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
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 fruitful 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 Equation (15); see Equations (3.8) in
Ref. [38
] 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:
- 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 Equation (17), is
divergence-free,
- 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).
- The source is post-Newtonian in the sense of the existence of the small parameter defined by
Equation (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.
- 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, Equation (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
.