Applications of the double and the dual numbers. The Bianchi models

Authors

DOI:

https://doi.org/10.31349/RevMexFisE.17.146

Keywords:

Double numbers, dual numbers, cosmology, Lie groups.

Abstract

We show that by using complex, double, and dual numbers one can find the invariant one-forms employed in the metrics of the Bianchi cosmological models. The result is equivalent to find, locally, all the Lie groups of dimension three.

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References

G.F. Torres del Castillo, {it Rev. Mex. Fis. E} {bf 65} (2019) 152.

V.K. Kisil, {it Geometry of M"obius Transformations, Elliptic, Parabolic and Hyperbolic Actions of SL$_{2}(mathbb{R})$} (Imperial College Press, London, 2012), chap. 3.

L.P. Hughston and K.P. Tod, {it An Introduction to General Relativity} (Cambridge University Press, Cambridge, 1990), chap. 26.

H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, {it Exact Solutions to Einstein's Field Equations}, 2nd ed. (Cambridge University Press, Cambridge, 2003), chap. 8.

{O}. Gr{o}n and S. Hervik, {it Einstein's General Theory of Relativity} (Springer, New York, 2007), chap. 15.

G.F.R. Ellis, R. Maartens and M.A.H. MacCallum, {it Relativistic Cosmology} (Cambridge University Press, Cambridge, 2012), sec. 18.3.

D.H. Sattinger and O.L. Weaver, {it Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics} (Springer-Verlag, New York, 1986), chaps. 7, 8.

G.F. Torres del Castillo, {it Differentiable Manifolds: A Theoretical Physics Approach}, 2nd edn. (Birkh"auser, New York, 2020) (to be published).

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Published

2020-07-01

How to Cite

[1]
G. F. Torres del Castillo, “Applications of the double and the dual numbers. The Bianchi models”, Rev. Mex. Fis. E, vol. 17, no. 2 Jul-Dec, pp. 146–149, Jul. 2020.

Issue

Vol. 17 No. 2 Jul-Dec (2020): Revista Mexicana de Física E

Section

Education in Physics

License

Copyright (c) 2020 Gerardo Francisco Torres del Castillo

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Authors retain copyright and grant the Revista Mexicana de Física E right of first publication with the work simultaneously licensed under a CC BY-NC-ND 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.