A
cosmological model: The twin bang. (p.6)
Consider now an infinite space,
filled by constant density material. How is the gravitational field in the vicinity
of a given point O ?
Immediately,
we think :
- Let us take the Poisson equation
:
(165)
DY = 4 p
G r
where Y
is the gravitational field
and r the mass-density.
First remark : a constant
density r does not fit a constant potential. Well.. let us solve the problem
in spherically symmetric conditions.
(166)
The gravitational field is :
(167)
The solution is :
(168)
The non-zero ( radial ) gravitational
field becomes :
(169)
which tends to infinite at the
infinite (...).
What is the
gravitational field ? In principle it is the force acting on a reference mass
m = + 1 .
(169 bis)
...O
is an arbitrary point. M another arbitrary point. I find that a test-mass
m = +1, located in M, is radially attracted by O. This makes possible to compute
the gravitational field in a spherical hole. We can use the following schema.
(170)
...We
can compute the field due to the sphere on the right, filled by constant density
material. Then we refind the preceeding result : the gravitational field is
zero in the spherical hole.
We say it's false.
- In the first case we have
assumed that the Newton's law is valid for infinite distances.
- In the second one we assume
that the Poisson equation is valid for a uniform medium.
...In the afore-mentioned paper we return
to the origin of the Poisson equation and Newton's law. It corresponds to
Newtonian approximation : strictly talking : weak field and small velocities
with respect to light velocity. As pointed out in the paper, the classical
analysis is based on steady state metrics (zeroth order term and perturbation
term are chosen time-independent). The zeroth order term on the metric is
identified to Minkowski space, which fits steady state condition (for it is
an empty space).
...But
it does not fit any longer, non-zero uniform mass distribution plus steady
conditions. Such a solution simply does not exist. If any matter is
present we get Friedmann models, not steady state model.
...Conclusion
: The classical analysis cannot be extended to constant density mass distributions
where it becomes impossible to define any gravitational potential. As a conclusion
:
The gravitational force in
an unbounded constant density mass distribution is zero everywhere.
...Corollary
:
The
gravitational field inside a spherical hole is non-zero.
Same thing if the hole looks like
a flat ellipsoid :
(171)
...Physically,
the frontier is not so abrupt. There is a matter density gradient, and a pressure
density gradient. If the galaxy is removed, such pressure gradient would make
the hole disappear. In the paper :
J.P.Petit and P.Midy : Repulsive dark matter. Geometrical Physics A , 3,
1998. Figure 4.
we have used such non-abrupt mass
distribution.
