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Asymptotic expansion of general relativity with Galilean covariance

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Abstract

The Galilean gravitation derives from a scalar potential and a vector one. Poisson’s equation to determine the scalar potential has not the expected Galilean covariance. Moreover, there are three missing equations to determine the potential vector. Besides, we require they have the Galilean covariance. These are the issues addressed in the paper. To avoid the drawbacks of the PPN approach and the NCT, we merge them into a new framework. The key idea is to take care that every term of the c expansion of the fields are Galilean covariants or invariants. The expected equations are deduced by variation of the Hilbert–Einstein functional. The contribution of the matter to the functional is derived from Souriau’s conformation tensor. We obtain a system of four non linear equations, solved by asymptotic expansion.

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Acknowledgements

I would like to acknowledge the Mainz Institute for Theoretical Physics (MITP) for enabling me to improve significantly this work during the workshop Applied Newton–Cartan Geometry (APPNC 2018).

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  1. UMR CNRS 9013 – LaMcube–Laboratoire de mécanique multiphysique multiéchelle, Univ. Lille, 59000, Lille, France

    G. de Saxcé

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de Saxcé, G. Asymptotic expansion of general relativity with Galilean covariance. Gen Relativ Gravit 52, 89 (2020). https://doi.org/10.1007/s10714-020-02738-3

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