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About "negative masses".
This question of negative masses is evoked in the book of
the French mathematician Jean-Marie Souriau (my neighbour and old good friend
).
"Structure des Systèmes Dynamiques", Dunod-France 1970, pp. 197-200
Recently translated into english :
"Structure of Dynamical Systems", Birkhauser Ed. 1997. (Unfortunately
very expansive : $ 124...).
Starting point : group theory. Dynamic groups run physics. Poincaré's
dynamic group runs relativistic world. This group is built from the Lorentz
group (see Geometrical Physics B). As the latter, the Poincaré group
has "four components". Two are "orthochron", refer to positive energy and
positive mass particles. The second set of two components, "antichron", runs
particles going backwards in time. Then Souriau showed (through "codajoint
action of these group's components on the momentum) that it corresponds to
negative energy, negative mass objects.
Basically, it is not forbiden that negative masses could exist. But it arises
an immediat problem : the result of a collisions of two particles, the first
with a positive mass (and positive energy mc2) and the other one
with negative mass and energy would give... nothing. No particles, and even
no space at all, for void is "made" of "joint photons".
What is the solution ?
Perhaps God, with infinite wisdom and foresight, decided to amputate the
antichron part of the dynamical Poincaré's group. Therefore the group
cannot generate negative masses.
Souriau does not exclude the possibility that negative matter could exist
somewhere in the universe. His suggestion :
-
Mutual attraction between positive masses (our matter, Newton's law)
-
Mutual repulsion between a positive and a negative mass (anti-Newton)
He adds :
- Negative masses repel each other.
According to Souriau's vision, negative masses should be completely autistic,
repel everything, including other negative masses. They should just fill all
available space. Some sort of panphobia .
Such negative matter could contribute to confine galaxies.
Between massive objects, the negative matter should be as uniform as possible
so that it would induce no gravitational lensing (in these wide regions).
We could not detect it, except through the confining effect, on galaxies and
clusters.
Anyway, if this negative matter is present in our universe, today, it would
have been closely mixed to the normal matter in the earlier time. Then a collision
in this dense mixture would have caused mutual complete annihilation :
- mc2 - mc2 = zero.......
Inverse gravitational lensing.
Positive gravitational lensing has been evoked on figure 44, in a preceeding
section. We used the crude model of the blunt (posicone).
A concentration of "negative mass" would correspond to the "blunt negacone".
See figures 88 and 89. We get a divergent system of geodesics.
This is an exact solution of field equation. See :
Jean-Pierre Petit and Pierre Midy : Matter ghost matter astrophysics.
2 : Conjugated steady state metrics. Exact solutions. Geometrical Physics
A , 5, 1998.
On figure 88 we see that it works as if negative mass repels photons".
(88)
NB : Uniform distribution of positive or negative mass
does not produce positive or negative lensing effect.
Confinement effect due to surrounding negative mass.
Positive masses attract each other, following Newton's
law. Gravitational (Jeans' instability) occurs in positive matter. This gives
mass-concentrations (galaxies, clusters of galaxies). According to Souriau's
idea, if negative matter is present, it would be located between positive
mass objects, filling all available, interstitial space. If negative matter
repels negative matter, the medium would be very uniform between positive
matter clumps. This hypothetic negative matter would be lacunar. See fig (89).
(89)
For example the galaxies could take place in that lacunar
negative mass distribution.
(90)
Counter-pressure due to the negative masses would participate
to the confinement of galaxies.
Let us imagine a didactical model : a large tent put on
pegs, which figure positive masses. The sharper the pegs, the denser are the
mass-concentrations. Conversely, the blunter the pegs, the larger are these
mass-concentrations.
(91)
The tent pegs
(92)
The portion of the tarpaulin that is in touch with the
pegs shows a positive curvature. Between these points the curvature is negative.
If the head of the peg is sharp it gives a point-like mass, a point of concentrated
positive curvature ( a conical point).
Globally, seen from a large distance, the tent is flat. It means that, here,
the amount of positive curvature is equal to the amount of negative one. Physically
it would correspond to a world where the amount of positive mass should be
equal and opposite to the amount of negative mass ( M* = - M ).
(93)
We have figured some geodesics (92) and their plane projections
(94).
(94)
Give up the didactif image. What about the 4d-hypersurface
?
