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Gauge Invariance for Gravitation and Gradient Continuum | SpringerLink

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Gauge Invariance for Gravitation and Gradient Continuum

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Covariance and Gauge Invariance in Continuum Physics

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Abstract

Geometrization of continuum physics that is the formulation of constitutive laws and conservation laws equations with respect to a reference spacetime involves some steps. First of all, physical measurable quantities should be identified with geometrical variables (metric, torsion, and curvature on the material manifold) and other additional variables if any. Second point, the spacetime is generally a dynamical background with its metric, torsion, and curvature, such is the case for general relativity. Then it is required to specify how all these geometrical variables are generated and modified by physical objects, namely material particle, material elements as line, surface, volume, defects, and how these physical objects evolve during the interaction of the continuum matter and the spacetime.

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Notes

  1. 1.

    Index 0 stands for time variable, then the Lagrangian \(\mathcal {L}\) also stands for dynamical situation.

  2. 2.

    Lagrangian density of type I as \(\mathcal {L}(g_{ij}, \partial _k g_{ij}, \partial _l \partial _k g_{ij})\) or of type II as \(\mathcal {L}(g_{ij}, \nabla _k g_{ij}, \nabla _l \nabla _k g_{ij})\) was considered in Manoff (1999) where three kinds of variational procedures, say the functional variation, the Lie variation, and the covariant variation, were used to derive the fields equations of Einstein’s gravitation theory.

  3. 3.

    The Euclidean connection derived from the metric tensor of a reference body was mostly the connection used in continuum mechanics for over two centuries, e.g. Rakotomanana (2003).

  4. 4.

    As shown by Kibble (1961), the Lagrangian density of the form \(\mathcal {L} (x^\mu , \varPhi , \partial _\alpha \varPhi , \partial _\beta \partial _\alpha \varPhi )\) should be replaced by a Lagrangian of the form \(\mathcal {L} (\varPhi , \nabla _\alpha \varPhi , \nabla _\beta \nabla _\alpha \varPhi ) \sqrt { \mathrm {Det} \mathbf {g}}\) which is invariant under a general coordinates transformations, which are nothing else than Poincaré’s local transformations.

  5. 5.

    The dependence on coordinates x μ is dropped according to results in e.g. Kibble (1961).

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  1. Institut de Recherche Mathématique, Université de Rennes, Rennes CX, France

    Lalaonirina R. Rakotomanana

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  1. Lalaonirina R. Rakotomanana
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R. Rakotomanana, L. (2018). Gauge Invariance for Gravitation and Gradient Continuum. In: Covariance and Gauge Invariance in Continuum Physics. Progress in Mathematical Physics, vol 73. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-91782-5_4

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