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3 Linearized Vacuum Equations

In what follows we solve the field equations (12View Equation, 13View Equation), 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 hab as a non-linear metric perturbation of Minkowski space-time. At the linearized level (or first-post-Minkowskian approximation), we write:
ab ab 2 hext = Gh 1 + O(G ), (22)
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 G as a book-keeping parameter, enabling one to label very conveniently the successive post-Minkowskian approximations. Since ab h is a dimensionless variable, with our convention the linear coefficient ab h1 in Equation (22View Equation) has the dimension of the inverse of G - a mass squared in a system of units where h = c = 1. In vacuum, the harmonic-coordinate metric coefficient hab1 satisfies
ab []h 1am = 0, (23) @mh 1 = 0. (24)
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 (r > a). 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 r > a. 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 G-expansion (or m-expansion in his notation) with the a-expansion (equivalent to the multipole expansion, since the lth order multipole moment scales like al with the source radius).

The multipolar-post-Minkowskian method will be implemented systematically, using STF-harmonics to describe the multipole expansion [210Jump To The Next Citation Point], and looking for a definite algorithm for the approximation scheme [26Jump To The Next Citation Point]. The solution of the system of equations (23View Equation, 24View Equation) takes the form of a series of retarded multipolar waves7

( ) + sum oo Kab (t - r/c) ha1b = @L --L---------- , (25) l=0 r
where r = |x |, and where the functions ab ab K L =_ K i1...il are smooth functions of the retarded time u =_ t- r/c [KL(u) (- Co o (R)], which become constant in the past, when t < - T. It is evident, since a monopolar wave satisfies [](KL(u)/r) = 0 and the d’Alembertian commutes with the multi-derivative @L, that Equation (25View Equation) represents the most general solution of the wave equation (23View Equation) (see Section 2 in Ref. [26Jump To The Next Citation Point] for a proof based on the Euler-Poisson-Darboux equation). The gauge condition (24View Equation), however, is not fulfilled in general, and to satisfy it we must algebraically decompose the set of functions K00 L, K0i L, Kij L into ten tensors which are STF with respect to all their indices, including the spatial indices i, ij. Imposing the condition (24View Equation) 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 [210Jump To The Next Citation Point26Jump To The Next Citation Point].

Theorem 1 The most general solution of the linearized field equations (23View Equation, 24View Equation), outside some time-like world tube enclosing the source (r > a), and stationary in the past (see Equation (19View Equation)), reads

ha1b = ka1b + @afb1 + @bfa1- jab@mfm1. (26)
The first term depends on two STF-tensorial multipole moments, IL(u) and JL(u), which are arbitrary functions of time except for the laws of conservation of the monopole: I = const, and dipoles: Ii = const, Ji = const. It is given by
( ) 00 4 sum (-)l 1 k 1 = - -2 ----@L -IL(u) , c l>0 l! r sum l { ( ) ( )} k01i= 4- (--)- @L -1 1I(i1L)-1(u) + --l--eiab@aL-1 1-JbL-1(u) , (27) c3 l>1 l! r l + 1 r { ( ) ( )} ij -4 sum (-)l- 1-(2) --2l- 1- (1) k1 = - c4 l! @L- 2 rIijL- 2(u) + l + 1@aL-2 reab(iJj)bL-2(u) . l>2
The other terms represent a linearized gauge transformation, with gauge vector fa 1 of the type (25View Equation), and parametrized for four other multipole moments, say W (u) L, X (u) L, Y (u) L and Z (u) L.

The conservation of the lowest-order moments gives the constancy of the total mass of the source, M =_ I = const, center-of-mass position8, Xi =_ Ii/I = const, total linear momentum (1) Pi =_ Ii = 0, and total angular momentum, Si =_ Ji = const. It is always possible to achieve Xi = 0 by translating the origin of our coordinates to the center of mass. The total mass M is the ADM mass of the Hamiltonian formulation of general relativity. Note that the quantities M, X i, P i and S i 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 IL(u) and JL(u), which thoroughly encode the physical properties of the source at the linearized level (because the other moments WL, ...,ZL 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 ab T 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 fa 1 entering Equation (26View Equation):

( ) 0 4 sum (- )l 1 f 1 = -3 -----@L --WL(u) , c l>0 l! r sum l ( ) fi1 = - 4- (--)-@iL 1XL(u) (28) c4 l>0 l! r l{ ( ) ( )} 4- sum (--)- 1- --l-- 1- - c4 l! @L-1 r YiL-1(u) + l + 1 eiab@aL-1 rZbL-1(u) . l>1
Because the theory is covariant with respect to non-linear diffeomorphisms and not merely with respect to linear gauge transformations, the moments W ,...,Z L L 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 IL and JL only, but that metric will not describe the same physical source as the one constructed from the six moments IL,...,ZL. In other words, the two non-linear metrics associated with the sets of multipole moments {IL,JL, 0,...,0} and {I ,J ,W ,...,Z } L L L L are not isometric. We point out in Section 4.2 below that the full set of moments {IL,JL, WL, ...,ZL} is in fact physically equivalent to some reduced set {ML, SL, 0,...,0}, but with some moments ML, SL that differ from IL, JL by non-linear corrections (see Equation (96View Equation)). All the multipole moments IL, JL, WL, XL, YL, ZL will be computed in Section 5.

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