Collective motions of the relativistic gravitational gas

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1. [1] R. Hakim, Einstein's Random Equations, to be published. 
2. [2] E.G. Tauber and J.W. Weinberg, Phys. Rev., t. 122, 1961, p. 1342. Zbl0103.22706MR122525
3. 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. 
4. R.W. Lindquist, Ann. Phys., t. 37, 1966, p. 487. Zbl0142.23902
5. [3] R. Hakim, J. Math. Phys., t. 8, 1967, p. 1153 ; Ibid. , t. 8, 1967, p. 1379. 
6. See also, Ann. Inst. H. Poincaré, t. 6, 1967, p. 225. Zbl0155.32601
7. [4] Phase space is always the tangent fibre bundle of the manifold configuration space. 
8. [5] Actually E is 6-dimensional if we bear in mind the constraint (2). 
9. [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. 
10. [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)). 
11. [8] Such as those given by P. Havas and J.N. Goldberg, Phys. Rev., t. 128, 1962, p. 398. Zbl0111.42104
12. [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)). Zbl0173.30402
13. [10] A. Lichnerowicz, Propagateurs et commutateurs en relativité générale (Publications. Mathématiques n° 10 de l'I. H. E. S.), p. 40. Zbl0098.42607
14. [11] We mainsly use the notations of Ref [10]. 
15. [12] Ref. [10], p. 43. 
16. [13] Ref. [10], p. 27. 
17. [14] Ref. [10], p. 39. 
18. [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. 
19. [16] Ref. [10], p. 33. 
20. [17] B. De Witt, Ann. Phys., t. 9, 1960, p. 220. Zbl0092.45003
21. [18] Ref. [10], p. 30. 
22. [19] Ref. [10], p. 35.