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The Origin of "Negative Masses".
...In the chapter "Could the Universe Be a Combination of Two Things?", we cite the work of Souriau: Structure of Dynamical Systems, Dunod 1970, pp. 197–200 (recently translated into English: Structure of Dynamical Systems, Birkhäuser Ed. 1997). Starting point: groups. By analyzing the action of the complete Poincaré group on its momentum, Souriau shows that the universe could simultaneously implement particles with positive energy and negative energy (thus negative mass). Group theory does not oppose this possibility. However, the potential encounter between two particles of opposite masses (unrelated to antimatter, which has positive mass, as mentioned earlier) raises a problem. The result would be complete annihilation. Not even photons would remain. Pure nothingness. An universe composed of equal parts of positive and negative masses would simply vanish. Souriau suggests the following solutions:
- Either God, in His infinite wisdom and insight, deliberately omitted creating negative masses.
- Or, for precaution, we truncate the Poincaré group by removing its two "antichronous" components, which not only reverse mass but also time, retaining only its two "orthochronous" components.
...Yet Souriau does not rule out the possibility of negative masses existing in the universe, and in that case he proposes the following dynamics:
- Positive masses attract each other according to Newton.
- Negative masses repel each other according to "anti-Newton."
- A positive mass and a negative mass repel each other according to "anti-Newton."
...By repelling each other, negative masses would not form structures, objects, stars, or galaxies. They would flee from each other and from everything else in the universe. A kind of panphobia, etymologically: fear of everything. This fundamentally isolationist behavior would then ensure their survival.
What might an universe containing negative masses look like?
...Negative masses would fill any region of space evacuated by matter. In these regions, the negative matter would adopt the most uniform distribution possible. Thus, light passing through these "no-matter" lands would experience no gravitational lensing effect. Therefore, from the standpoint of detection via observation: zero.
Inverse Gravitational Lensing Effect.
...We previously discussed gravitational lensing—the bending of light rays due to the presence of matter concentration. See Figure 44. The standard 2D didactic model is the familiar old "dull cone."
...What about the effect of a concentration of negative mass on the trajectory of photons? This corresponds to a region of negative curvature, a "dull anti-cone," see Figures 88 and 89. Geodesics diverge.
This geometry is also a solution of Einstein's equation.
See: "Jean-Pierre Petit and Pierre Midy: Matter ghost matter astrophysics. 2: Conjugated steady state metrics. Exact solutions. [See on site: Geometrical Physics A, 2- 5], 1998."
...It suffices to take the Schwarzschild solutions (interior and exterior) and reverse the sign of the mass. Below is the 2D didactic image. Negative masses "repel" light rays.
However, a uniform distribution of mass—positive or negative—does not produce any gravitational lensing effect, positive or negative.
Confinement Effect Due to Negative Masses.
...Positive masses are self-attractive and sensitive to gravitational instability. They give rise to condensations that expel negative masses, whose distribution then becomes sparse.
Galaxies, for example, could settle into such a sparse distribution.
...The gravitational counter-pressure exerted by negative masses on positive masses could then contribute to their confinement.
...A brief note on the 2D didactic model, which might evoke the geometry of an universe populated by a mixture of positive and negative masses. Imagine a vast tent stretched over stakes. If the stakes are very sharp, they represent point-like positive masses. If they are blunt, they represent concentrations of positive masses.
...The portion of the tent that conforms to the rounded shape of the stake has positive curvature. Beyond that, curvature is negative. If the stake is sharp, the region near the tip resembles a cone (the envelope of the tangent plane). The point represents concentrated curvature. If the tent is stretched "over a flat ground," the overall curvature is zero. This means there is as much positive curvature in the "more curved" regions as negative curvature in the "less curved" regions.
On this tent we have drawn some geodesics. If we project all of this onto the flat ground, we obtain:
But let us leave aside the didactic image. What about a "4D hypersurface"?
Einstein's equation was written as:
S = c T
where S is a geometric tensor and T the "energy-matter tensor." When explicitly written, under certain conditions and in a specific form, the matter energy density ρ and pressure p (which is an energy density per unit volume: a pascal is also a joule per cubic meter) appear explicitly.
...Let us denote ρ+ and p+ the contributions to density and pressure due to positive masses. Let T+ be the tensor constructed from these quantities. The contributions ρ− and p−, due to negative masses, would be negative. Using these quantities, we construct the tensor T−.
The corresponding field equation is then:
S = c (T+ + T−)
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