a206 A cosmological model: The twin bang. (p.6) Consider now an infinite space filled with material of constant density. What is the gravitational field near 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 under spherically symmetric conditions. (166)
The gravitational field is: (167)
The solution is: (168)
The non-zero (radial) gravitational field becomes: (169)
which tends to infinity at infinity (...). 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 is another arbitrary point. I find that a test mass m = +1, located at M, is radially attracted by O. This makes it possible to calculate the gravitational field in a spherical hole. We can use the following scheme. (170)
...We can calculate the field due to the sphere on the right, filled with material of constant density. Then we find the previous result: the gravitational field is zero in the spherical hole.
We say it is false.
- In the first case, we have assumed that Newton's law is valid at infinite distances.
- In the second case, we assume that the Poisson equation is valid in a uniform medium.
...In the aforementioned paper, we return to the origin of the Poisson equation and Newton's law. It corresponds to the Newtonian approximation: strictly speaking, weak field and low velocities compared to the speed of light. As pointed out in the paper, the classical analysis is based on steady-state metrics (the zeroth-order term and the perturbation term are chosen time-independent). The zeroth-order term of the metric is identified with Minkowski space, which fits the steady-state condition (as it is an empty space).
...But this no longer fits when there is a non-zero uniform mass distribution combined with steady-state conditions. Such a solution simply does not exist. If any matter is present, we get Friedmann models, not a steady-state model.
...Conclusion: The classical analysis cannot be extended to constant density mass distributions, where it becomes impossible to define any gravitational potential. In 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.
The same thing if the hole has the shape of a flat ellipsoid: (171)
...Physically, the boundary is not so abrupt. There is a matter density gradient, as well as a pressure gradient. If the galaxy is removed, such a 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 a non-abrupt mass distribution.