3 Linearized Vacuum Equations
In what follows we solve the field equations (12, 13), in the vacuum region outside the compact-support
source, in the form of a formal non-linearity or post-Minkowskian expansion, considering the field variable
as a non-linear metric perturbation of Minkowski space-time. At the linearized level (or
first-post-Minkowskian approximation), we write:
where the subscript “ext” reminds us that the solution is valid only in the exterior of the source, and where
we have introduced Newton’s constant
as a book-keeping parameter, enabling one to label very
conveniently the successive post-Minkowskian approximations. Since
is a dimensionless variable, with
our convention the linear coefficient
in Equation (22) has the dimension of the inverse of
- a
mass squared in a system of units where
. In vacuum, the harmonic-coordinate metric coefficient
satisfies
We want to solve those equations by means of an infinite multipolar series valid outside a time-like world
tube containing the source. Indeed the multipole expansion is the correct method for describing the physics
of the source as seen from its exterior (
). On the other hand, the post-Minkowskian series is
physically valid in the weak-field region, which surely includes the exterior of any source, starting at a
sufficiently large distance. For post-Newtonian sources the exterior weak-field region, where both
multipole and post-Minkowskian expansions are valid, simply coincides with the exterior
.
It is therefore quite natural, and even, one would say inescapable when considering general
sources, to combine the post-Minkowskian approximation with the multipole decomposition.
This is the original idea of the “double-expansion” series of Bonnor [54], which combines the
-expansion (or
-expansion in his notation) with the
-expansion (equivalent to the
multipole expansion, since the
th order multipole moment scales like
with the source
radius).
The multipolar-post-Minkowskian method will be implemented systematically, using STF-harmonics to
describe the multipole expansion [210
], and looking for a definite algorithm for the approximation scheme [26
].
The solution of the system of equations (23, 24) takes the form of a series of retarded multipolar
waves
where
, and where the functions
are smooth functions of the retarded time
[
], which become constant in the past, when
. It is evident, since
a monopolar wave satisfies
and the d’Alembertian commutes with the multi-derivative
, that Equation (25) represents the most general solution of the wave equation (23) (see
Section 2 in Ref. [26
] for a proof based on the Euler-Poisson-Darboux equation). The gauge
condition (24), however, is not fulfilled in general, and to satisfy it we must algebraically decompose
the set of functions
,
,
into ten tensors which are STF with respect to all
their indices, including the spatial indices
,
. Imposing the condition (24) reduces the
number of independent tensors to six, and we find that the solution takes an especially simple
“canonical” form, parametrized by only two moments, plus some arbitrary linearized gauge
transformation [210
, 26
].
Theorem 1 The most general solution of the linearized field equations (23, 24), outside some time-like
world tube enclosing the source (
), and stationary in the past (see Equation (19)), reads
The first term depends on two STF-tensorial multipole moments,
and
, which are arbitrary
functions of time except for the laws of conservation of the monopole:
, and dipoles:
,
. It is given by
The other terms represent a linearized gauge transformation, with gauge vector
of the type (25), and
parametrized for four other multipole moments, say
,
,
and
.
The conservation of the lowest-order moments gives the constancy of the total mass of the source,
, center-of-mass
position,
, total linear momentum
, and total angular momentum,
. It is always possible to achieve
by translating the origin of our coordinates to
the center of mass. The total mass
is the ADM mass of the Hamiltonian formulation of general
relativity. Note that the quantities
,
,
and
include the contributions due to the waves
emitted by the source. They describe the “initial” state of the source, before the emission of gravitational
radiation.
The multipole functions
and
, which thoroughly encode the physical properties of the
source at the linearized level (because the other moments
parametrize a gauge
transformation), will be referred to as the mass-type and current-type source multipole moments. Beware,
however, that at this stage the moments are not specified in terms of the stress-energy tensor
of the
source: the above theorem follows merely from the algebraic and differential properties of the vacuum
equations outside the source.
For completeness, let us give the components of the gauge-vector
entering Equation (26):
Because the theory is covariant with respect to non-linear diffeomorphisms and not merely with respect to
linear gauge transformations, the moments
do play a physical role starting at the non-linear
level, in the following sense. If one takes these moments equal to zero and continues the calculations one
ends up with a metric depending on
and
only, but that metric will not describe the same
physical source as the one constructed from the six moments
. In other words, the
two non-linear metrics associated with the sets of multipole moments
and
are not isometric. We point out in Section 4.2 below that the full set of moments
is in fact physically equivalent to some reduced set
,
but with some moments
,
that differ from
,
by non-linear corrections (see
Equation (96)). All the multipole moments
,
,
,
,
,
will be computed in
Section 5.