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The Boltzmann Equations in General Relativity

Published online by Cambridge University Press:  20 January 2009

A. G. Walker
A. G. Walker
Affiliation:
University of Oxford.
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In a recent paper, J. L. Synge gives an interesting derivation of the conservation equations Tij,j = 0 satisfied by the energy tensor Tij of a continuous medium. Previous to the appearance of this paper, these equations were generally obtained by assuming the classical equations of motion and continuity, after which it was necessary to appeal to the Principle of Equivalence. It then follows that the path of a free particle is a geodesic. Synge however starts with the hypothesis that the path of a particle between collisions is a geodesic and that the proper mass is constant. The conservation equations are then deduced exactly from the law of conservation of momentum for collisions.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 4 , Issue 4 , April 1936 , pp. 238 - 253
Copyright
Copyright © Edinburgh Mathematical Society 1936

References

page 238 note 1 Trans. Roy. Soc., Canada (3), 28 (1934), 127–171.Google Scholar

page 238 note 2 See, for example, Eddington, , “Mathematical Theory of Relativity,” §§ 53, 54.Google Scholar

page 240 note 1 With this convention, the signature of the quadratic (1) is – 2.

page 240 note 2 The subscript 1 attached to dΩ indicates that we are dealing with material particles. The subscript 2 will occur when we discuss photons.

page 241 note 1 It will be understood that Latin suffixes h, i, j , k, l take the values 0, 1, 2, 3, and Greek suffixes μ,ν take the values 1, 2, 3.

page 244 note 1 See Eisenhart, , “Riemannian Geometry,” §39.Google Scholar

page 246 note 1 The vector p i will refer to photons in the remainder of this paper, and should not be confused with the momentum vector associated with a material particle.

page 247 note 1 This expression for dΩ2 may be compared with the expression for dΩ1 given by (23).

page 250 note 1 This function R(t) must not be confused with the scalar curvature R = gij Rij.