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4.4 Equations of motion and gravitational waveform

Among the results of these approaches are formulae for the equations of motion and gravitational waveform of binary systems of compact objects, carried out to high orders in a PN expansion. Here we shall only state the key formulae that will be needed for this review. For example, the relative two-body equation of motion has the form
dv- m- a = dt = r2 {− ˆn + A1PN + A2PN + A2.5PN + A3PN + A3.5PN + ...} , (65 )
where m = m1 + m2 is the total mass, r = |x1 − x2 |, v = v1 − v2, and ˆn = (x1 − x2)∕r. The notation AnPN indicates that the term is n 𝒪 (ε ) relative to the Newtonian term −nˆ. Explicit and unambiguous formulae for non-spinning bodies through 3.5PN order have been calculated by various authors (see [34Jump To The Next Citation Point] for a review). Here we quote only the first PN corrections and the leading radiation-reaction terms at 2.5PN order:
{ m 3 } A1PN = (4 + 2η)-- − (1 + 3η)v2 + --η˙r2 ˆn + (4 − 2η)˙rv, (66 ) r 2 8 m {( 2 m ) ( 2 m ) } A2.5PN = − 15-η-r 9v + 17-r ˙rˆn − 3v + 9r- v , (67 )
where η = m1m2 ∕(m1 + m2 )2. These terms are sufficient to analyze the orbit and evolution of the binary pulsar (see Section 5.1). For example, the 1PN terms are responsible for the periastron advance of an eccentric orbit, given by 2 ω˙ = 6πfbm ∕a (1 − e), where a and e are the semi-major axis and eccentricity of the orbit, respectively, and fb is the orbital frequency, given to the needed order by Kepler’s third law 2πfb = (m ∕a3)1∕2.

Another product is a formula for the gravitational field far from the system, written schematically in the form

2m { ij ij ij ij ij } hij = ---- Qij + Q 0.5PN + Q 1PN + Q 1.5PN + Q 2PN + Q 2.5PN + ... , (68 ) R
where R is the distance from the source, and the variables are to be evaluated at retarded time t – R. The leading term is the so-called quadrupole formula
ij 2-¨ij h (t,x) = R I (t − R ), (69 )
where ij I is the quadrupole moment of the source, and overdots denote time derivatives. For a binary system this leads to
( ) ij ij m ˆniˆnj Q = 2η v v − ------- . (70 ) r
For binary systems, explicit formulae for the waveform through 2PN order have been derived (see [40Jump To The Next Citation Point] for a ready-to-use presentation of the waveform for circular orbits; see [34Jump To The Next Citation Point] for a full review).

Given the gravitational waveform, one can compute the rate at which energy is carried off by the radiation (schematically ∫ h˙˙hd Ω, the gravitational analog of the Poynting flux). The lowest-order quadrupole formula leads to the gravitational wave energy flux

8 2m4 ( 2 2) E˙ = ---η --4 12v − 11˙r . (71 ) 15 r
This has been extended to 3.5PN order beyond the quadrupole formula (see [34Jump To The Next Citation Point] for a review). Formulae for fluxes of angular and linear momentum can also be derived. The 2.5PN radiation-reaction terms in the equation of motion (65View Equation) result in a damping of the orbital energy that precisely balances the energy flux (71View Equation) determined from the waveform. Averaged over one orbit, this results in a rate of increase of the binary’s orbital frequency given by
˙fb = 192π-f2b(2π ℳfb )5∕3F (e), 5 ( ) (72 ) 2− 7∕2 73- 2 37-4 F (e) = (1 − e) 1 + 24 e + 96e ,
where ℳ is the so-called “chirp” mass, given by ℳ = η3∕5m. Notice that by making precise measurements of the phase Φ (t) = 2 π∫ tf(t′) dt′ of either the orbit or the gravitational waves (for which f = 2fb for the dominant component) as a function of the frequency, one in effect measures the “chirp” mass of the system.

These formalisms have also been generalized to include the leading effects of spin-orbit and spin-spin coupling between the bodies [145144289].

Another approach to gravitational radiation is applicable to the special limit in which one mass is much smaller than the other. This is the method of black hole perturbation theory. One begins with an exact background spacetime of a black hole, either the non-rotating Schwarzschild or the rotating Kerr solution, and perturbs it according to gμν = g(0μ)ν + hμν. The particle moves on a geodesic of the background spacetime, and a suitably defined source stress-energy tensor for the particle acts as a source for the gravitational perturbation and wave field hμν. This method provides numerical results that are exact in v, as well as analytical results expressed as series in powers of v, both for non-rotating and for rotating black holes. For non-rotating holes, the analytical expansions have been carried to 5.5PN order, or ε5.5 beyond the quadrupole approximation. All results of black hole perturbation agree precisely with the m → 0 1 limit of the PN results, up to the highest PN order where they can be compared (for reviews see [188235]).

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