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Some conformally separable metrics in flat space and their application to accelerated coordinate frameworks in special relativity

Published online by Cambridge University Press:  09 April 2009

N. W. Taylor
Affiliation:
Department of MathematicsUniversity of New EnglandArmidale New South Wales
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If the metric of an n-dimensional space is taken in the form ds2 = u22+dσ2, where 2 and 2 are cartesian metrics of r and (n−r) dimensions, respectively, the various forms of u for flat space are quite simple. The study of accelerated motions in special relativity by various authors has led to four dimensional metrics of this form. Those in which u = ±1 at the space origin for all values of time are of particular interest. They are locally cartesian at the accelerated observer, and so the coordinates in the neighbourhood of the observer correspond directly to physical measurements. Hence, such metrics provide convenient means of describing physical conditions experienced by accelerated observers. If the τ-space contains the time direction and is of one or two dimensions, arbitrary rectilinear motions are allowed.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1968

References

[1]Forsyth, A. R., Theory of Differential Equations. Vols. 5 and 6, Partial Differential Equations. Dover, New York, (1959).Google Scholar
[2]Gupta, S. N., “Conformal transformations and space travel’, Science 134 (1961), 13601361.CrossRefGoogle ScholarPubMed
[3]Lass, H., “Accelerating frames of reference and the clock paradox”. Am. J. Phys. 31 (1963), 274276.CrossRefGoogle Scholar
[4]Marsh, L. McL., “Relativistic accelerated systems”. Am. J. Phys. 33 (1965), 934938.CrossRefGoogle Scholar
[5]Møller, C., The Theory of Relativity. Oxford (1952).Google Scholar
[6]Page, L., “A new relativity”. Phys. Rev. 49 (1936) 254268.CrossRefGoogle Scholar
[7]Page, L. and Adams, N. I., Electrodynamics. Dover, New York (1965).Google Scholar
[8]Sokolnikoff, I. S., Tensor Analysis. Theory and Applications to Geometry and Mechanics of Continua. John Wiley & Sons, Inc., New York (1964).Google Scholar
[9]Wong, Y. C., “Some Einstein spaces with conformally separable fundamental tensors”. Trans. Am. Math. Soc. 53 (1943), 157194.CrossRefGoogle Scholar
[10]Yano, K., “Conformally separable quadratic differential forms”. Proc. Imp. Acad. Tokyo. 16 (1940), 8386.Google Scholar