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Astrophysical Black Holes pp 273–312Cite as

The Effective-One-Body Approach to the General Relativistic Two Body Problem

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Part of the book series: Lecture Notes in Physics ((LNP,volume 905))

Abstract

The two-body problem in General Relativity has been the subject of many analytical investigations. After reviewing some of the methods used to tackle this problem (and, more generally, the N-body problem), we focus on a new, recently introduced approach to the motion and radiation of (comparable mass) binary systems: the Effective One Body (EOB) formalism. We review the basic elements of this formalism, and discuss some of its recent developments. Several recent comparisons between EOB predictions and Numerical Relativity (NR) simulations have shown the aptitude of the EOB formalism to provide accurate descriptions of the dynamics and radiation of various binary systems (comprising black holes or neutron stars) in regimes that are inaccessible to other analytical approaches (such as the last orbits and the merger of comparable mass black holes). In synergy with NR simulations, post-Newtonian (PN) theory and Gravitational Self-Force (GSF) computations, the EOB formalism is likely to provide an efficient way of computing the very many accurate template waveforms that are needed for Gravitational Wave (GW) data analysis purposes.

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Notes

  1. 1.

    This is related to an idea emphasized many times by John Archibald Wheeler: quantum mechanics can often help us in going to the essence of classical mechanics.

  2. 2.

    We consider, for simplicity, ‘equatorial’ motions with m = ℓ, i.e., classically, \(\theta = \frac{\pi } {2}\).

  3. 3.

    It is convenient to write the ‘effective metric’ in Schwarzschild-like coordinates. Note that the effective radial coordinate R differs from the two-body ADM-coordinate relative distance \(R^{\mathrm{ADM}} =\vert \boldsymbol{q}\vert\). The transformation between the two coordinate systems has been determined in [32, 61].

  4. 4.

    Indeed \(E_{\mathrm{real}}^{\mathrm{total}} = Mc^{2} + E_{\mathrm{real}}^{\mathrm{relative}} = Mc^{2} + \mbox{ Newtonian terms} + \mathrm{1PN}/c^{2} + \cdots \), while \(\mathcal{E}_{\mathrm{effective}} =\mu c^{2} + N + \mathrm{1PN}/c^{2} + \cdots \).

  5. 5.

    The PN-expanded EOB building blocks \(A_{3\mathrm{PN}}(R),B_{3\mathrm{PN}}(R),\ldots\) already represent a resummation of the PN dynamics in the sense that they have “condensed” the many terms of the original PN-expanded Hamiltonian within a very concise format. But one should not refrain to further resum the EOB building blocks themselves, if this is physically motivated.

  6. 6.

    We recall that the coefficients n 1 and \((d_{1},d_{2},d_{3})\) of the (1, 3) Padé approximate \(P_{3}^{1}[A_{\mathrm{3PN}}(u)]\) are determined by the condition that the first four terms of the Taylor expansion of A 3 1 in powers of \(u = GM/(c^{2}R)\) coincide with A 3PN.

  7. 7.

    The new, resummed EOB waveform discussed above was not available at the time, so that these comparisons employed the coarser “Newtonian-level” EOB waveform \(h_{22}^{(N,\epsilon )}(x)\).

  8. 8.

    The two “pinching” frequencies used for this comparison are M ω 1 = 0. 047 and M ω 2 = 0. 31.

  9. 9.

    Including the effective-field-theory reformulation of the computation of the PN-expanded Fokker-action [47, 89].

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    Thibault Damour & Alessandro Nagar

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  1. Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Como, Italy

    Francesco Haardt

  2. Dipartimento di Scienza e Alta Tecnologia, Università degli Studi dell’Insubria, Como, Italy

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    Aldo Treves

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    Monica Colpi

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Damour, T., Nagar, A. (2016). The Effective-One-Body Approach to the General Relativistic Two Body Problem. In: Haardt, F., Gorini, V., Moschella, U., Treves, A., Colpi, M. (eds) Astrophysical Black Holes. Lecture Notes in Physics, vol 905. Springer, Cham. https://doi.org/10.1007/978-3-319-19416-5_7

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