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Note on a distance invariant and the calculation of Ruse's invariant

Published online by Cambridge University Press:  20 January 2009

A. G. Walker
A. G. Walker
Affiliation:
The University, Liverpool.
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Several papers on the subject of spatial distance in General Relativity appeared a few years ago, and a simple extension of this idea to any pair of points in any Riemannian space was given by me in a thesis. A distance invariant was defined, and this was found to depend upon a certain two-point invariant which was first introduced by H. S. Ruse in a study of Laplace's Equation. This invariant, now written ρ and defined in (3), has lately re-appeared, and it may now be of interest to publish the results found earlier. These include a geometrical interpretation of ρ, a simple method of calculation, and an expansion as a power series in the geodesic arc. The dependence of ρ upon the geodesic arc is also considered.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 7 , Issue 1 , October 1942 , pp. 16 - 26
Copyright
Copyright © Edinburgh Mathematical Society 1942

References

page 16 note 1 For the Senior Mathematical Scholarship, Oxford, 1934.Google Scholar

page 16 note 2 Proc. Edinburgh Math. Soc. (2), 2 (1931), 137.Google Scholar

page 16 note 3 Copson, and Ruse, , Proc. Royal Soc. Edinburgh, 60 (1940), 117.CrossRefGoogle Scholar

This theorem on null geodesies was given by me in Quart. J. of Math., 4 (1933), 72,Google Scholar and has since been generalised by Prof. Temple, G., Proc. Royal soc. A, 168 (1938), 122.CrossRefGoogle Scholar

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page 18 note 1 Walker, A. G., Proc. Royal Soc. Edinburgh, 52 (1932), 351.Google Scholar

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page 20 note 1 This proof, which replaces a longer one of my own, is due to H. S. Ruse.

page 20 note 2 See, for example, Veblen, , “Invariants of Quadratic Differential Forms,” Cambridge Tract, 24 (1927), 94.Google Scholar

page 26 note 1 Journal de Math., 18 (1939), 225.Google Scholar