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Equivalence Principle, Covariance, and Curvature Tensor

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A Primer in Tensor Analysis and Relativity

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

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Abstract

The previous part of the book started from the Maxwell equations, one of which is the Gauss law. We already know that this law of electrostatics is nicely embedded into relativistic framework, being part of the full set of equations, which are invariant under the Lorentz transformations. At the same time, there is one more case when we meet the same Gauss equation.

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Notes

  1. 1.

    We will not distinguish between different versions of the equivalence principle; the interested reader can look, e.g., into the books [1, 2].

  2. 2.

    Note two alternative notations for covariant derivative. Similar notations are also used for partial derivative, e.g., \(\partial _\alpha {\mathbf{T}} = {\mathbf{T}}_{,\,\alpha }\).

  3. 3.

    We also met this condition in Part 1, but then it was somehow trivial, since in flat space it immediately follows from the existence of a global orthonormal baisis. In the case of a general curved manifold this condition is completely nontrivial, as we shall see briefly.

  4. 4.

    Based on the pedagogical paper [5].

  5. 5.

    It is customary to call such a particle free, because gravity can be compensated by inertia in a special coordinate system.

References

  1. C.W. Misner, K.S. Thorn, J.A. Wheeler, Gravitation (W.H. Freeman and Co., San Francisco, 1973)

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  2. R. D’Inverno, Introducing Einstein’s Relativity (Oxford University Press, Oxford, 1992–1998)

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  3. A.T. Fomenko, S.P. Novikov, B.A. Dubrovin, Modern Geometry-Methods and Applications, Part I: The Geometry of Surfaces, Transformation Groups, and Fields (Springer, Berlin, 1992)

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  4. S. Weinberg, Gravitation and Cosmology (Wiley, Hoboken, 1972)

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  5. D.F. Carneiro, E.A. Freitas, B. Gonçalves, A.G. de Lima, I.L. Shapiro, On useful conformal tranformations in general relativity. Gravit. Cosmol. 40, 305 (2004). gr-qc/0412113

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  6. F.W. Hehl, P. Heide, G.D. Kerlick, J.M. Nester, General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys. 48, 393 (1976)

    Article  ADS  MathSciNet  Google Scholar 

  7. I.L. Shapiro, Physical aspects of the space-time torsion. Phys. Repts. 357, 113 (2002)

    Article  ADS  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departamento de Física, Universidade Federal de Juiz de Fora, Juiz de Fora, Minas Gerais, Brazil

    Ilya L. Shapiro

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  1. Ilya L. Shapiro
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Correspondence to Ilya L. Shapiro .

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Shapiro, I.L. (2019). Equivalence Principle, Covariance, and Curvature Tensor. In: A Primer in Tensor Analysis and Relativity. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-26895-4_14

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