Go to previous page Go up Go to next page

4.8 Body zone boundary dependent terms

As explained in Section 4.5 we discard the body zone boundary dependent terms in the field, since we expect that they cancel out between the body zone contribution and the N∕B contribution. Before moving on to the higher order calculations, however, it is instructive to see that such a cancellation really occurs in the field and consequently the equations of motion up to 0.5 PN order.

First, we show the independence of the body zone radius RA in the 0.5 PN field. Returning back to the derivation of the 1 PN h ττ with care for the RA dependence (see Equation (128View Equation)), we get

∑ Pτ ∫ Λ ττ(τ,yk) ∂2 ∑ hττ = 4ε4 -A-+ 4ε6 d3y 6-N--------+ 2ε6---2 PAτrA + 𝒪 (ε7) A=1,2 rA C∕B |⃗x − ⃗y| ∂τ A=1,2 ∑ ( 2 ) = 4ε4 1-- P τ− ε2 7m-A- + 𝒪(ε6). (133 ) A=1,2 rA A 2εRA
Now we split PτA as PAτ= ¯PτA + P˜τA where P¯τA does not depend on RA while ˜PτA does. In order to evaluate the RA dependent part of PAτ, P˜τA, we use the fact that the integral of Λ τNτ over the near zone does not depend on the size of the body zone. (Notice that Pτ A is defined as the volume integral of ττ Λ N over BA.) In fact, from the relevant expression of the pseudotensor, we obtain at the lowest order
∫ ∫ 2 3 6 ττ 2 3 ττ ε d α ε 6ΛN = ε d y6 ΛN N ∕B N ∕B 2 27-∑ m-A- (134 ) = − ε 2 εRA A=1,2 + (terms independent of RA, or terms having positive power (s) of RA ).
Hence we find ˜Pτ = ε27m2 ∕(2εR ) + 𝒪 (ε2). A A A The above equation shows us that the 0.5 PN field is independent of RA and fully expressed by the RA independent energy ¯τ PA, or mass up to 0.5 PN order.

In the similar manner we split PAi as P iA = ¯PAi+ P˜iA and obtain P˜iA = ε211m2AviA∕(3εRA ) + 𝒪(ε2) from the definition of P i A. Thus from the fact that Qi = ε2m2 vi∕(6εRA ) + 𝒪 (ε2) A A A, we find that the 0.5 PN momentum-velocity relation does not depend on R A: ¯P i= ¯P τvi + 𝒪 (ε2) A A A. Finally evaluating the surface integrals in the general form of equations of motion using the “renormalized” (barred) moments, we find that the equations of motion are independent of RA as was expected.

As remarked in Section 4.5 from now on we use the same symbol for the renormalized moments as for the “bare” moments henceforth as before for notational simplicity.

View Image

Figure 4: Flowchart of the post-Newtonian iteration.

  Go to previous page Go up Go to next page