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Self-action in gravity

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Abstract

A particle moving with variable acceleration in the flat space-time affects the self-force due to outgoing radiation. The gravitational fields bring an additional contribution to self-force due to scattering waves on the curved backgrounds. This force is not zero even for a particle at rest. A review of the self-interaction in the gravitational field is presented. We consider the self-force for the particle connected with the vector and scalar fields. Different backgrounds are considered—black holes, stars, topological defects, and wormholes.

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Notes

  1. The signature \((-,+,+,+)\) is used. The geometrical objects are that as in Hawking and Ellis book [1] and we use calligraphic notations for them \(\mathcal {R}, \mathcal {R}_{\mu \nu }\). The Greek letters include time \(\alpha ,\beta , \ldots = 0,1,2,3\) and Latin—the space \(i,j,\ldots = 1,2,3\), only. The \(\text {g}^{(3)} = \det \text {g}_{ij}\) is determinant of 3D metric and \(\cos \gamma = \cos \theta \cos \theta ' + \sin \theta \sin \theta ' \cos (\varphi -\varphi ')\) is cosine of spherical angle. The line element of unite 2-sphere: \(d\varOmega _2 = d\theta ^2 + \sin ^2\theta d\varphi ^2\). \(\square = \text {g}^{\mu \nu }\nabla _\mu \nabla _\nu \) and \(\triangle = \text {g}^{ik}\nabla _i\nabla _k\) are corresponding 4D and 3D Laplace–Beltrami operators. The semicolon denotes the covariant derivative, e.g., \(u^\mu _{;\nu } = \nabla _\nu u^\mu \).

  2. Synge [31] called this quantity a “world function.”

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Acknowledgements

I would like to thank A. Popov, P. Taylor, D. Vassilevich, M. Volkov, and A. Zelnikov for helpful comments. The work was supported in parts by the Grants 2016/03319-6 and 2019/10719-9 of the São Paulo Research Foundation (FAPESP), by the RFBR Projects 19-02-00496-a.

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  1. Centro de Matemática, Computação e Cognição, Universidade Federal do ABC, Santo André, SP, 09210-170, Brazil

    Nail Khusnutdinov

  2. Regional Scientific and Educational Mathematical Center of Kazan Federal University, Kremlevskaya 18, Kazan, Russia, 420008

    Nail Khusnutdinov

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Khusnutdinov, N. Self-action in gravity. Eur. Phys. J. Plus 136, 669 (2021). https://doi.org/10.1140/epjp/s13360-021-01640-4

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