We now turn attention to the gravitational self-force problem, for which we shall draw many parallels to the electromagnetic case just described. To begin with, consider the perturbative split defined in Eqn. (2.30); starting with a background geometrygµν, we seek an approximate solution gµν+pretµν to the Einstein field
equations, or, schematically,
G(g+pret) = 8πT+O(p2). (2.46)
Here, the perturbation is thought to arise from the motion of some small massm, which at zeroth order in this scheme moves along a geodesic ofgµν.
We now define the regular fieldpR µν via
pretµν =pSµν+pRµν. (2.47)
This is known as the Detweiler-Whiting decomposition [77]. It can then be shown that, at this order of approximation,mmoves along a geodesic not ofgµν, but ofgµν+pRµν.
We may go further: the purelydissipative part of the field is given by
pdissµν = 1 2 p ret µν−padvµν , (2.48)
whereas the conservative parts of the field are given by
pconµν =pRµν−pdissµν =1 2 p ret µν +padvµν −pSµν. (2.49) Clearly, pR µν =pconµν +pdissµν . (2.50)
Though we’ve expressed some preference for thinking of the radiation-reaction effect as being geodesic motion in a perturbed background, there does indeed arise a force on the particle when viewed from an initial background geodesic. From this point of view, the resulting acceleration off the background worldline is
uβ∇βuα=− gαβ+uαuβuγuδ ∇γpRδβ− 1 2∇βp R γδ . (2.51)
the gravitational self-force was obtained by Mino, Sasaki, and Tanaka [158], and separately by Quinn and Wald [185]. It is now known as the MiSaTaQuWa equation, and there is general consensus about its validity for point particle motion in the approximation that spin and higher multipole moments may be neglected. However, numerical calculation of the self-force is nontrivial. The Detweiler-Whiting field is defined in terms of a subtraction of two singular fields, the full retarded metric perturbation and the singular piece of that perturbation. Of course, it is not possible to simply compute the two divergent pieces separately and then subtract them to obtain the regular field. Barack and Ori [24] proposed the now widely-used mode-sum regularization scheme to circumvent this difficulty. The retarded field is decomposed into multipole modes, and the singular blow-up extracted by considering the large `-mode behavior. Then, the singular field can be subtracted from the retarded field in a mode-by-mode by fashion in order to reconstruct the regular part of the metric perturbation. We shall have much more to say about these matters in Ch. 4, in which we detail a method (and its results) for computing the self-force for a point mass in eccentric motion about a Schwarzschild black hole.
CHAPTER 3: PN theory: background and development Section 3.1: Assumptions and applicability
The post-Newtonian approximation has a rich history, dating back to just after Einstein’s first publication of the full theory, when Einstein himself developed the quadrupole moment wave generation formalism [88], later refined by Landau and Lifshitz [145]. Even after nearly a century of effort, the current state-of-the-art for the PN equations of motion (EOM) is 4PN (i.e. fourth-order in a PN expansion). The 1PN EOM were obtained by Lorentz and Droste [149], Einstein, Infeld, and Hoffman [2], and Fock [95]. At 2PN order, the EOM were worked out by Ohta, Okamura, Kimura, and Hiida [164, 166, 165], and the 2.5PN EOM were determined by Damour and Deruelle [63, 67, 66, 62] and Itoh, Futamase, and Asada[129]. Non- conservative–ordissipative–effects, related to the emission of gravitational radiation, enter the approximation at 2.5PN. The 3PN EOM were derived by Jaranowski, Sch¨afer, and Damour [133, 134, 132, 68, 69], by Blanchet and Faye [42, 41, 43, 44], by Futamase and Itoh [106], and by Foffa and Sturani [97]. To obtain the 3.5PN EOM, the 1PN correction to the radiation-reaction force is needed. These were obtained by Iyer and Will [130, 131] for point-mass binaries. The 2PN correction was then obtained by Gopakumar, Iyer, and Iyer [113]. Gravitational wave tail effects arise at 4PN order, and modify the radiation damping force as a 1.5PN correction [33, 98, 107]. Partial progress was made on the 4PN EOM by Jaranowski and Sch¨afer [135, 136, 137] and by Foffa and Sturani [96], with the complete dynamics being worked out by Damour, Jaranowski, and Sch¨afer [70]. Work done by Bini, Damour, and Geralico [31], as well as work by Hopper, Kavanagh, and Ottewill [125], have confirmed these results in the test body limit.
In addition to the problem of the equations of motion, there is the matter of post-Newtonian gravitational wave generation, in which the energy flux and gravitational waveform are related to the nature and the motion of the gravitating source. Wagoner and Will were the first to go beyond the original quadrupole moment formalism [213], and their calculations were redone with greater rigor by Blanchet and Sch¨afer [47]. The binary inspiral waveform and orbital parameter evolution was computed to 2PN order by Blanchet, Damour, Iyer, and Gopakumar [40, 112], as well as Will and Wiseman [228, 227]. Hereditary terms, which depend on the entire past history of the source, appear at 1.5PN in the radiative field. The 1.5PN hereditary contribution was arrived at by Wiseman [229], Poisson [173], and Blanchet and Sch¨afer [37], while the 2.5PN and 3.5PN tail effects, and the 3PN “tail of tails,” were worked out by Blanchet [38, 39]. At 3PN, there are
instantaneous (non-tail) contributions from relativistic corrections to the source moments, which have been developed by Blanchet, Iyer, Joguet, and Faye [36, 46, 45]. The extension of these results to the eccentric case was accomplished by Arun, Blanchet, Iyer, and Qusailah [14, 13, 15],
Comparisons between PN and BHP theory calculations were made by Poisson [173], Tagoshi and Naka- mura [201], and by Sasaki, Tagoshi, and Tanaka [194, 202, 204]. More recent work by Fujita [103] and Shah, Friedman, Whiting, and Johnson-McDaniel [196, 198, 139] made use of advances in BHP theory to compute PN corrections to extremely high order (up to 22PN). This work has been done primarily in the context of circular orbits. In 2015, Sago and Fujita [191] computed the rates of change of the orbital parameters under radiation reaction for eccentric orbits to 4PN order in the test mass limit.
In this chapter, we attempt to give a flavor for the PN approximation in GR, as well as the related, but “upstream” (i.e. more fundamental) post-Minkowskian (PM) expansion. We largely follow [177], [32], and [225] in our development of the following material.
When describing an approximation as being “post-Newtonian,” we sometimes interchangeably say that we are using a “slow-motion” assumption. However, we can be a bit more precise: the PN approximation is valid in the near-zone of a slowly-moving, weakly self-gravitating source. Given a source stress-energy tensor Tαβ and a source Newtonian potentialU, we define the small PN parameteras
≡max ( T 0i T00 , T ij T00 1/2 , cU2 1/2) 1. (3.1)
So again, where we might often say thatv/c1 in the PN regime, it is more accurate to refer to Eqn. (3.1). Note also that for this chapter, we temporarily depart from our unit convention which setsc=G= 1, since we use these constants as book-keeping parameters to keep track of the size of expansion terms.
In the “classic formulation” of PN theory, one starts with the standard Einstein equation, and then expands the tensor components in powers of. However, one eventually runs into trouble with this approach. The PN approximation is not a priori valid everywhere, and in particular it cannot take into account boundary conditions at infinity, and therefore the radiation reaction effect. So, as we shall see, the radiation problem must be handled in this domain by re-expanding the PM expansion in a PN fashion, and then performing a matching between the PN source and the far-zone PM metric. Therefore, we first describe the PM approximation, before moving on to the PN field equations and the problem of gravitational radiation from PN sources. These topics are very dense, and so we attempt mainly to give a high-level description and a summary of the results.