List Of Footnotes

List of Footnotes

1		Jaranowski and Schäfer [101 ] have shown that the 3PN ADM Hamiltonian with $ωstatic = 0$ corresponds to the Brill–Lindquist initial data.
2		Henceforth we call the $(τ,xi)$ coordinates the near zone coordinates.
3		Historically, it was said that in harmonic coordinates there was a “breakdown” at 3 PN order (see e.g. [106 ]). The divergences at 3 PN order may be gauge effects.
4		As was defined in the introduction, $ε$ represents the smallness of the typical velocity of the system (divided by the velocity of light).
5		Our formalism is sometimes called an extended body approach, since we set $ε$ to some finite value when we apply our method to real binary systems.
6		We assume that the star is pressure supported. This means that the pressure $P$ scales as $ε−4$ . If we consider polytropes, $P(ρ) = K ργ$ for instance, the polytropic constant $K$ must scale as $ε4(γ− 1)$ . No such a sequence of solutions parametrized by $ε$ seems to have been obtained so far. We simply assume the existence of such a sequence.
7		It is just for simplicity to call $PμA$ a four-momemtum. $P μA$ is not defined (general, nor even Lorentz) covariantly nor as a volume integral of tensor density of weight $− 1$ . The integrand $Λ μν$ is a tensor density of weight $− 2$ . In Section 6.1, we will find that $P μA$ equals $√ − gmAu μA$ in a certain sense. $PτA$ is an energy of the star (modulo the contribution from the integrand $χμναβ,αβ$ ) if the star considered were isolated, the space-time is asymptotically flat, and finally, if we integrate over a spacelike hypersurface, rather than integrate only over the body zone of the star [115 ].
8		Notice that when solving a Poisson equation $Δg(⃗x) = f(⃗x)$ , a particular solution suffices for our purpose. By virtue of the surface integral term in Equation (101 ), it is not necessary to care for a homogeneous solution of the Poisson equation.
9		Every spatial three vector is treated as a Euclidean flat space three-vector. In the Cartesian coordinates the norm is evaluated with a Kronecker delta.
10		A super-potential here is a particular solution of a Poisson equation whose source term is non-compact.
11		In fact, we need the 2.5 PN field and a part of the 3 PN field ( $8h τ[i,j]$ ) to evaluate the 3 PN mass-energy relation. This does not mean that the Hadamard partie finie regularization is sufficient to derive the 3 PN equations of motion. Indeed, when a Dirac delta distribution is adopted to achieve the point particle limit instead of the strong field point particle limit, one needs another regularization scheme such as the dimensional regularization to derive the 3 PN equations of motion in an unambiguous form [22 ].
12		Defining the $Θ$ parts of $PμA$ and $DiA$ in the same way as for $QiA$ , it turns out that $PiAχ = PτAχviA +QiAχ + ε2d2DiAχ∕dτ2$ is a trivial identity [91 ]. Note that $Pμ Aχ$ and $Di Aχ$ can be converted into surface integral forms and thus can be evaluated explicitly.
13		It is natural to expect that $⃗zA$ is in star $A$ unless the star is, say, crescent-shaped. One may not choose the representative point of the sun to be in the earth.
14		Unlike in our case, their coordinate transformation does not remove the logarithmic dependence of their free parameters completely. The remaining logarithmic dependence was used to make their equations of motion conservative.
15		In [25 , 26 , 27 ], to treat Poisson integrals which could not be computed in a closed form, Blanchet and Faye first substitute the Poisson integrals by a geodesic equation and evaluate the partie finie of the integral, and then compute the remaining volume integrals.
16		On the other hand, the equations of motion (171 ) themselves consist of one term at the Newtonian order, $8$ terms at 1 PN order, $31$ terms at 2 PN order, $6$ terms at 2.5 PN order, and about $100$ terms at 3 PN order. The 4 PN equations of motion may have about $1000$ terms.
17		Notice that if we recover the expansion parameter $ε$ , $z < ε$ for the field point in the far zone.
18		Here we call the coupling between temporal derivatives of the quadrupole moment and the orbital motion the tidal-orbit coupling
19		We could evaluate the $χ$ part of the spin by virtue of its antisymmetric nature, and the result is $ij 3 M Aχ = 𝒪 (ε )$ . Thus $ij M Aχ$ does not affect the 3 PN equations of motion for two point masses up to 3 PN order even if we treat $ij MA χ$ separately from $M iAjΘ$ .