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) and change the integral region to a 
spatial hypersurface,
Note that the above integral depends on the arbitrary length 
in general. The cancellation between the
dependent terms in the far zone contribution and those in the near zone contribution was shown
by [129
] through all the post-Newtonian order. In the following, we shall omit the terms which have
negative powers of 
(
). In other words, we simply let 
whenever it gives a
convergent result. On the other hand, we shall retain terms having positive powers of 
(
), and logarithmic terms (
) to confirm that the final result, in the end of
calculation, is independent of 
(and for logarithmic terms to keep the arguments of logarithm
non-dimensional).
Second we split the integral into two parts: a contribution from the body zone 
, and from
elsewhere, 
. Schematically we evaluate the following two types of integrals (we omit indices of the
field),
. We shall deal with these two contributions successively.
As for the body zone contribution, we make a multipole expansion using the scaling of the integrand,
i.e. 
in the body zone. For example, the 
part in Equation (76), 
, gives
denotes a symmetric and tracefree (STF) operation on the indices in the brackets.
See [129, 150, 165] for some useful formulas of STF tensors. Also we define 
. To derive the
3 PN equations of motion, 
up to 
and 
as well as 
up to 
are
required.
In the above equations we define the multipole moments as
where we introduced a collective multi-index
and 
. Then 
, 
,
and 
. We simply call 
the four-momentum of the star 
, 
the momentum, and 
the
energy7.
Also we call 
the dipole moment of the star 
, and 
the quadrupole moment of the star
.
Then we transform these moments into more convenient forms. By the conservation law (67), we have
, an overdot denotes a time derivative with respect to 
, and 
. Using these
equations and noticing that the body zone remains unchanged (in the near zone coordinates),
i.e. 
, we have
where
and
The operators ![[ ]](article385x.gif){kind=small}
and 
attached to the indices denote anti-symmetrization and symmetrization.
is the spin of the star 
and Equation (86) is the momentum-velocity relation. Thus our
momentum-velocity relation is a direct analog of the Newtonian momentum-velocity relation (see
Section 3.4). In general, we have
and
where 
is the number of indices in the multi-index 
.
Now, from the above equations, especially Equation (88), we find that the body zone contributions,
, are of order 
. Note that if we can not or do not assume the (nearly) stationarity of the
initial data for the stars, then, instead of Equation (88) we have
(see Section 3.3). In this case the lowest order metric differs from
the Newtonian form. From our (almost) stationarity assumption the remaining motion inside a star, apart
from the spinning motion, is caused only by the tidal effect by the companion star and from Equation (88);
it appears at 3 PN order [81].
To obtain the lowest order 
, we have to evaluate the surface integrals 
. Generally, in
the moments 
and 
appear formally at the order 
and 
. Thus 
and 
appear as
and 
appear at the order 
for any 
and we have to calculate an infinite number of moments. In fact, this is
not the case and it was shown in [91] that only 
terms of 
contribute to the 3 PN
equations of motion. The important thing here is that 
and 
are at most 
in
.
Finally, since the order of 
is higher than that of 
, we conclude that
.
contributionAs far as the 
contribution is concerned, since the integrand 
is at least
quadratic in the small deviation field 
, we make the post-Newtonian expansion in the integrand.
Then, basically, with the help of (super-)potentials 
which satisfy 
, 
denoting the Laplacian, we have for each integral (see e.g. the 
term in Equation (77))
) can be derived without using Dirac delta distributions (see Appendix B of [91]). For the 
terms in
Equation (77), we use potentials many times to convert all the volume integrals into surface integrals and “
”
terms8.
In fact finding the super-potentials is one of the most formidable tasks especially when we proceed to
high post-Newtonian orders. Fortunately, up to 2.5 PN order, all the required super-potentials are
available. At 3 PN order, there appear many integrands for which we could not find the required
super-potentials. To obtain the 3 PN equations of motion, we devise an alternative method similar to
the method employed by Blanchet and Faye [28]. The details of the method will be explained
later.
Now the lowest order integrands can be evaluated with the body zone contribution 
, and since
is 
, we find
in an 
series,
Similarly, in the following we expand 
in an 
series. From these equations we find that
the deviation field in 
, 
, is 
. (It should be noted that in the body zone
is assumed to be of order unity and within our method we can not calculate 
there
explicitly. To obtain 
in the body zone, we have to know the internal structure of the
star.)
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| http://www.livingreviews.org/lrr-2007-2 | 
This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 2.0 Germany License. Problems/comments to |
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![∫ f (⃗y) ∮ [ 1 ∂g (⃗y) ∂ ( 1 )] d3y------- = − 4πg(⃗x) + dSk -----------k-− g(⃗y)--k- ------- . (101 ) N∕B |⃗x − ⃗y| ∂(N∕B) |⃗x − ⃗y| ∂y ∂y |⃗x − ⃗y|](article428x.gif){kind=small}
