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VII.—On the Measurement of Spatial Distance in a Curved Space-time

Published online by Cambridge University Press:  15 September 2014

H. S. Ruse
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Extract

The problem of defining the concept of spatial distance in a general riemannian space-time has been discussed by E. T. Whittaker, who showed how to translate into the terms of four-dimensional geometry the method adopted by astronomers in determining the distance of a star from the earth by means of a comparison of its absolute and apparent luminosities. The problem was further considered by the present writer, who derived a different formula for spatial distance by a method which was essentially one of partitioning the whole of space-time into space and time relative to the given observer. It was suggested that the new spatial distance might be that determined practically by parallax-measurements, though the evidence in support of this suggestion was perhaps hardly sufficient to carry conviction.

Type
Proceedings
Information
Proceedings of the Royal Society of Edinburgh , Volume 53 , 1934 , pp. 79 - 88
Copyright
Copyright © Royal Society of Edinburgh 1934

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References

page 79 note * Proc. Roy. Soc., A, vol. cxxxiii, 1931, p. 93.

page 79 note † Proc. Roy. Soc. Edin., vol. lii, 1932, p. 183.

page 79 note ‡ The word distance will be used to mean spatial distance. Displacements between point-events in the four-dimensional world will be called intervals.

page 81 note * Eddington, Mathematical Theory of Relativity, 1924, p. 77.

page 83 note * See Veblen, Invariants of Quadratic Differential Forms (Camb. Math. Tract, No. 24, 1927), chap. vi.

page 83 note † This notation differs slightly from that of my previous paper, in which the coordinates of S were denoted by unaccented letters. In this paper x 0, x 1, x 2, x 3 are used exclusively for current coordinates.

page 85 note * Ruse, , Proc. London Math. Soc., vol. xxxii, 1931, p. 90.Google Scholar

page 86 note * Kermack, , M'Crea, , and Whittaker, , Proc. R.S.E., vol. liii, 1933, p. 35.Google Scholar

page 88 note * Eddington, op. cit., p. 156, equation (67.12) with R sin x=r.