# Part A: Post-Newtonian Sources

## 2 Einstein’s Field Equations

The field equations of general relativity form a system of ten second-order partial differential equations obeyed by the space-time metric ,
where the Einstein curvature tensor is generated, through the gravitational coupling , by the matter stress-energy tensor . Among these ten equations, four govern, via the contracted Bianchi identity, the evolution of the matter system,
The space-time geometry is constrained by the six remaining equations, which place six independent constraints on the ten components of the metric , leaving four of them to be fixed by a choice of a coordinate system.

In most of this paper we adopt the conditions of harmonic, or de Donder, coordinates. We define, as a basic variable, the gravitational-field amplitude

where denotes the contravariant metric (satisfying ), where is the determinant of the covariant metric, , and where represents an auxiliary Minkowskian metric. The harmonic-coordinate condition, which accounts exactly for the four equations (10) corresponding to the conservation of the matter tensor, reads
The equations (11, 12) introduce into the definition of our coordinate system a preferred Minkowskian structure, with Minkowski metric . Of course, this is not contrary to the spirit of general relativity, where there is only one physical metric without any flat prior geometry, because the coordinates are not governed by geometry (so to speak), but rather are chosen by researchers when studying physical phenomena and doing experiments. Actually, the coordinate condition (12) is especially useful when we view the gravitational waves as perturbations of space-time propagating on the fixed Minkowskian manifold with the background metric . This view is perfectly legitimate and represents a fructuous and rigorous way to think of the problem when using approximation methods. Indeed, the metric , originally introduced in the coordinate condition (12), does exist at any finite order of approximation (neglecting higher-order terms), and plays in a sense the role of some “prior” flat geometry.

The Einstein field equations in harmonic coordinates can be written in the form of inhomogeneous flat d’Alembertian equations,

where . The source term, , can rightly be interpreted as the stress-energy pseudo-tensor (actually, is a Lorentz tensor) of the matter fields, described by , and the gravitational field, given by the gravitational source term , i.e.
The exact expression of , including all non-linearities, reads
As is clear from this expression, is made of terms at least quadratic in the gravitational-field strength and its first and second space-time derivatives. In the following, for the highest post-Newtonian order that we consider (3PN), we need the quadratic, cubic and quartic pieces of . With obvious notation, we can write them as
These various terms can be straightforwardly computed from Eq. (15); see Eqs. (3.8) in Ref. [22] for explicit expressions.

As said above, the condition (12) is equivalent to the matter equations of motion, in the sense of the conservation of the total pseudo-tensor ,

In this article, we look for the solutions of the field equations (13, 14, 15, 17) under the following four hypotheses:
1. The matter stress-energy tensor is of spatially compact support, i.e. can be enclosed into some time-like world tube, say , where is the harmonic-coordinate radial distance. Outside the domain of the source, when , the gravitational source term, according to Eq. (17), is divergence-free,
2. The matter distribution inside the source is smooth: . We have in mind a smooth hydrodynamical “fluid” system, without any singularities nor shocks (a priori), that is described by some Eulerian equations including high relativistic corrections. In particular, we exclude from the start any black holes (however we shall return to this question when we find a model for describing compact objects).
3. The source is post-Newtonian in the sense of the existence of the small parameter defined by Eq. (1). For such a source we assume the legitimacy of the method of matched asymptotic expansions for identifying the inner post-Newtonian field and the outer multipolar decomposition in the source’s exterior near zone.
4. The gravitational field has been independent of time (stationary) in some remote past, i.e. before some finite instant in the past, in the sense that

The latter condition is a means to impose, by brute force, the famous no-incoming radiation condition, ensuring that the matter source is isolated from the rest of the Universe and does not receive any radiation from infinity. Ideally, the no-incoming radiation condition should be imposed at past null infinity. We shall later argue (see Section 6) that our condition of stationarity in the past (19), although much weaker than the real no-incoming radiation condition, does not entail any physical restriction on the general validity of the formulas we derive.

Subject to the condition (19), the Einstein differential field equations (13) can be written equivalently into the form of the integro-differential equations

containing the usual retarded inverse d’Alembertian operator, given by
extending over the whole three-dimensional space .