For a more detailed description, we refer the reader to one of the many reviews that have been written on the subject; see e. In both cases, multipolar expansions are performed. The two solutions, in the near and in the wave zone, are then matched. These approaches yield the equations of motion of the bodies, i. Here, E which depends on terms of integer PN orders can be considered as the energy of the system , and depending on terms of half-integer PN orders is the emitted GW flux. The lowest PN order in the GW flux is given by the quadrupole formula [ ] see also [ ] , where Q ab is the traceless quadrupole moment of the source.
The leading term in C is then of 2. Once the energy and the GW flux are known with this accuracy, the gravitational waveform can be determined, in terms of them, at n -PN order. Presently, PN schemes determine the motion of a compact binary, and the emitted gravitational waveform, up to 3. It should also be remarked that the state-of-the-art PN waveforms have been compared with those obtained with NR simulations, showing a remarkable agreement in the inspiral phase i. An alternative to the schemes discussed above is the ADM-Hamiltonian approach [ ], in which using the ADM formulation of GR, the source is described as a canonical system in terms of its Hamiltonian.
This framework has been extended to spinning binaries see [ ] and references therein.
Recently, an alternative way to compute the Hamiltonian of a post-Newtonian source has been developed, the effective field theory approach [ , , , ], in which techniques originally derived in the framework of quantum field theory are employed. This approach was also extended to spinning binaries [ , ]. ADM-Hamiltonian and effective field theory are probably the most promising approaches to extend the accuracy of PN computations for spinning binaries. The effective one-body EOB approach developed at the end of the last century [ ] and recently improved [ , ] see, e. This approach maps the dynamics of the two compact objects into the dynamics of a single test particle in a deformed Kerr spacetime.
It is a canonical approach, so the Hamiltonian of the system is computed, but the radiative part of the dynamics is also described. After this calibration, the waveforms reproduce with good accuracy those obtained in NR simulations see, e. In the same period, a different approach has been proposed to extend PN templates to the merger phase, matching PN waveforms describing the inspiral phase, with NR waveforms describing the merger [ 17 , ]. To conclude this section, we mention that PN schemes originally treated compact objects as point-like, described by delta functions in the stress-energy tensor, and employing suitable regularization procedures.
This is appropriate for BHs, and, as a first approximation, for NSs, too. Indeed, finite size effects are formally of 5-PN order see, e. However, their contribution can be larger than what a naive counting of PN orders may suggest [ ]. PN schemes are also powerful tools to study the nature of the gravitational interaction, i. They have been applied either to build general parametrizations, or to determine observable signatures of specific theories two kinds of approaches that have been dubbed top-down and bottom-up , respectively [ ].
Let us discuss top-down approaches first. Nearly fifty years ago, Will and Nordtvedt developed the PPN formalism [ , ], in which the PN metric of an N -body system is extended to a more general form, depending on a set of parameters describing possible deviations from GR. This approach which is an extension of a similar approach by Eddington [ ] facilitates tests of the weak-field regime of GR. It is particularly well suited to perform tests in the solar system.
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All solar-system tests can be expressed in terms of constraints on the PPN parameters, which translates into constraints on alternative theories of gravity. For instance, the measurement of the Shapiro time-delay from the Cassini spacecraft [ 99 ] yields the strongest bound on one of the PPN parameters; this bound determines the strongest constraint to date on many modifications of GR, such as Brans-Dicke theory.
More recently, a different parametrized extension of the PN formalism has been proposed which, instead of the PN metric, expands the gravitational waveform emitted by a compact binary inspiral in a set of parameters describing deviations from GR [ , ]. As mentioned above, PN approaches have also been applied bottom-up , i. For instance, the motion of binary pulsars has been studied, using PN schemes, in specific alternative theories of gravity, such as scalar-tensor theories [ ].
The most promising observational quantity to look for evidence of GR deviations is probably the gravitational waveform emitted in compact binary inspirals, as computed using PN approaches. In the case of theories with additional fundamental fields, the leading effect is the increase in the emitted gravitational flux arising from the additional degrees of freedom.
This increase typically induces a faster inspiral, which affects the phase of the gravitational waveform see, e. For instance, in the case of scalar-tensor theories a dipolar component of the radiation can appear [ ]. For further details, we refer the interested reader to [ ] and references therein. The post-Newtonian approach has mainly been used to study the relativistic two-body problem, i.
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The first computation of this kind, at leading order, was done by Peters and Mathews for generic eccentric orbits [ , ]. It took about thirty years to understand how to extend this computation at higher PN orders, consistently modeling the motion and the gravitational emission of a compact binary [ , ].
The state-of-the-art computations give the gravitational waveform emitted by a compact binary system, up to 3.
An alternative approach, based on the computation of the Hamiltonian [ ], is currently being extended to higher PN orders [ , , ]; however, in this approach the gravitational waveform is computed with less accuracy than the motion of the binary. PN and EOB approaches have also been extended to include the effects of tidal deformation of NSs [ , , , ]. PN approaches have been extended to test GR against alternative theories of gravity. Some of these extensions are based on a parametrization of specific quantities, describing possible deviations from GR. This is the case in the PPN approach [ , ], most suitable for solar-system tests see [ , ] for extensive reviews on the subject , and in the parametrized post-Einsteinian approach [ , ], most suitable for the analysis of data from GW detectors.
Other extensions, instead, start from specific alternative theories and compute — using PN schemes — their observational consequences. In particular, the motion of compact binaries and the corresponding gravitational radiation have been extensively studied in scalar-tensor theories [ , , 30 ]. The PN expansion is less successful at describing strong-field, relativistic phenomena. Different tools have been devised to include this regime and one of the most successful schemes consists of describing the spacetime as a small deviation from a known exact solution. In this approach, the spacetime is assumed to be, at any instant, a small deviation from the background geometry, which, in the cases mentioned above, is described by the Schwarzschild or the Kerr solution here denoted by.
The deformed spacetime metric can then be decomposed as. The simple expansion 15 implies a deeper geometrical construction see, e.
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Therefore, in the spacetime perturbation approach it is the spacetime manifold itself to be perturbed and expanded. However, once the perturbations are defined and the gauge choice, i. In particular, the linearized Einstein equations can be considered as linear equations on the background spacetime, and all the tools to solve linear differential equations on a curved manifold can be applied.
This was first addressed by Regge and Wheeler in their seminal paper [ ], where they showed that in the case of a Schwarzschild background, the metric perturbations can be expanded in tensor spherical harmonics [ ], in terms of a set of perturbation functions which only depend on the coordinates t and r.
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They also noted that the terms of this expansion belong to two classes even and odd perturbations, sometimes also called polar and axial , with different behaviour under parity transformations i. The linearized Einstein equations, expanded in tensor harmonics, yield the dynamical equations for the perturbation functions. Furthermore, perturbations corresponding to different harmonic components or different parities decouple due to the fact that the background is spherically symmetric. After a Fourier transformation in time, the dynamical equations reduce to ordinary differential equations in r.
Regge and Wheeler worked out the equations for axial perturbations of Schwarzschild BHs; later on, Zerilli derived the equations for polar perturbations [ ]. It turns out to be possible to define a specific combination of the axial perturbation functions , and a combination of the polar perturbation functions , K lm which describe completely the propagation of GWs.
This approach was soon extended to general spherically symmetric BH backgrounds and a gauge-invariant formulation in terms of specific combinations of the perturbation functions that remain unchanged under perturbative coordinate transformations [ , ].
We further note that the Weyl and Riemann tensors are identical in vacuum. In this framework, the perturbation equations reduce to a wave equation for the perturbation of which is called the Teukolsky equation [ ]. The main advantage of the Bardeen-Press-Teukolsky approach is that it is possible to separate the angular dependence of perturbations of the Kerr background, even though such background is not spherically symmetric.
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Its main drawback is that it is very difficult to extend it beyond its original setup, i. The tensor harmonic approach is much more flexible. In particular, spacetime perturbation theory with tensor harmonic decomposition has been extended to spherically symmetric stars [ , , , ] the extension to rotating stars is much more problematic [ ]. As we discuss in Section 5. It is not clear whether such generalizations are possible with the Bardeen-Press-Teukolsky approach.
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The sources describe the objects that excite the spacetime perturbations, and can arise either directly from a non-vanishing stress-energy tensor or by imposing suitable initial conditions on the spacetime. In the point particle limit the source term is a nontrivial perturbing stress-tensor, which describes for instance the infall of a small object along generic geodesics. Thus, the spacetime perturbation approach is in principle able to describe qualitatively, if not quantitatively, highly dynamic BHs under general conditions.
The original approach treats the small test particle moving along a geodesic of the background spacetime.