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Motion in Alternative Theories of Gravity

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Mass and Motion in General Relativity

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 162))

Abstract

Although general relativity (GR) passes all present experimental tests with flying colors, it remains important to study alternative theories of gravity for several theoretical and phenomenological reasons that we recall in these lecture notes. The various possible ways of modifying GR are presented, and we notably show that the motion of massive bodies may be changed even if one assumes that matter is minimally coupled to the metric as in GR. This is illustrated with the particular case of scalar-tensor theories of gravity, whose Fokker action is discussed, and we also mention the consequences of the no-hair theorem on the motion of black holes. The finite size of the bodies modifies their motion with respect to pointlike particles, and we give a simple argument showing that the corresponding effects are generically much larger in alternative theories than in GR. We also discuss possible modifications of Newtonian dynamics (MOND) at large distances, which have been proposed to avoid the dark matter hypothesis. We underline that all the previous classes of alternatives to GR may a priori be used to predict such a phenomenology, but that they generically involve several theoretical and experimental difficulties.

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Notes

  1. 1.

    One needs to compute the integrals represented by the various diagrams to derive expression (6). See Ref. [41] for explicit diagrammatic calculations.

  2. 2.

    Two different (though related) meanings of “post-Keplerian” exist in the literature. We are here considering a post-Keplerian expansion in powers of v orbital 2 ∕ c 2, while keeping the full nonperturbative dependence in the gravitational self-energy Gm ∕ Rc 2. On the other hand, post-Keplerian deviations mean relativistic effects modifying the lowest-order Keplerian motion, like those described in Section 4.3. Only this latter meaning is used in GR, because its strong equivalence principle implies that the internal structure of a body does not influence its motion up to order \(\mathcal{O}(1/{c}^{10})\), as recalled in Section 5.

  3. 3.

    The precise definitions of these multipoles and their explicit expressions may be found for instance in Section 6 of Ref. [39].

  4. 4.

    These larger finite-size effects are due to the fact that a spin-0 scalar field can couple to the spherical inertia moment of a body, contrary to a spin-2 graviton. They should not be confused with the violation of the strong equivalence principle, which also occurs in scalar-tensor theories because all form factors m(φ), N(φ), … depend on the body’s self-energy. Regardless of its finite size, the motion of a self-gravitating body in a uniform exterior gravitational field depends thus on its internal structure.

  5. 5.

    See Section II.B of Ref. [30] for a critical discussion of various such mass-dependent models.

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  1. gℝεℂo, Institut d’Astrophysique de Paris, UMR 7095-CNRS, Université Pierre et Marie, Curie-Paris 6, 98bis boulevard Arago, F-75014, Paris, France

    Gilles Esposito-Farèse

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  1. Institut d'Astrophysique de Paris, boulevard Arago 98bis, Paris, 75014, France

    Luc Blanchet

  2. Observatoire de la Cote d'Azur, Nice, France

    Alessandro Spallicci

  3. Dept. Physics, University of Florida, Gainesville, 32611-8440, Florida, USA

    Bernard Whiting

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Esposito-Farèse, G. (2009). Motion in Alternative Theories of Gravity. In: Blanchet, L., Spallicci, A., Whiting, B. (eds) Mass and Motion in General Relativity. Fundamental Theories of Physics, vol 162. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3015-3_17

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