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Elementary Development of the Gravitational Self-Force

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Mass and Motion in General Relativity

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

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Abstract

The gravitational field of a particle of small mass m moving through curved spacetime, with metric g ab , is naturally and easily decomposed into two parts each of which satisfies the perturbed Einstein equations through O(m). One part is an inhomogeneous field h ab S which, near the particle, looks like the Coulomb m ∕ r field with tidal distortion from the local Riemann tensor. This singular field is defined in a neighborhood of the small particle and does not depend upon boundary conditions or upon the behavior of the source in either the past or the future. The other part is a homogeneous field h ab R. In a perturbative analysis, the motion of the particle is then best described as being a geodesic in the metric g ab + h ab R. This geodesic motion includes all of the effects which might be called radiation reaction and conservative effects as well.

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Notes

  1. 1.

    If the acceleration of gravity \(\vec{g}\) differs significantly across a large object, then the center of mass moves responding to some average, over the object, of \(\vec{g}\) which does not necessarily match a free-fall trajectory.

  2. 2.

    Following Dirac’s [31] usage, I prefer to use the word “actual” to refer to the complete, and total field that might be measured at some location. Often in self-force treatises the “retarded field” plays this central role. But, this obscures the fact that, viewed from near by, a local observer unaware of boundary conditions could make no measurement which would reveal just what part of the field is the retarded field. This confusion is increased if the spacetime is not flat, so that the retarded field could be determined only if the entire spacetime geometry were known.

  3. 3.

    A terse but adequate description of perturbative tidal effects on a Newtonian, self-gravitating, non-rotating, incompressible fluid is given on p. 467 of [16].

  4. 4.

    Expansions for the somewhat related “direct” field are also available [3, 4, 7, 37, 38, 43, 45], though their use is, similarly, not at all elementary.

  5. 5.

    In fact the singular field was discovered first [24] using matched asymptotic expansions. And the Green’s function appeared only later during an attempt to show consistency with the usual DeWitt-Brehme [30] approach to radiation reaction.

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Acknowledgements

My understanding of gravitational self-force effects has evolved over the past decade in large part in discussions with colleagues during the annual Capra meetings. I am deeply indebted to the organizers and participants of these fruitful meetings. And I am particularly pleased to have had recent collaborators Leor Barack, Peter Diener, Eric Poisson, Norichika Sago, Wolfgang Tichy, Ian Vega, and Bernard Whiting, who individually and as a group have kept me on track and moving forward. This work was supported in part by the National Science Foundation, through grant number PHY-0555484 with the University of Florida. Some of the numerical results described here were preformed at the University of Florida High-Performance Computing Center (URL: http://hpc.ufl.edu).

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Authors and Affiliations

  1. Institute of Fundamental Theory, Department of Physics, University of Florida, Gainesville, FL, 32611, USA

    Steven Detweiler

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  1. Steven Detweiler
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Correspondence to Steven Detweiler .

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Editors and Affiliations

  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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Detweiler, S. (2009). Elementary Development of the Gravitational Self-Force. 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_10

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