Comparison

8.5 Comparison

By comparing Equation (181

) with the Blanchet and Faye 3 PN equations of motion [27

], we find the following relation:

where $this work m1⃗a1$ is the 3 PN acceleration given in Equation (181

), $BF (⃗a1 )λ= −1987∕3080$ is the Blanchet and Faye 3 PN acceleration with $λ = − 1987 ∕3080$ , and $m1 ⃗a1|δAln$ is given in Equation (175

) with $εRA$ replaced by $r′A$ for notational consistency with the Blanchet and Faye 3 PN equations of motion shown in [27

]. $m1 ⃗a1|δA,BF$ is an acceleration induced by the following dipole moments of the stars:

We can compute $m1ai |δ 1 A,BF$ by substituting $δi A,BF$ instead of $δi AΘ$ into Equation (166

). Thus, by choosing the dipole moments,

we have the 3 PN equations of motion in completely the same form as $(⃗aB1F )λ= −1987∕3080$ . In other words, our 3 PN equations of motion physically agree with $(⃗aBF1 )λ=−1987∕3080$ modulo the definition of the dipole moments (or equivalently, the coordinate transformation under the harmonic coordinates condition). In [93 ], we have shown some arguments that support this conclusion.

The value of $λ$ that we found, $λ = − 1987∕3080$ , is perfectly consistent with the relation (1 ) and the result of [54 ] ( $ωstatic = 0$ ).

Finally, let us discuss the ambiguity in the 3 PN equations of motion previously derived by Blanchet and Faye in [25 , 27 ]. In their formalism, Dirac delta distributions are used to achieve the point particle limit. The (Lorentz invariant generalized) Hadamard partie finie regularization has been extensively employed to regularize divergences caused by their use of a singular source. In fact, unless regularization is employed, divergences occur both in the evaluation of the 3 PN field where (Poisson) integrals diverge at the location of the stars and in their derivation of the equations of motion where a substitution of the metric into a geodesic equation causes divergences.

When we regularize some terms at a point, say, $ziA$ , where the terms are singular, using the Hadamard partie finie regularization, roughly speaking we take an angular average of the finite part of the terms in the neighborhood of the singular point. Then if there are logarithmic terms such as $ln (|⃗x − ⃗zA|∕r12)$ , we should take an angular average over some sphere centered on $i zA$ with a finite radius. The radius of the sphere is arbitrary but we do not ignore it because we should ensure the argument of the logarithms to be dimensionless.

The problem here is that there is a priori no reason to expect that the radius for each star introduced to regularize the field and another radius for that star introduced to regularize the geodesic equation coincide with each other. Thus the Blanchet and Faye 3 PN equations of motion have four arbitrary constants instead of two in our equations of motion. In our framework, we can see the origin of the number if we assume that we have defined a different body zone $B′ A$ in the derivation of the equations of motion from $BA$ used in the derivation of the 3 PN field. However, in reality, we have only one body zone for each star. In our formalism the field is expressed in terms of the four-momentum (and multipole moments) which are defined as volume integrals over the body zone. On the other hand our general form of the equations of motion has been derived based on the conservation law of the four-momentum, and thus we evaluate the surface integrals in the general form of the equations of motion over the boundary of the body zone.

In fact, [25 , 27 ] have shown that two of the four arbitrary constants can be removed by using a gauge freedom remaining in the harmonic gauge condition; the two places where the singular points exist are in some sense ambiguous. The remaining two turn out to appear as the ratios $ζA = r′A∕sA$ ( $A = 1,2$ ) where $r′A$ and $sA$ are the four regularization parameters (roughly speaking, the radii of $B ′ A$ and $BA$ in the terminology in the previous paragraph). Blanchet and Faye [25 , 27 ] then proved that assuming the equations of motion are polynomials of the two masses of the stars, those two ratios should satisfy $ln ζA = σ + λ (m1 + m2 )∕mA$ where $σ$ and $λ$ are pure numbers. Then they showed that in order for their equations of motion to admit conserved energy, then $σ = 159∕308$ , while no argument was found to fix $λ$ .

The above argument in turn means that the Blanchet and Faye 3 PN equations of motion do not give a conserved energy unless $′ B A$ is different from $BA$ . Damour, Jaranowski, and Schäfer [54 ] pointed out that there is an unsatisfactory feature in the generalized partie finie regularization which contradicts with the mathematical structure of general relativity. Indeed, by using dimensional regularization which is pointed out by them to be more satisfactory in this regard, [54 ] derived an unambigous ADM Hamiltonian in the ADM transeverse traceless gauge. Later, Blanchet et al. [22 ] used dimensional regularization and found that their new equations of motion physically agree with ours and admit a conserved energy.