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On the concept of gravitational force and Gauss's theorem in general relativity

  • J. L. Synge (a1) and H. S. Ruse

Extract

Whittaker and Ruse have developed forms of Gauss's theorem in general relativity, their theorems connecting integrals of normal force taken over a closed 2-space V2 with integrals involving the distribution of matter taken over an open 3-space bounded by V2. The definition of force employed by them involves the introduction of a normal congruence (with unit tangent vector λi), the “force” relative to the congruence being the negative of the first curvature vector of the congruence (– δλi/δs). This appears at first sight a natural enough definition, because – δλi/δs at an event P represents the acceleration relative to the congruence of a free particle travelling along a geodesic tangent to the congruence at P. In order to give physical meaning to this definition of force it is necessary to specify the congruence λi physically, and it would seem most natural to choose the congruence of world-lines of flow of the medium. Supposing certain conditions satisfied by this congruence (cf. Ruse, loc. cit.), the theory of Ruse is applicable, and from this follows a form of Gauss's theorem.

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References

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page 93 note 1 Whittaker, E. T., Proc. Roy. Soc. (A) 149 (1935), 384395.

page 93 note 2 Ruse, H. S., Proc. Edin. Math. Soc. (2) 4 (1935), 144158.

page 93 note 3 A form of Gauss's theorem inapplicable to such a fundamental case is not satisfactory. We cannot apply Ruse's form of the theorem to this case with λi = θi, because the special conditions which he imposes on λi require p = 0, as we can see from inspection of equation (5.7) with g i = 0

page 94 note 1 Temple, G., Proc. Roy. Soc. (A) 154 (1936), 354363.

page 94 note 2 CfSynge, J. L., “A criticism of the method of expansion in powers of the gravitational constant in general relativity,” to appear shortly in Proc. Roy. Soc. (A).

page 95 note 1 Prof Barnes, C. has drawn my attention to Maxwell's remarks regarding this point: J. C. Maxwell, Mattel- and Motion (London, 1894), 85.

page 95 note 2 CfSynge, J. L., Annals of Mathematics 35 (1934), 705713, for a simple proof by a method applicable also to the deviation of geodesic null-lines.

page 97 note 1 Eisenhart, L. P., Riemannian Geometry (Princeton, 1926), 113.

page 99 note 1 For this form, see Synge, J. L., “Integral electromagnetic theorems in general relativity” (Proc. Roy. Soc. A, 157 (1936), 434443), equation (2.25).

page 100 note 1 Eisenhart, L. P., Riemannian Geometry (Princeton, 1926), 96.

page 101 note 1 CfEisenhart, L.P., Riemannian Geometry (Princeton 1926), 113.

page 101 note 2 It is possible to adopt two definitions of the energy tensor, differing by a factor c 2. That here employed reduces for a stream of unstressed matter to T ij = pθi θj, where θi is a unit vector and p is energy-density, not mass-density. This form is to be preferred, because in general relativity energy should be regarded as the more primitive concept, from which mass is a convenient conventional derivative.

page 102 note 1 Synge, J. L., Trans. Roy. Soc. Canada, Sect. III, 28 (1934), 163, where however a factor c 2 enters because there the concept of mass was taken as fundamental.

page 102 note 2 The factor c 2 is present in the denominator on the right hand side of (3.20) because we have used proper time s instead of the usual time in our definition of acceleration, so that our X i is the usual force (comparable to that of Newtonian mechanics) divided by c 2.

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