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Mass and Motion in General Relativity pp 415–442Cite as

High-Accuracy Comparison Between the Post-Newtonian and Self-Force Dynamics of Black-Hole Binaries

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

Abstract

The relativistic motion of a compact binary system moving in circular orbit is investigated using the post-Newtonian (PN) approximation and the perturbative self-force (SF) formalism. A particular gauge-invariant observable quantity is computed as a function of the binary’s orbital frequency. The conservative effect induced by the gravitational SF is obtained numerically with high precision, and compared to the PN prediction developed to high order. The PN calculation involves the computation of the 3PN regularized metric at the location of the particle. Its divergent self-field is regularized by means of dimensional regularization. The poles \(\propto {(d - 3)}^{-1}\) that occur within dimensional regularization at the 3PN order disappear from the final gauge-invariant result. The leading 4PN and next-to-leading 5PN conservative logarithmic contributions originating from gravitational wave tails are also obtained. Making use of these exact PN results, some previously unknown PN coefficients are measured up to the very high 7PN order by fitting to the numerical SF data. Using just the 2PN and new logarithmic terms, the value of the 3PN coefficient is also confirmed numerically with very high precision. The consistency of this cross-cultural comparison provides a crucial test of the very different regularization methods used in both SF and PN formalisms, and illustrates the complementarity of these approximation schemes when modeling compact binary systems.

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Notes

  1. 1.

    As usual the nPN order refers to terms equivalent to (v ∕ c)2n beyond Newtonian theory, where v is a typical internal velocity of the material system and c is the speed of light.

  2. 2.

    Since we are interested in the motion of the small particle m 1, we remove the index 1 from u 1 α.

  3. 3.

    In all of this section we shall set \(G = c = 1\).

  4. 4.

    Here g αβ is the contravariant metric, inverse of the covariant metric g αβ of determinant g = det(g αβ), and \({\eta }^{\alpha \beta } =\mathrm{ diag}(-1, 1, 1, 1)\) represents an auxiliary Minkowski metric in Cartesian coordinates.

  5. 5.

    We use shorthands such as x ab = x a x b; \(\hat{{x}}^{abc} = {x}^{abc} -\frac{1} {5}({\delta }^{ab}{x}^{c} + {\delta }^{ac}{x}^{b} + {\delta }^{bc}{x}^{a}){r}^{2}\) denotes the symmetric and trace-free part of x abc; ε abc is the Levi-Civita antisymmetric symbol.

  6. 6.

    The Landau o symbol for remainders takes its standard meaning.

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Acknowledgements

The authors acknowledge the 2008 Summer School on Mass and Motion, organized by A. Spallicci and supported by the University of Orléans and the CNRS, through which we experienced an extensive opportunity to understand each other’s perspective and make rapid progress on this work. SD and BFW acknowledge support through grants PHY-0555484 and PHY-0855503 from the National Science Foundation. LB and ALT acknowledge support from the Programme International de Coopération Scientifique (CNRS–PICS).

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Authors and Affiliations

  1. gℝεℂo, Institut d’Astrophysique de Paris, C.N.R.S. & Université Pierre et Marie Curie, 98bis Boulevard Arago, 75014, Paris, France

    Luc Blanchet & Alexandre Le Tiec

  2. Institute for Fundamental Theory, Department of Physics, University of Florida, Gainesville, FL, 32611-8440, USA

    Steven Detweiler & Bernard F. Whiting

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  1. Luc Blanchet
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  3. Alexandre Le Tiec
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  4. Bernard F. Whiting
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Correspondence to Alexandre Le Tiec .

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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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Blanchet, L., Detweiler, S., Le Tiec, A., Whiting, B.F. (2009). High-Accuracy Comparison Between the Post-Newtonian and Self-Force Dynamics of Black-Hole Binaries. 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_15

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  • DOI: https://doi.org/10.1007/978-90-481-3015-3_15

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