@article{AIHPA_1968__9_1_17_0,
author = {Droz-Vincent, Ph. and Hakim, R\'emi},
title = {Collective motions of the relativistic gravitational gas},
journal = {Annales de l'institut Henri Poincar\'e. Section A, Physique Th\'eorique},
pages = {17--33},
publisher = {Gauthier-Villars},
volume = {9},
number = {1},
year = {1968},
language = {en},
url = {http://www.numdam.org/item/AIHPA_1968__9_1_17_0/}
}
TY - JOUR AU - Droz-Vincent, Ph. AU - Hakim, Rémi TI - Collective motions of the relativistic gravitational gas JO - Annales de l'institut Henri Poincaré. Section A, Physique Théorique PY - 1968 SP - 17 EP - 33 VL - 9 IS - 1 PB - Gauthier-Villars UR - http://www.numdam.org/item/AIHPA_1968__9_1_17_0/ LA - en ID - AIHPA_1968__9_1_17_0 ER -
%0 Journal Article %A Droz-Vincent, Ph. %A Hakim, Rémi %T Collective motions of the relativistic gravitational gas %J Annales de l'institut Henri Poincaré. Section A, Physique Théorique %D 1968 %P 17-33 %V 9 %N 1 %I Gauthier-Villars %U http://www.numdam.org/item/AIHPA_1968__9_1_17_0/ %G en %F AIHPA_1968__9_1_17_0
Droz-Vincent, Ph.; Hakim, Rémi. Collective motions of the relativistic gravitational gas. Annales de l'institut Henri Poincaré. Section A, Physique Théorique, Tome 9 (1968) no. 1, pp. 17-33. http://www.numdam.org/item/AIHPA_1968__9_1_17_0/
[1] , Einstein's Random Equations, to be published.
[2] and , Phys. Rev., t. 122, 1961, p. 1342. | MR | Zbl
, 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.
, Ann. Phys., t. 37, 1966, p. 487. | Zbl
[3] , 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
[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 and , Phys. Rev., t. 128, 1962, p. 398. | Zbl
[9] This would be only a simple generalization of previous results where the electromagnetic radiation was dealt with (See Ref. [3] and also and , Relativistic kinetic equations including radiation effects I. Vlasov approximation (to appear in J. Math. Phys., 9, 116 (1968)). | Zbl
[10] , Propagateurs et commutateurs en relativité générale (Publications. Mathématiques n° 10 de l'I. H. E. S.), p. 40. | Numdam | Zbl
[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] , Ann. Phys., t. 9, 1960, p. 220. | Zbl
[18] Ref. [10], p. 30.
[19] Ref. [10], p. 35.
