f31003 Matter-ghost matter astrophysics. 7: Confinement of spheroidal galaxies by surrounding ghost matter. (p3) .
3) The gravitational field in a spherical hole.
We compute it by combining two systems. The first is an infinite medium with constant density ρ. The second is a simple sphere filled with the same constant density ρ material. Let us compute the field created by such a sphere, considered in isolation. Inside:
(19)
whose solution is:
(20)
Outside:
(21)
This field is attractive. See figure 12.
Fig. 12: Gravitational field due to a constant density sphere whose density is ρ > 0
If we subtract this from a constant mass density medium, we obtain the same field, plus a spherical hole. Since the constant density medium produces a zero field, this resulting field remains unchanged. See figure 13.
Fig. 13: Gravitational field in a hole.
If elliptic galaxies are surrounded by dense ghost matter, spheroidal galaxies must be too. As a conclusion, this ghost matter environment must confine them as well, produce a missing mass effect, and strong lensing due to the negative lensing effect.
References.
[1] R. Adler, M. Bazin & M. Schiffer: Introduction to general relativity, Mac Graw Hill book, 1975, chapter 10, section 10.5: Classical limit of gravitational equations, p. 345.
Acknowledgements:
This work is supported by the French CNRS and by the A. Dreyer Brevets et Développement company.
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