Collective motions of the relativistic gravitational gas
Annales de l'I.H.P. Physique théorique, Volume 9 (1968) no. 1, p. 17-33
@article{AIHPA_1968__9_1_17_0,
     author = {Droz-Vincent, Philippe and Hakim, R\'emi},
     title = {Collective motions of the relativistic gravitational gas},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     publisher = {Gauthier-Villars},
     volume = {9},
     number = {1},
     year = {1968},
     pages = {17-33},
     language = {en},
     url = {http://www.numdam.org/item/AIHPA_1968__9_1_17_0}
}
Droz-Vincent, Ph.; Hakim, Rémi. Collective motions of the relativistic gravitational gas. Annales de l'I.H.P. Physique théorique, Volume 9 (1968) no. 1, pp. 17-33. http://www.numdam.org/item/AIHPA_1968__9_1_17_0/

[1] R. Hakim, Einstein's Random Equations, to be published.

[2] E.G. Tauber and J.W. Weinberg, Phys. Rev., t. 122, 1961, p. 1342. | MR 122525 | Zbl 0103.22706

N.A. Chernikov, Soviet Phys. Dokl., t. 1, 1956, p. 103; t. 2, 1957, p. 248; t. 5, 1960, p. 764; t. 5, 1960, p. 786; t. 7, 1962, p. 397; t. 7, 1962, p. 428 ; Phys. Letters, t. 5, 1963, p. 115; Acta Phys. Polonica, t. 23, 1963, p. 629; t. 26, 1964, p. 1069; t. 27, 1964, p. 465.

R.W. Lindquist, Ann. Phys., t. 37, 1966, p. 487. | Zbl 0142.23902

[3] R. Hakim, J. Math. Phys., t. 8, 1967, p. 1153 ; Ibid. , t. 8, 1967, p. 1379.

See also, Ann. Inst. H. Poincaré, t. 6, 1967, p. 225. | Numdam | Zbl 0155.32601

[4] Phase space is always the tangent fibre bundle of the manifold configuration space.

[5] Actually E is 6-dimensional if we bear in mind the constraint (2).

[6] Since μ is the tangent bundle of a metric manifold (i. e. U4), then on this space one can construct a canonical metric tensor GAB. See the article by Lindquist (Ref. [2]) and references quoted therein.

[7] By « effective volume » we mean a 6-dimensional volume. This conservation law, i. e. the Liouville theorem, means that if Δ1 ⊂ Σ1 is such a 6-dimensional volume, then mes (Δ1) = mes (Δ2) where Δ2 is the « volume » in Σ2 obtained from the transformation of Δ1 under the group motion (i. e. Eq. (3)).

[8] Such as those given by P. Havas and J.N. Goldberg, Phys. Rev., t. 128, 1962, p. 398. | Zbl 0111.42104

[9] This would be only a simple generalization of previous results where the electromagnetic radiation was dealt with (See Ref. [3] and also R. Hakim and A. Mangeney, Relativistic kinetic equations including radiation effects I. Vlasov approximation (to appear in J. Math. Phys., 9, 116 (1968)). | Zbl 0173.30402

[10] A. Lichnerowicz, Propagateurs et commutateurs en relativité générale (Publications. Mathématiques n° 10 de l'I. H. E. S.), p. 40. | Numdam | Zbl 0098.42607

[11] We mainsly use the notations of Ref [10].

[12] Ref. [10], p. 43.

[13] Ref. [10], p. 27.

[14] Ref. [10], p. 39.

[15] In the same way as neglecting correlations of electromagnetic field or of particles amounts to dealing with a kinetic equation valid at order ∼ e2, neglecting correlations of the gravitational field is expected to provide a kinetic equation valid at order χ. We verify this statement on the resulting equation.

[16] Ref. [10], p. 33.

[17] B. De Witt, Ann. Phys., t. 9, 1960, p. 220. | Zbl 0092.45003

[18] Ref. [10], p. 30.

[19] Ref. [10], p. 35.