Connections Between Metric Coefficients, Basis-  And Unit Vectors

David A de Wolf

David A de Wolf

Abstract

Abstract
This work gathers in one place what is pertinent about the connections between metric coefficients, basis- and unit vectors in a four-dimensional relativistic manifold. Some of this material can be found scattered elsewhere; its collection into one place reveals connections that either are not known or are obscure, for example, that the metric coefficients are not all independent of each other. It at least should serve as a useful tutorial for those who are not thoroughly familiar with this material.

References

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[2] Hartle J. B., Gravity, An Introduction to Einstein's General Relativity, Addison-
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[3] M. P. Hobson, G. Efstathiou, and A. N. Lasenby, General Relativity, An
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[5] Pauli W., Theory of Relativity, Pergamon Press, 1958.
[6] Bergmann P. G., Introduction To The Theory of Relativity, Dover Publications, 1976.
[7] De Wolf D.A. , Basis Vectors in Relativity, Eur. J. Phys. Ed., 12, 2021.
[8] See ref. 1, §9.7ref. 2, §2.1, or https://en.wikipedia.org/wiki/Manifold
[9] See ref. 8 and in particular ref. 3, §2.1
[10] https://en.wikipedia.org/wiki/Curvilinear_coordinates, https://www.physicsforums.com/threads/general-relativity-and-curvilinear-coordinates.818066/
[11] Covariant vectors are indicated with subscripts and contravariant ones with super-
scripts. See also ref. 3, §3.4 and §3.5, or https://www.seas.upenn.edu/~amyers/
DualBasis.pdf.
[12] See ref. 2. §20.2.
[13] https://en.wikipedia.org/wiki/Kerr_metric.
[14] Ref. [2], section 8.4