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Gravitational Self-force in the Schwarzschild Spacetime

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Einstein Equations: Local Energy, Self-Force, and Fields in General Relativity

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Abstract

Gravitational self-force techniques will be shortly reviewed along the lines of the two lectures presented by D. Bini at the 2019 edition of the “Domoschool.” The most important application of gravitational self-force concerns metric and curvature perturbations in black hole spacetimes due to moving particles or evolving fields. However, from a practical point of view (and for teaching purposes) we have chosen to perform the whole discussion at the level of a (massless) scalar field. In fact, in this simple case one can look at the various steps implicit in any self-force computation without facing with the additional difficulties of implementing them in a more involved tensorial background.

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Notes

  1. 1.

    Note that PN solutions “do not know” the horizon at r = 2M, but they are sensitive to the origin r = 0 only. The horizon, in fact, will be a regular singular point of the radial equation only when all PN terms will be summed. Practically, we may say that high-PN orders already start “seeing” the horizon, marked by the presence of successive l − n diverging coefficients at various PN n orders.

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Acknowledgements

We thank Thibault Damour for helpful discussions on analytical self-force computations.

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Authors and Affiliations

  1. Istituto per le Applicazioni del Calcolo “M. Picone”, Rome, Italy

    Donato Bini & Andrea Geralico

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  1. Donato Bini
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  2. Andrea Geralico
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  1. Department of Science and High Technology, University of Insubria, Como, Italy

    Sergio Luigi Cacciatori

  2. Department of Physics and Astronomy, University of Bologna, Bologna, Italy

    Alexander Kamenshchik

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Bini, D., Geralico, A. (2022). Gravitational Self-force in the Schwarzschild Spacetime. In: Cacciatori, S.L., Kamenshchik, A. (eds) Einstein Equations: Local Energy, Self-Force, and Fields in General Relativity. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-21845-3_2

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