f31001 Astrophysics of dark matter. 7 : Confinement of spheroidal galaxies by surrounding dark matter. (p1)
Comment.
...This work is the subject of passionate discussions with the mathematician Jean-Marie Souriau, my neighbor and friend. We did not reach an agreement and each remained firm in their positions.
Souriau :
- From Newton, you get Poisson. But from Poisson, you get Newton.
- Certainly, but where do you get the Poisson equation from, out of your hat?
- Well, I prefer to decide that the universe obeys the Poisson equation, that's all. That's how it is. --- * * * Astrophysics of dark matter. 7 :*
Confinement of spheroidal galaxies by surrounding dark matter.
Jean-Pierre Petit & Pierre Midy Marseille Observatory France. --- * *
**Abstract **:
...This is a new perspective on the origin of the Poisson equation. We show that for an infinite constant density mass distribution, this equation simply does not exist, because no gravitational potential can be defined. Building the Poisson equation from general relativity requires a steady-state zeroth-order metric solution and a steady-state metric perturbation term. In a uniform and unlimited medium, these elements are missing. In conclusion, the dark matter surrounding a spheroidal galaxy confines it, even though it is a spherically symmetric system.
1) Introduction.
...In a previous article, we considered the confinement of a galaxy due to its surrounding dark matter. What happens if the galaxy is spheroidal? One would answer: this confinement cannot exist, because it contradicts Gauss's theorem. Any matter that creates a Newtonian field, if located outside a sphere, contributes nothing in that region of space.
If, as the Lacedaemonian used to say...
...The starting point is that you assume a priori that the gravitational field is Newtonian at any distance, which should be proven. The Newtonian force varies as 1/r². Consider a medium with spherical symmetry, and successive layers, of the same thickness Dr. See figure 1.
Fig. 1 : The contribution of successive layers to the Newtonian force.
These two volumes correspond to masses:
(1)
M = r s Dr and M' = r s' Dr
The corresponding contributions to the total Newtonian force at O are:
(2)
...But s » r², so that F » F'. If one wants to calculate the value of the gravitational field at a point in an infinite constant matter density field, one must take into account the matter located at infinite distance. Its contribution cannot be neglected.
...Consider a fundamental problem. We have an infinite distribution of matter in space, and a single spherical hole. We want to calculate the field inside. The basic method consists of starting from the field due to an infinite, constant density distribution of matter. What does it look like?
...Simple, says the reader, let's apply the Poisson equation. One computes the flux of the field through a closed surface:
Fig. 2 : The flux through a closed surface due to a Newtonian field.
then applies the Green theorem:
(3)
Writing:
(4)
we obtain the Poisson equation. We assume that this local law is valid everywhere in space. Then, considering a constant density r medium, we construct the solution:
(4bis)
D Y = 4 p G r = constant
In spherical symmetry:
(5)
whose solution is:
(6)
...Conclusion: if we take any arbitrary point in space, it is associated with a radial field, which tends to infinity at infinite distance!
Fig.3 : The "classical" gravitational field, in a constant density medium, around any arbitrarily chosen point M.
...Isn't that strange? Physically, any given point P is equally attracted by all the points located in its vicinity. The resultant of the forces acting on that point should be zero. If based on that Poisson equation, it is not. Why?
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