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1.1 Gravitational wave detection and post-Newtonian approximation

The motion and associated emission of gravitational waves (GW) of self-gravitating systems have been a main research interest in general relativity. The problem is complicated conceptually as well as mathematically because of the nonlinearity of the Einstein equations. There is no hope in any foreseeable future to have exact solutions describing motions of arbitrarily shaped, massive bodies, so we have to adopt some sort of approximation schemes for solving the Einstein equations to study such problems. In the past years many types of approximation schemes have been developed depending on the nature of the system under consideration. Here we shall focus on a particular scheme called the post-Newtonian (PN) approximation. There are many systems in astrophysics where Newtonian gravity is dominant, but general relativistic gravity plays also an important role in their evolution. For such systems it would be nice to have an approximation scheme which gives a Newtonian description in the lowest order and general relativistic effects as higher order perturbations. The post-Newtonian approximation is perfectly suited for this purpose. Historically Einstein computed first the post-Newtonian effects, e.g. the precession of the perihelion [72]. Studies of the post-Newtonian approximation were made by Lorentz and Droste [117Jump To The Next Citation Point], Einstein, Infeld, and Hoffmann [73Jump To The Next Citation Point], Fock [77Jump To The Next Citation Point], Plebansky and Bazanski [131], and Chandrasekhar and associates [404142Jump To The Next Citation Point43].

Now it is widely known that the post-Newtonian approximation is important in analyzing a number of relativistic problems, such as the equations of motion of binary pulsars [34617489Jump To The Next Citation Point], solar-system tests of general relativity [159Jump To The Next Citation Point160Jump To The Next Citation Point161Jump To The Next Citation Point164Jump To The Next Citation Point], and gravitational radiation reaction [3942]. Any approximation scheme necessitates one or several small parameters characterizing the nature of the system under consideration. A typical parameter which most of the schemes adopt is the magnitude of the metric deviation from a certain background metric. In particular if the background is Minkowski spacetime and there is no other parameter, the scheme is sometimes called the post-Minkowskian approximation in the sense that the constructed spacetime reduces to Minkowski spacetime in the limit that the parameter tends to zero. This limit is called the weak field limit. In the case of the post-Newtonian approximation the background spacetime is also Minkowski spacetime, but there is another small parameter, that is, the typical velocity of the system divided by the speed of light. We introduce a non-dimensional parameter ε to express the “slowness” of the system. These two parameters (the deviation from the flat metric and the velocity) have to have a certain relation in the following sense. As the gravitational field gets weaker, all velocities and forces characteristic of the material systems become smaller, in order to permit the weakening of gravity to remain an important effect in the system’s dynamics. For example in the case of a binary system, the typical velocity would be the orbital velocity v ∕c ∼ ε and the deviation from the flat metric would be the Newtonian potential, say Φ. Then these are related by Φ∕c2 ∼ v2∕c2 ∼ ε2 which guarantees that the system is bounded by its own gravity.

In the post-Newtonian approximation, the equations of general relativity take the form of Newton’s equations in an appropriate limit as ε → 0. Such a limit is called the Newtonian limit and it will be the basis of constructing the post-Newtonian approximation. However, the limit is not in any sense trivial since it may be thought of as two limits tied together as just described. It is also worth noting that the Newtonian limit cannot be uniform everywhere for all time. For example any compact binary system, no matter how weak the gravity between its components and slow its orbital motion is, will eventually spiral together due to backreaction from the emission of gravitational waves. As the result the effects of its Newtonian gravity will be swamped by those of its gravitational waves. This will mean that higher order effects of the post-Newtonian approximation eventually dominate the lowest order Newtonian dynamics and thus if the post-Newtonian approximation is not carefully constructed, this effect can lead to many formal problems, such as divergent integrals [71Jump To The Next Citation Point]. It has been shown that such divergences may be avoided by carefully defining the Newtonian limit [79Jump To The Next Citation Point]. Moreover, the use of such a limit provides us a strong indication that the post-Newtonian hierarchy is an asymptotic approximation to general relativity [82Jump To The Next Citation Point]. Therefore we shall first discuss in this paper the Newtonian limit and how to construct the post-Newtonian hierarchy before attacking practical problems in later sections.

Before going into the details, we mention the reason for the growth of interest in the post-Newtonian approximation in recent years. Certainly the discovery of the binary neutron star system PSR 1913+16 was a strong reason to have renewed interest in the post-Newtonian approximation, since it is the first system found in which general relativistic gravity plays a fundamental role in its evolution [89]. Particularly the indirect discovery of gravitational waves by the observation of the period shortening led to many fruitful studies of the equations of motion with gravitational radiation emission in 1980s (see [47Jump To The Next Citation Point48Jump To The Next Citation Point49Jump To The Next Citation Point] for a review). The effect of radiation reaction appears in the form of a potential force at the order of 5 ε higher than the Newtonian force in the equations of motion. Ehlers and colleagues [71] critically discussed the foundation of the so-called quadrupole formula for the radiation reaction (see also the introduction of [129Jump To The Next Citation Point]). There have been various attempts to show the validity of the quadrupole formula [5Jump To The Next Citation Point47Jump To The Next Citation Point79Jump To The Next Citation Point90104Jump To The Next Citation Point105Jump To The Next Citation Point106Jump To The Next Citation Point107135Jump To The Next Citation Point138Jump To The Next Citation Point139155157156]. Namely, Damour [46Jump To The Next Citation Point] proved the formula for compact binaries with the help of the “dominant Schwarzschild condition” [47Jump To The Next Citation Point]. Blanchet and Damour [21] proved it for general fluid systems.

At that time, however, no serious attempts with direct detection of gravitational waves in mind had been made for the study of higher order effects in the equations of motion. The situation changed gradually in the late 1980s because of the increasing expectation of a direct detection of gravitational waves by kilometer-size interferometric gravitational wave detectors, such as LIGO [1Jump To The Next Citation Point116], VIRGO [35153], GEO [6283], and TAMA [114149]. Coalescing binary neutron stars are the most promising candidates of sources of gravitational waves for such detectors. The reasons are that (i) we expect, say, the (initial) LIGO to detect the signal of coalescence of binary neutron stars about once per year to once per hundreds of years [38103102], and (ii) the waveform from coalescing binaries can be predicted with high accuracy compared to other sources [1151161Jump To The Next Citation Point]. Information carried by gravitational waves tells us not only various physical parameters of neutron stars [45], but also the cosmological parameters [75119141142158] if and only if we can make a detailed comparison between the observed signal with theoretical predictions during the epoch of the so-called inspiraling phase where the orbital separation is much larger than the radius of the component stars [44Jump To The Next Citation Point]. This is the place where the post-Newtonian approximation may be applied to make theoretical templates for gravitational waves. The problem is that in order to make any meaningful comparison between theory and observation we need to know the detailed waveforms generated by the motion up to, say, 4 PN order which is of order 8 ε higher than the Newtonian order [2310147]. This request from gravitational wave astronomy forces us to construct higher order post-Newtonian equations of motion and waveform templates.

Replying to this request, there have been various works studying the equations of motion for a compact binary system and developing higher order post-Newtonian gravitational waveform templates for such a system. The most systematic among those works that have succeeded in achieving higher order iteration are the ones by Blanchet, Damour, and Iyer who have developed a scheme to calculate the waveform at a higher order, where the post-Minkowskian approximation is used to construct the external field and the post-Newtonian approximation is used to construct the field near the material source. They and their collaborators have obtained the waveform up to 3.5 PN order which is of order 7 ε higher than the lowest quadrupole wave [2324293132] by using the equations of motion up to that order [22Jump To The Next Citation Point91Jump To The Next Citation Point93Jump To The Next Citation Point111Jump To The Next Citation Point123Jump To The Next Citation Point130Jump To The Next Citation Point]. The 3.5 PN waveform includes tail terms which manifest nonlinearity of general relativity. Blanchet and Schäfer [33] have investigated a spectral (Fourier) decomposition of the tail and computed the contribution of the tail to the gravitational wave luminosity emitted by a binary system having a general eccentric orbit. Asada and Futamase [12Jump To The Next Citation Point] showed that the dominant part of the tail term originates from the phase shift of the wave due to the Coulomb part of the gravitational field.

In this paper, we mainly discuss the foundation of the Newtonian limit and the post-Newtonian equations of motion for relativistic compact binaries in an inspiralling phase. In the next Section 1.2 we briefly give an historical introduction on the latter topic.

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