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General Relativity, Cosmology and Astrophysics pp 111–145Cite as

The General Relativistic Two Body Problem and the Effective One Body Formalism

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Part of the book series: Fundamental Theories of Physics ((FTPH,volume 177))

Abstract

A new analytical approach to the motion and radiation of (comparable mass) binary systems has been introduced in 1999 under the name of 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.

    For gravitational waveforms, one conventionally defines the PN accuracy as the fractional PN accuracy with respect to the leading-order, \(O(c^{-5})\), quadrupolar emission. E.g., a \(1\)PN-accurate waveform retains next-to-leading order terms, i.e. terms smaller than the quadrupolar waveform by a factor \(O(c^{-2})\).

  2. 2.

    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.

  3. 3.

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

  4. 4.

    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 \(R^\mathrm{{ADM}} = \vert {\varvec{q}} \vert \). The transformation between the two coordinate systems has been determined in Refs. [57, 59].

  5. 5.

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

  6. 6.

    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.

  7. 7.

    We recall that the coefficients \(n_1\) and \((d_1,d_2,d_3)\) of the \((1,3)\) Padé approximant \(P_3^1 [A_\mathrm{3PN} (u)]\) are determined by the condition that the first four terms of the Taylor expansion of \(A_3^1\) in powers of \(u=\textit{GM}/(c^2R)\) coincide with \(A_\mathrm{3PN}\).

  8. 8.

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

  9. 9.

    The two “pinching” frequencies used for this comparison are \(M\omega _1=0.047\) and \(M\omega _2=0.31\).

  10. 10.

    See Ref. [113], which quoted and used some combinations of the logarithmic contributions to \(a(u)\) and \(\bar{d}(u)\). given in Ref. [119].

  11. 11.

    See [73] for a recent extension of the EOB formalism to non-circular (ellipticlike or hyperboliclike) motions.

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    Thibault Damour

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    Jiří Bičák

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    Tomáš Ledvinka

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Damour, T. (2014). The General Relativistic Two Body Problem and the Effective One Body Formalism. In: Bičák, J., Ledvinka, T. (eds) General Relativity, Cosmology and Astrophysics. Fundamental Theories of Physics, vol 177. Springer, Cham. https://doi.org/10.1007/978-3-319-06349-2_5

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