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Special Relativity pp 229–242Cite as

\(^{\star }\)Special Relativity in Arbitrary Coordinates

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Part of the book series: Undergraduate Lecture Notes in Physics ((ULNP))

Abstract

Thus far we have limited ourselves to studying Special Relativity using Cartesian coordinates to cover Minkowski spacetime and you are probably sick of the phrase “the equation is ... in Cartesian coordinates” and are wondering why we are not discussing equations in general coordinate systems as we did in Chap.4. There is no a priori reason to adopt Cartesian coordinates, only they allow greater mathematical simplicity tailored to the beginner. It is time to re-examine Special Relativity using arbitrary coordinates \({\displaystyle \left\{ x^{\mu }\right\} }\). They involve a little more mathematics and this is the reason why they were put off until now. After generalizing the geometrical formulation of Minkowski spacetime to arbitrary coordinate systems, we will revisit the laws of motion of particles and the physics of various forms of mass-energy in this spacetime. We will conclude by briefly introducing some of the basic ideas of General Relativity.

The important thing is not to stop questioning.

—Albert Einstein.

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Notes

  1. 1.

    For timelike curves the parameter \(\lambda \) has the dimensions of a time while for null curves it has the dimension of a length.

  2. 2.

    The curve itself is a geometric object defined independently of the particular parametrization used (it is an equivalence class of parametrizations, with the equivalence relation obtained by identifying parametric representations which have the same image).

  3. 3.

    One can obtain Eq. (10.30) also by considering a frame on which the disk is rolling. Then this equation is the result of time dilation for the time taken by the disk (deformed to an ellipse) to perform a complete roll [6].

  4. 4.

    Cf. Ref. [7].

  5. 5.

    By contrast a perfect fluid with radiation equation of state\({\displaystyle P=\rho c^2 /3}\) describes waves with random phases, polarizations, and propagation directions.

References

  1. R.M. Wald, General Relativity (Chicago University Press, Chicago, 1984)

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  2. S.M. Carroll, Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley, San Francisco, 2004)

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  3. B.L. Moiseiwitsch, Variational Principles (Interscience, London, 1966)

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  4. H.J. Weber, G.B. Arfken, Essential Mathematical Methods for Physicists (Elsevier, Amsterdam, 2004)

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  5. H. Goldstein, Classical Mechanics, 2nd edn. (Addison-Wesley, Reading, 1972)

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  6. K. Vøyenli, Am. J. Phys. 45, 876 (1977)

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  7. R.V. Buny, S.D.H. Hsu, Phys. Lett. B 632, 543 (2006)

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Authors and Affiliations

  1. Physics Department, Bishop’s University, College Street 2600, Sherbrooke, QC, J1M 1Z7, Canada

    Valerio Faraoni

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  1. Valerio Faraoni
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Correspondence to Valerio Faraoni .

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© 2013 Springer International Publishing Switzerland

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Faraoni, V. (2013). \(^{\star }\)Special Relativity in Arbitrary Coordinates. In: Special Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01107-3_10

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