## 7 The Third Post-Newtonian Metric

The detailed calculations that are called for in applications necessitate having ar one’s disposal some
explicit expressions of the metric coefficients , in harmonic coordinates, at the highest possible
post-Newtonian order. The 3PN metric that we present below is expressed by means of some particular
retarded-type potentials, , , , etc., whose main advantages are to somewhat minimize the
number of terms, so that even at the 3PN order the metric is still tractable, and to delineate the different
problems associated with the computation of different categories of terms. Of course, these
potentials have no physical significance by themselves. The basic idea in our post-Newtonian
iteration is to use whenever possible a “direct” integration, with the help of some formulas like
. The 3PN harmonic-coordinates metric (issued from Ref. [22]) reads
All the potentials are generated by the matter stress-energy tensor through the definitions (analogous
to Eqs. (81))
and represent some retarded versions of the Newtonian and gravitomagnetic potentials,
From the 2PN order we have the potentials
Some parts of these potentials are directly generated by compact-support matter terms, while other parts
are made of non-compact-support products of -type potentials. There exists also a very important
cubically non-linear term generated by the coupling between and , the second term in the
-potential. At the 3PN level we have the most complicated of these potentials, namely
which involve many types of compact-support contributions, as well as quadratic-order
and cubic-order parts; but, surprisingly, there are no quartically non-linear
terms.
The above potentials are not independent. They are linked together by some differential
identities issued from the harmonic gauge conditions, which are equivalent, via the Bianchi
identities, to the equations of motion of the matter fields (see Eq. (17)). These identities read
It is important to remark that the above 3PN metric represents the inner post-Newtonian field of an
isolated system, because it contains, to this order, the correct radiation-reaction terms corresponding to
outgoing radiation. These terms come from the expansions of the retardations in the retarded-type
potentials (113, 114, 115).