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VI.—On the Definition of Spatial Distance in General Relativity

Published online by Cambridge University Press:  15 September 2014

H. S. Ruse
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Extract

In a recent paper, Professor E. T. Whittaker discussed the problem of defining, in a general riemannian space-time, the concept of spatial distance between material particles. It is the object of this paper to give an alternative definition, and to compare the new formula with that of Whittaker.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 52 , 1933 , pp. 183 - 194
Copyright
Copyright © Royal Society of Edinburgh 1933

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References

page 183 note * Proc. Boy. Soc., A, 133 (1931), 93.

page 183 note † The instantaneous three-dimensional space of the observer consists of those world-points in his immediate neighbourhood which he regards as simultaneous. Geometrically, it is a small portion near O of the hypersurface formed by the geodesics through O which are perpendicular at this point to the observer's world-line.

page 186 note * The η's are in fact a particular set of Riemann normal coordinates. Systems of this type have recently been employed by Thomas, T. Y., Proc. Nat. Acad. Sci., 16 (1930), 761.CrossRefGoogle Scholar

page 187 note * Eddington, , Mathematical Theory of Relativity (1924), 163.Google Scholar

page 187 note † See, for example, Eisenhart, , Riemannian Geometry (1926)Google Scholar, ch. iii, (29.3).

page 188 note * Veblen, , Invariants of Quadratic Differential Forms (Camb. Math. Tract No. 24, 1927)Google Scholar, ch. vi.

page 188 note † Ruse, , Proc. London Math. Soc., 32 (1931), 90.Google Scholar

page 189 note * It is necessary to assume that he measures time by a clock in his possession, so that the physical time is identical with his proper-time τ.

page 191 note * x = x, y = y, z = z are then the equations of a geodesic, since the equations of geodesies in this space are all of the form x = au + b, y = a′u + b′, z=a″u + b″, where the a's and b's are constants.

page 191 note † Whittaker, loc. cit., equation (4).

page 193 note * Whittaker, , loc. cit., 96Google Scholar, and Eddington, , Mathematical Theory of Relativity (1924), 161.Google Scholar

page 194 note * Op. cit., 163.