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4.3 Lorentz contraction and multipole moments

In Equations (81View Equation, 82View Equation, 83View Equation), we have defined multipole moments of a star. The definition of those multipole moments are operational ones and are not necessarily equal to “intrinsic” multipole moments of the star. This is clear, for example, if we remember that a moving ball has spurious multipole moments due to Lorentz contraction.

We have adopted the generalized Fermi normal coordinates (GFC) [13Jump To The Next Citation Point] as the star’s reference coordinates to address this problem. A question specific to our formalism is whether the differences between the multipole moments defined in Equations (81View Equation, 82View Equation, 83View Equation) and the multipole moments in GFC give purely monopole terms. If so, we of course have to take into account such terms in the field to compute the equations of motion for two intrinsically spherical star binaries. This problem is addressed in the Appendix C of [91Jump To The Next Citation Point] and the differences are mainly attributed to the shape of the body zone. The body zone BA which is spherical in the near zone coordinates (NZC) is not spherical in the GFC mainly because of a kinematic effect (Lorentz contraction). To derive the 3 PN equations of motion, it turns out that it is sufficient to compute the difference in the STF quadrupole moment up to 1 PN order. The result is

δI⟨ij⟩≡ I⟨ij⟩ − I⟨ij⟩ A A,NZC ∮A,GFC = ε−61vk vl dS yl y⟨iyj⟩Λττ + 𝒪 (ε3) 2 A A ∂BA k A A A 3 = − ε24m-Av⟨Aivj⟩A + 𝒪 (ε3), (106 ) 5
where IijA,NZC ≡ IiAj. IiAj,GFC are the quadrupole moments defined in the generalized Fermi normal coordinates. Note that the difference ⟨ij⟩ δIA is expressed in a surface integral form.

As is obvious from Equation (106View Equation), this difference appears even if the companion star does not exist. We note that we could derive the 3 PN metric for an isolated star A moving at a constant velocity using our method explained in this section by simply letting the mass of the companion star be zero. Actually, the ⟨ij⟩ δIA above is a necessary term which makes the so-obtained 3 PN metric equal to the Schwarzschild metric boosted at the constant velocity ⃗vA in harmonic coordinates.

The reason why the influence of a body’s Lorentz contraction appears starting from 3 PN order is as follows. The body’s Lorentz contraction appears as an apparent deformation of the body, namely, apparent quadrupole moment as the leading order in the frame where the body is moving. The body’s (real) quadrupole affects the field from 2 PN order through (see Equation (78View Equation))

∑ 3I⟨ij⟩rirj hτqτuad = ε8 --A--5A-A-. (107 ) A 2rA
The radius of a compact body A is of order of its mass mA, and the Lorentz contraction deforms the body so that the radius of the body changes by the amount ε2mAv2A + 𝒪 (ε4). So the apparent quadrupole moments due to Lorentz contraction is of order of
I⟨Aij⟩apparent ∼ ε2m3Av2A. (108 )
This field is the 3 PN field and thus the Lorentz contraction of a body affects the equations of motion starting from 3 PN order.
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