1 | In this article Greek indices take the values ![]() ![]() ![]() ![]() |
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2 | The TT coordinate system can be extended to the near zone of the source as well; see for instance Ref. [151]. | |
3 | See Ref. [81![]() |
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4 | Let us mention that the 3.5PN terms in the equations of motion are also known, both for point-particle
binaries [136![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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5 | See also Equation (140![]() ![]() |
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6 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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7 | Our notation is the following: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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8 | The constancy of the center of mass ![]() ![]() ![]() |
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9 | This assumption is justified because we are ultimately interested in the radiation field at some given finite
post-Newtonian precision like 3PN, and because only a finite number of multipole moments can contribute at
any finite order of approximation. With a finite number of multipoles in the linearized metric (26![]() ![]() ![]() ![]() ![]() ![]() |
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10 | The ![]() ![]() |
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11 | In this proof the coordinates are considered as dummy variables denoted ![]() ![]() |
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12 | Recall that in actual applications we need mostly the mass-type moment ![]() ![]() |
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13 | This function approaches the Dirac delta-function (hence its name) in the limit of large multipoles:
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14 | An alternative approach to the problem of radiation reaction, besides the matching procedure, is to work only within a post-Minkowskian iteration scheme (which does not expand the retardations): see, e.g., Ref. [69]. | |
15 | Notice that the normalization ![]() ![]() ![]() ![]() ![]() |
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16 | At the 3PN order (taking into account the tails of tails), we find that ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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17 | The computation of the third term in Equation (106![]() ![]() ![]() |
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18 | The function ![]() ![]() ![]() ![]() |
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19 | Equation (112![]() ![]() ![]() ![]() ![]() |
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20 | Actually, such a metric is valid up to 3.5PN order. | |
21 | It has been possible to “integrate directly” all the quartic contributions in the 3PN metric. See the terms composed of
![]() ![]() ![]() |
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22 | The function ![]() ![]() ![]() ![]() ![]() |
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23 | It was shown in Ref. [38![]() ![]() |
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24 | Note also that the harmonic-coordinates 3PN equations of motion as they have been obtained in Refs. [37![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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25 | One may wonder why the value of ![]() ![]() ![]() ![]() ![]() ![]() |
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26 | See some comments on this work in Ref. [84], pp. 168 - 169. | |
27 | The result for ![]() ![]() ![]() ![]() |
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28 | The work [34] provided also some new expressions for the multipole moments of an isolated post-Newtonian source, alternative to those given by Theorem 6, in the form of surface integrals extending on the outer part of the source’s near zone. | |
29 | We have ![]() ![]() ![]() ![]() ![]() ![]() |
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30 | When working at the level of the equations of motion (not considering the metric outside the world-lines), the effect of
shifts can be seen as being induced by a coordinate transformation of the bulk metric as in Ref. [38![]() |
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31 | Notice also the dependence upon ![]() ![]() ![]() |
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32 | Note that in the result published in Ref. [95![]() ![]() ![]() |
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33 | Actually, in the present computation we do not need the radiation-reaction 2.5PN term in these relations; we give it only for completeness. | |
34 | In this section we pose ![]() ![]() ![]() |
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35 | We are following the discussion in Ref. [24![]() |
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36 | Actually, the post-Newtonian series could be only asymptotic (hence divergent), but nevertheless it should give excellent results provided that the series is truncated near some optimal order of approximation. In this discussion we assume that the 3PN order is not too far from that optimum. | |
37 | When computing the gravitational-wave flux in Ref. [45![]() ![]() ![]() ![]() ![]() |
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38 | For circular orbits there is no difference at this order between ![]() ![]() ![]() ![]() |
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39 | All formulas incorporate the changes in some equations following the published Errata (2005) to the
works [16, 19, 45, 40, 4![]() |
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40 | Generalizing the flux formula (231![]() |
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41 | Notice the “strange” post-Newtonian order of this time variable: ![]() |
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42 | We neglect the non-linear memory (DC) term present in the Newtonian plus polarization ![]() ![]() |
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