. In this form, it actually emits more energy than later, when it draws energy from fusion reactions. It is its surface that radiates. It must "sweat" its energy away. Otherwise, it could not contract, increase its core temperature, and initiate the fusion process (minimum 700,000 degrees).
The compactness of the object makes it a poor radiator. At equal temperature, thermal energy scales with the cube of the radius, while emissive surface area scales with the square.
On the other hand, a flat plate constitutes an optimal radiator. By pushing our matter outward, the ghost matter conglomerates would compress it into plate-like structures (the walls of "jointed soap bubbles"). See the cited paper and figures above.
Calculations would be needed, but one can suppose that this geometry would favor intense radiative cooling, thus destabilizing the medium with respect to gravitational instability (for such gravitational instability issues, see my comic book "A Thousand Billion Suns," Ed. Belin, 8 rue Férou, Paris 75006, or on the "CD-Lanturlu").
The matter would then tend to fragment into proto-galaxies. Immediately, ghost matter would infiltrate the available space, leading to a scenario where galaxies are embedded within voids of ghost matter. This yields the same pattern as that produced by the presence of negative masses in our universe (Souriau's hypothesis). Let us re-examine the schema of galaxies surrounded by "negative matter" (ghost matter, twin matter, matter with negative mass—whatever name one chooses).
According to Souriau's proposed schema, negative masses repel each other. Under these conditions, they would not explain the large-scale structure of the universe.
An explanation for galaxy confinement.
Thus, we obtain a scenario in which ghost matter exerts counter-pressure on galaxies, ensuring their confinement. This offers an alternative to the idea of dark matter residing within galaxies.
See
J.P. Petit and P. Midy: Repulsive dark matter. [
See on this site: Geometrical Physics A,
3
, 1998, section 2
]. But there are also spherical galaxies. These would therefore be located within cavities of similar geometry, carved out within the nearly uniform distribution of surrounding ghost matter (recall that ghost matter is hotter than ours). Would these cavities then be confining?
Does this contradict Gauss's theorem?
All physics students know that if a sphere is uniformly charged electrically, the field inside is zero. One might then think of decomposing the gravitational field within a spherical cavity into contributions from successive concentric shells, each contributing zero.
This seems... obvious. But this theorem relies on an assumption: that the gravitational force follows an inverse-square law (1/r²) at any distance, including... at infinity.
A Newtonian field yields what is called Poisson's equation, via Green's theorem:
ΔY = 4πGρ
Einstein's field equation, at small distances, for weak curvatures, in quasi-stationary (cosmologically speaking) conditions, and for velocities much smaller than light speed, reduces to Newton's law and Poisson's equation.
Can this equation handle a uniform (ρ = constant) and infinite distribution of matter? This has been assumed until now. But doing so leads to a paradox. Consider spherical symmetry and an arbitrary point O, taken as the origin of coordinates. Poisson's equation then becomes:
where r is the radial distance and Y is the gravitational potential, from which the gravitational force g (radial in spherical symmetry) derives:
The equation does not admit a solution Y = constant for r ≠ 0. There is therefore a gravitational force, which seems paradoxical: one might expect that each particle, subject to the attractive force of all its neighbors, would experience a net zero force.
This solution is:
The gravitational field, centered on point O, is non-zero and corresponds to:
Not only is the field non-zero, but it tends toward infinity...