Alternative method for matching post-Newtonian expansion to post-Minkowskian field
aa r X i v : . [ g r- q c ] F e b Alternative method for matching post-Newtonianexpansion to post-Minkowskian field
Abbas Mirahmadi
Department of Physics, University of Tehran, North Kargar Ave., Tehran, IranE-mail: [email protected]
February 2021
Abstract.
In 2002, Poujade and Blanchet succeeded in matching the post-Newtoniansolution to the Einstein field equation to the post-Minkowskian field up to any arbi-trary order as well as reproducing, in a different way, the results of the 1998 paperby Blanchet in which he showed how to match the post-Minkowskian series to thepost-Newtonian expansion. Comparing these two papers, it might be asked whether itis possible to match the post-Newtonian field to the post-Minkowskian one by meansof a method similar to the one used in the 1998 paper. The answer is affirmative, andin the present paper we provide this alternative method. Furthermore, detailed proofsof several properties and results stated in previous papers are given.
Keywords : gravitational waves, post-Minkowskian expansion, post-Newtonianexpansion, matching
1. Introduction
Due to the nonlinearities of the Einstein field equation, employing approximate solutionmethods for the analytical study of the gravitational waves is unavoidable. Einsteinhimself solved the problem of the linear approximation to the gravitational radiationgenerated by a localized time-dependent source and, by approximating | x − x ′ | appearingin the retarded argument and the denominator of the integrand as constant for largedistances and applying the conservation equation, found his famous quadrupole formula[1, 2]. More systematically, in the cases of self-gravitating sources with internal speedsvery small compared with the speed of light, the post-Minkowskian and post-Newtonianexpansions are used. By substituting these two asymptotic expansions into the Einsteinfield equation, we find the post-Minkowskian and post-Newtonian equations governingthe gravitational field. Although the post-Minkowskian expansion, if one combines itwith a multipole expansion [3, 4], can provide approximate solutions up to a certainfinite order, there is no guarantee that higher-order approximations are obtainable [5].Moreover, there are several works, such as [6, 7], showing that the post-Newtonian lternative method for matching ... λ ≪ | x | (where λ is a typicalwavelength of the emitted gravitational wave), breaks down beyond a specific order dueto the divergence of integrals, which itself is owing to the behavior of the lower-orderapproximations in the far zone.The inefficiency of the usual methods of taking the retarded and Poisson integralsdoes not mean that the solutions to the post-Minkowskian and post-Newtonianequations cannot be obtained up to any order. As suggested by Fock, the problemof describing the gravitational field everywhere in R needs to be split into twosubproblems, one concerning the near zone and the other outside the source, andthen, a matching procedure must be employed in order to determine the unknownterms of the general solutions to the subproblems [8]. Based on the Fock’s idea,Blanchet and Damour established their own algorithm in which they employed the post-Minkowskian expansion to describe the gravitational field outside the source, a regionwhere T µν ( t, x ) = 0, considered the post-Newtonian expansion as the approximatesolution to the Einstein field equation in the near zone and assumed that the domainsof validity of both expansions under discussion overlap (the mathematical statementof this last assumption is called the matching equation ) [9]. Thanks to the methodintroduced by Riesz [10], which depends deeply on analytic continuation, Blanchet andDamour could provide the most general solution to each order of the post-Minkowskianapproximation in 1986 [5]. Then, in the following years, they matched the post-Minkowskian series to the post-Newtonian field order by order [11–13].Instead of proceeding with the order-by-order matching procedure in which acoordinate transformation between the fields is looked for, in 1998 Blanchet matchedthe resummation of the post-Minkowskian expansion (containing all the coefficients from n = 1 to infinity) to the post-Newtonian field [14]. Roughly speaking, the method usedwas to convert the near-zone integrals brought about in the course of computations intothe far-zone integrals by means of a process of analytic continuation and then to applythe matching equation. Later, in 2002, Poujade and Blanchet matched the resummationof the post-Newtonian expansion to the post-Minkowskian field with a method differentfrom the one employed in the 1998 paper [15]. They substituted the near-zone expansionof the post-Minkowskian solution and the far-zone expansion of the post-Newtonianfield into the matching equation, and then, after the identical terms cancelling out, theydetermined the unknown post-Minkowskian and post-Newtonian moments of the generalsolutions by comparing the remaining unknown and known terms. In this paper, weintend to reproduce the 2002 paper results of matching the post-Newtonian expansion tothe post-Minkowskian expansion by virtue of a method similar to the one used in [14]. Aswe will see later, in contrast to [14], here we need to transform the far-zone integrals intothe near-zone integrals, and this makes the computations more sophisticated. Beforeproceeding further, we devote the next subsection to stating the results ‡ of the previouspapers that we need to achieve our aim. ‡ The detailed proofs of these results can be found in [16]. lternative method for matching ... In this paper, we use the Landau-Lifshitz formulation of the Einstein field equation inwhich the main variables are the components of h µν = √− gg µν − η µν , where g = det[ g µν ],[ g µν ] = [ g µν ] − and η µν = diag ( − , , , ✷ h µν ( t, x ) = 16 πGc τ µν ( t, x ) , (1.1) ∂ µ h µν ( t, x ) = 0 . (1.2)The first of the above equations is called the relaxed Einstein field equation and τ µν appearing in it the effective energy-momentum pseudotensor . ✷ = ∂ α ∂ α = η αβ ∂ α ∂ β and τ µν is given by τ µν = ( − g ) T µν + c πG Λ µν , (1.3)where, defining [ g µν ] = [ η µν + h µν ] and [ g µν ] = [ η µν + h µν ] − , Λ µν readsΛ µν = ∂ α h µβ ∂ β h να − h αβ ∂ αβ h µν + 12 g µν g λα ∂ ρ h λβ ∂ β h αρ − g µλ g αβ ∂ ρ h νβ ∂ λ h αρ − g νλ g αβ ∂ ρ h µβ ∂ λ h αρ + g λα g βρ ∂ β h µλ ∂ ρ h να + 18 (cid:0) g µλ g να − g µν g λα (cid:1) (2 g βρ g στ − g ρσ g βτ ) ∂ λ h βτ ∂ α h ρσ . (1.4)Moreover, (1.1), together with (1.2) called the harmonic gauge condition , imply that τ µν is conserved, i.e., ∂ µ τ µν ( t, x ) = 0 . (1.5)We restrict our attention to the gravitational waves sources that can be treatedas perfect fluids with internal speeds very small in comparison to the speed of light.We also choose a coordinate system in which the origin of the spatial coordinates islocated within the source and assume that the material energy-momentum tensor ofthe source T µν ( t, x ) is compactly supported (i.e., there is a positive constant d suchthat T µν ( t, x ) = 0 for | x | > d ) and belongs to C ∞ ( R ). Furthermore, we consider thefollowing conditions for h µν ( t, x ) [5]: h µν ( t, x ) ∈ C ∞ (cid:0) R (cid:1) , (1.6) ∂ t h µν ( t, x ) = 0 when t ≤ −T , (1.7)lim | x |→∞ t =const h µν ( t, x ) = 0 when t ≤ −T , (1.8)where by −T we mean an instant in the past.We denote the post-Minkowskian and post-Newtonian expansions of a generalfunction by M ( f )( t, x ) = P ∞ n = n G n f ( n ) ( t, x ) and ¯ f ( t, x ) = P ∞ m = m (1 /c ) m ¯ f ( m ) ( t, x )respectively, where n and m depend on the function. Having introduced thenotation and taking the domains of validity of the two approximations into account, lternative method for matching ... ✷ h µν ( n ) ( t, x ) = Λ µν ( n ) ( t, x ) for n ≥ , (1.9) ∂ µ h µν ( n ) ( t, x ) = 0 for n ≥ , (1.10) ∂ µ Λ µν ( n ) ( t, x ) = 0 for n ≥ , (1.11) h µν ( n ) ( t, x ) is smooth at | x | > d for n ≥ , (1.12) ∂ t h µν ( n ) ( t, x ) = 0 when t ≤ −T for n ≥ , (1.13)lim | x |→∞ t =const h µν ( n ) ( t, x ) = 0 when t ≤ −T for n ≥ , (1.14)while substitution of the post-Newtonian expansions results in∆¯ h µν ( n ) ( t, x ) = 16 πG ¯ τ µν ( n − ( t, x ) + ∂ t ¯ h µν ( n − ( t, x ) for n ≥ , (1.15) ∂ t ¯ h ν ( n − ( t, x ) + ∂ i ¯ h iν ( n ) ( t, x ) = 0 for n ≥ , (1.16) ∂ t ¯ τ ν ( n − ( t, x ) + ∂ i ¯ τ iν ( n − ( t, x ) = 0 for n ≥ , (1.17)¯ h µν ( n ) ( t, x ) is smooth at | x | < R for n ≥ , (1.18) ∂ t ¯ h µν ( n ) ( t, x ) = 0 when t ≤ −T for n ≥ , (1.19)where R is the radius at which we take the boundary of the near zone to be ( d < R ≪ λ ). In both cases, the general solution at each order is written as the sumof the general solution to the corresponding homogeneous equation (which is subjectto some restrictions mainly brought about by the harmonic gauge condition) and aparticular solution to the inhomogeneous equation derived from the relaxed Einsteinfield equation [5, 15]. That particular solution is given with the use of the followingtheorem [5, 15]: Theorem 1.1.
A particular solution to the equation L f ( t, x ) = g ( t, x ), where L is either∆ or ✷ , is FP B =0 A L − h ( | x ′ | /r ) B g ( t ′′ , x ′ ) i , where L − is either ∆ − (with t ′′ = t ) or ✷ − (with t ′′ = t ′ = t − | x − x ′ | /c ). In this particular solution, B is a complex number and r an arbitrary constant, and by A and FP B =0 we mean the analytic continuation of ... and the coefficient of the zeroth power of B in the Laurent expansion of ... about B = 0,respectively. Evidently, for this to be a particular solution, L − h ( | x ′ | /r ) B g ( t ′′ , x ′ ) i needs to be analytic in some original domain and analytically continuable to some(punctured) neighborhood of B = 0.In terms of the post-Minkowskian approximation, the only results that we need areas follows. It can be proven that the structure of M ( h µν ) ( t, x ) and Λ µν ( M ( h )) ( t, x ),the (untruncated) post-Minkowskian expansions of h µν ( t, x ) and Λ µν ( t, x ), read [5] M ( h µν ) ( t, x ) = M ( h µν AS ) ( x ) + M ( h µν PZ ) ( t, x )= ∞ X ℓ =0 − X a = −∞ ˆ n L | x | k ˆ C µνL,k + ∞ X q =0 ∞ X a = −∞ ∞ X p =0 ˆ n Q | x | a (cid:18) ln | x | r (cid:19) p ˆ F µνQ,a,p ( t )+ R µν ( t, x ) , (1.20) lternative method for matching ... µν ( M ( h ))( t, x ) = Λ µν AS ( M ( h )) ( x ) + Λ µν PZ ( M ( h )) ( t, x )= ∞ X ℓ =0 − X a = −∞ ˆ n L | x | k ˆ C ′ µνL,k + ∞ X q =0 ∞ X a = −∞ ∞ X p =0 ˆ n Q | x | a (cid:18) ln | x | r (cid:19) p ˆ F ′ µνQ,a,p ( t )+ R ′ µν ( t, x ) , (1.21)where in each case the term with the subscript AS denotes the first term on the RHSof the second equality and the one with the subscript PZ the remaining terms. ˆ n L isthe symmetric-trace-free part of n L = n I ℓ = n i · · · n i ℓ , the constant r is the same as intheorem 1.1 given earlier, and ˆ C µνL,k and ˆ C ′ µνL,k are constant, while ˆ F µνQ,a,p ( t ), ˆ F ′ µνQ,a,p ( t ), R µν ( t, x ) and R ′ µν ( t, x ) are past-zero . § Furthermore, R µν ( t, x ) and R ′ µν ( t, x ) areO (cid:0) | x | N (cid:1) (called the big-O ) as | x | → N tends to infinity, and hence, | x | m R µν ( t, x )and | x | m R ′ µν ( t, x ) with any m are well-behaved in any punctured neighborhood of | x | = 0. The set of equations governing M ( h µν ) ( t, x ) and Λ µν ( M ( h )) ( t, x ) are ✷ M ( h µν ) ( t, x ) = Λ µν ( M ( h ))( t, x ) , (1.22) ∂ µ M ( h µν ) ( t, x ) = 0 , (1.23) ∂ µ Λ µν ( M ( h ))( t, x ) = 0 . (1.24)With regard to the post-Newtonian approximation, it can be shown that, for anyarbitrary n , ¯Λ µν ( n ) ( t, x ) is smooth in the near zone while outside the near zone its structureis of the form ¯Λ µν ( n ) ( t, x ) = ∞ X q =0 a max ( n ) X a = −∞ p max ( n ) X p =0 ˆ n Q | x | a (cid:18) ln | x | r (cid:19) p ˆ E µν ( n ) Q,a,p ( t ) , (1.25)where a max ( n ) > E µν ( n ) Q,a,p ( t ) is a past-stationary k function. These structuralproperties of ¯Λ µν ( n ) ( t, x ) in the near zone and outside of it guarantee that the generalsolution to n th-order problem is given by [15]¯ h µν ( n ) ( t, x ) ⋆ = FP B =0 A∆ − (cid:20) (cid:18) | x ′ | r (cid:19) B (cid:16) πG ¯ τ µν ( n − ( t, x ′ ) + ∂ t ¯ h µν ( n − ( t, x ′ ) (cid:17) (cid:21) + ∞ X ℓ =0 ˆ n L | x | ℓ ˆ B µν ( n ) L ( t ) , (1.26)where by the sign ⋆ = we mean that the LHS of this sign equals its RHS provided that R R appearing on the RHS is written as the sum of R | x ′ | < R and R R < | x ′ | ; otherwise, itcannot be used in computations [16]. Similar to ¯Λ µν ( n ) ( t, x ), ¯ h µν ( n ) ( t, x ) is smooth in the § We denote the multi-index i ...i ℓ − m by L − m , and by ˆ T Q we mean the symmetric-trace-free part of T Q . In addition, past-zero functions are the ones that equate to zero at t ≤ −T . Last but not least,the subscripts AS and PZ stand for always-stationary and past-zero respectively. k Past-stationarity means being stationary at t ≤ −T . lternative method for matching ... B µν ( n ) L ( t )’swith ℓ > ℓ max ( n ) are taken to be zero ¶ , can be written as¯ h µν ( n ) ( t, x ) = ∞ X q =0 a ′ max ( n ) X a = −∞ p ′ max ( n ) X p =0 ˆ n Q | x | a (cid:18) ln | x | r (cid:19) p ˆ G µν ( n ) Q,a,p ( t ) , (1.27)where ˆ G µν ( n ) Q,a,p ( t ) is past-stationary. In contrast to [15] in which this structure hasbeen assumed, it has been obtained in [16]. Moreover, as it can be seen in the aboveequation, no remainder term appears in the structure. By summing the post-Newtoniancoefficients ¯ h µν ( n ) ( t, x ) (multiplied by 1 /c n ) over all values of n and using (3.9) in [15], onecan readily reach¯ h µν ( t, x ) ⋆ = 16 πGc ∞ X k =0 c k ∂ kt FP B =0 A (cid:20) − π Z R (cid:18) | x ′ | r (cid:19) B | x − x ′ | k − (2 k )! ¯ τ µν ( t, x ′ ) d x ′ (cid:21) + ∞ X k =0 ∞ X ℓ =0 c k (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L ∂ kt ˆ B µνL ( t ) . (1.28)It is also obvious that, since all the post-Newtonian coefficints ¯ h µν ( n ) ( t, x ) and ¯Λ µν ( n ) ( t, x ) aresmooth in the near zone, so do the (untruncated) post-Newtonian expansions ¯ h µν ( t, x )and ¯Λ µν ( t, x ). Additionally, considering the structures given in (1.25) and (1.27), thestructures of ¯ h µν ( t, x ) and ¯Λ µν ( t, x ) outside the near zone can be expressed as¯ h µν ( t, x ) = ∞ X q =0 ∞ X a = −∞ ∞ X p =0 ˆ n Q | x | a (cid:18) ln | x | r (cid:19) p ˆ G µνQ,a,p ( t ) , (1.29)¯Λ µν ( t, x ) = ∞ X q =0 ∞ X a = −∞ ∞ X p =0 ˆ n Q | x | a (cid:18) ln | x | r (cid:19) p ˆ E µνQ,a,p ( t ) . (1.30)Finally, the set of equations governing them are as follows: ✷ ¯ h µν ( t, x ) = 16 πGc ¯ τ µν ( t, x ) , (1.31) ∂ µ ¯ h µν ( t, x ) = 0 , (1.32) ∂ µ ¯ τ µν ( t, x ) = 0 . (1.33)Last but not least, let us provide the matching equation. It is clear that we canwrite [14] M ( h µν )( t, x ) = lim | x |→ t =const M ( h µν ) ( t, x ) = ∞ X q =0 ∞ X a = −∞ ∞ X p =0 ˆ n Q | x | a (cid:18) ln | x | r (cid:19) p ˆ F µνQ,a,p ( t ) , (1.34) M (cid:0) ¯ h µν (cid:1) ( t, x ) = lim | x |→∞ t =const ¯ h µν ( t, x ) = ∞ X q =0 ∞ X a = −∞ ∞ X p =0 ˆ n Q | x | a (cid:18) ln | x | r (cid:19) p ˆ G µνQ,a,p ( t ) . (1.35) ¶ This assumption is technically inevitable as in the post-Minkwskian solution [17]. lternative method for matching ... rescaled variables, it would havebeen obvious that M ( h µν )( t, x ) and M (cid:0) ¯ h µν (cid:1) ( t, x ) are in fact the post-Minkowskian andpost-Newtonian expansions of h µν ( t, x ) within the common region of validity of theseexpansions. Therefore, recalling the assumption of existence of the matching regiondiscussed before, we have [14] M ( h µν )( t, x ) = M (cid:0) ¯ h µν (cid:1) ( t, x ) , (1.36)which is in full agreement with the similarity of the structures of M ( h µν )( t, x ) and M (cid:0) ¯ h µν (cid:1) ( t, x ). (1.36) is the matching equation and we use it to match M ( h µν ) ( t, x )and ¯ h µν ( t, x ) together.Having reviewed the past results, we are now in a position to match the post-Newtonian series to the post-Minkowskian field. This is what we will do in section 2. Insection 3, it will be shown in great detail that ¯ h µν ( t, x ) obtained in section 2 indeed fulfillsthe harmonic gauge condition. In section 4, we will determine the moments ˆ B µνL ( t ), andin section 5, provide the detailed computation of the closed form + of ¯ h µν ( t, x ). Finally,in section 6, a brief summary of the work will be given. At the end of this paper,two appendices have also been included. Appendix A contains a collection of usefuldefinitions, lemmas and formulae, and appendix B has been devoted to proving that¯ h µν ( t, x ) is independent of the constant r .
2. Determination of ¯ h µν ( t, x )We start our investigation with the trivial equality ✷ (cid:20) (cid:18) | x | r (cid:19) B ¯ h µν ( t, x ) (cid:21) = (cid:18) | x | r (cid:19) B ✷ ¯ h µν ( t, x )+ (cid:18) | x | r (cid:19) B h B | x | − ∂ | x | ¯ h µν ( t, x ) + B ( B + 1) | x | − ¯ h µν ( t, x ) i . (2.1)As we saw in subsection 1.2, the structure of ¯ h µν ( t, x ) outside the near zone is of theform P ∞ q =0 P ∞ a = −∞ P ∞ p =0 ˆ n Q | x | a (ln ( | x | /r )) p ˆ G µνQ,a,p ( t ). Since ¯ h µν ( t, x ) is smooth insidethe near zone, it is reasonable to assume that there exists a radius of convergence r < R within which, by using the Taylor expansion for functions of three variables (seeappendix A), we can write¯ h µν ( t, x ) = ∞ X j =0 j ! x J h ∂ J ¯ h µν ( t, x ) (cid:12)(cid:12)(cid:12) x = i = ∞ X j =0 j ! | x | j n J h ∂ J ¯ h µν ( t, x ) (cid:12)(cid:12)(cid:12) x = i , (2.2)and by means of (A.9), we reach¯ h µν ( t, x )= ∞ X j =0 [ j ] X k =0 j ! (2 j − k + 1)!!(2 j − k + 1)!! | x | j δ { i i · · · δ i k − i k ˆ n i k +1 ...i j } h ∂ J ¯ h µν ( t, x ) (cid:12)(cid:12)(cid:12) x = i + We will later state what this means. lternative method for matching ... ∞ X j =0 [ j ] X k =0 j ! (2 j − k + 1)!!(2 j − k + 1)!! j !2 k k ! ( j − k )! | x | j δ i i · · · δ i k − i k ˆ n i k +1 ...i j h ∂ J ¯ h µν ( t, x ) (cid:12)(cid:12)(cid:12) x = i = ∞ X j =0 [ j ] X k =0 k k ! ( j − k )! (2 ( j − k ) + 1)!!(2 ( j − k ) + 2 k + 1)!! | x | j ˆ n J − k h ∂ J − k ∆ k ¯ h µν ( t, x ) (cid:12)(cid:12)(cid:12) x = i = ∞ X ℓ =0 ∞ X k =0 k k ! ℓ ! (2 ℓ + 1)!!(2 ℓ + 2 k + 1)!! | x | ℓ +2 k ˆ n L h ∂ L ∆ k ¯ h µν ( t, x ) (cid:12)(cid:12)(cid:12) x = i = ∞ X q =0 ∞ X k =0 ˆ n Q | x | q +2 k ¯ h µνQ,k ( t ) , (2.3)where to obtain the second equality, we have used the fact that, since all the indicesof J are dummy and ∂ J is a symmetric tensor, we can replace δ { i i · · · δ i k − i k ˆ n i k +1 ...i j } by only one of its terms times the number of its terms. Due to ˆ n J − k andthe Kronecker delta being totally symmetric tensors, the number of the terms of δ { i i · · · δ i k − i k ˆ n i k +1 ...i j } is equal to the number of ways in which one can select 2 k objects from j distinguishable ones and put them in k indistinguishable boxes such thateach contains two objects, i.e., (cid:0) j (cid:1)(cid:0) j − (cid:1) · · · (cid:0) j − k +22 (cid:1) /k ! = j ! / (cid:2) k k ! ( j − k )! (cid:3) where k ! hasappeared in the denominator because of the indistinguishability of the boxes. Hence, wehave replaced δ { i i · · · δ i k − i k ˆ n i k +1 ...i j } by (cid:2) j ! / k k ! ( j − k )! (cid:3) δ i i · · · δ i k − i k ˆ n i k +1 ...i j .Moreover, to write the fourth equality, or in other words, to relabel the summations, ithas been noted that, since j takes all nonnegative integer values, so does k because itsmaximum value is [ j/ j − k due to the inequality j ≥ k .For any arbitrary values of a , q , p and k , one is able to connectˆ n Q | x | a (ln ( | x | /r )) p ˆ G µνQ,a,p ( t ) to ˆ n Q | x | q +2 k ¯ h µνQ,k ( t ) (the indices of Q are not summed over)smoothly. (Their functional dependence on θ and ϕ are the same. Therefore, oneonly needs to connect ˆ G µνQ,a,p ( t ) to ¯ h µνQ,k ( t ) and | x | a (ln ( | x | /r )) p to | x | q +2 k in a smoothmanner. It is trivially obvious that a number of mappings are available for doing so.)For a ≤
0, we connect each ˆ n Q | x | a (ln ( | x | /r )) p ˆ G µνQ,a,p ( t ) to A a,p ˆ n Q | x | q ¯ h µνQ, ( t ) where P a = −∞ P ∞ p =0 A a,p = 1, and for a >
0, to B p ˆ n Q | x | q +2 a ¯ h µνQ,a ( t ) where P ∞ p =0 B p = 1. Wedenote the functions constructed in this way by f µνq,a,p ( t, x ). It is clear that in the regions | x | < r and | x | > R we have P ∞ q =0 P ∞ a = −∞ P ∞ p =0 f µνq,a,p ( t, x ) = ¯ h µν ( t, x ). We demandthis equation also hold in the region r < | x | < R . Now note that an equality similar to(2.1) can also be written for f µνq,a,p ( t, x ), that is ✷ (cid:20) (cid:18) | x | r (cid:19) B f µνq,a,p ( t, x ) (cid:21) = (cid:18) | x | r (cid:19) B ✷ f µνq,a,p ( t, x )+ (cid:18) | x | r (cid:19) B h B | x | − ∂ | x | f µνq,a,p ( t, x ) + B ( B + 1) | x | − f µνq,a,p ( t, x ) i . (2.4) lternative method for matching ... ✷ (cid:20) Re (cid:20) (cid:18) | x | r (cid:19) B (cid:21) f µνq,a,p ( t, x ) (cid:21) = Re (cid:20) (cid:18) | x | r (cid:19) B (cid:21) ✷ f µνq,a,p ( t, x )+ 2 Re (cid:20) B (cid:18) | x | r (cid:19) B (cid:21) | x | − ∂ | x | f µνq,a,p ( t, x )+ Re (cid:20) B ( B + 1) (cid:18) | x | r (cid:19) B (cid:21) | x | − f µνq,a,p ( t, x ) , (2.5) ✷ (cid:20) Im (cid:20) (cid:18) | x | r (cid:19) B (cid:21) f µνq,a,p ( t, x ) (cid:21) = Im (cid:20) (cid:18) | x | r (cid:19) B (cid:21) ✷ f µνq,a,p ( t, x )+ 2 Im (cid:20) B (cid:18) | x | r (cid:19) B (cid:21) | x | − ∂ | x | f µνq,a,p ( t, x )+ Im (cid:20) B ( B + 1) (cid:18) | x | r (cid:19) B (cid:21) | x | − f µνq,a,p ( t, x ) . (2.6)Outside the near zone, f µνq,a,p ( t, x ) is equal to ˆ n Q | x | a (ln ( | x | /r )) p ˆ G µνQ,a,p ( t ), andhence, due to the past-stationarity of ˆ G µνQ,a,p ( t ), both real and imaginary parts of( | x | /r ) B f µνq,a,p ( t, x ) fulfill the no-incoming radiation condition (see appendix A) providedthat Re( B ) + a <
0. Thus, if the retarded integrals converge, we can writeRe (cid:20) (cid:18) | x | r (cid:19) B (cid:21) f µνq,a,p ( t, x ) = ✷ − (cid:20) Re (cid:20) (cid:18) | x ′ | r (cid:19) B (cid:21) ✷ ′ f µνq,a,p ( t ′ , x ′ ) (cid:21) + ✷ − (cid:20) (cid:20) B (cid:18) | x ′ | r (cid:19) B (cid:21) | x ′ | − (cid:0) ∂ | x ′ | f µνq,a,p ( t ′ , x ′ ) (cid:1) t ′ (cid:21) + ✷ − (cid:20) Re (cid:20) B ( B + 1) (cid:18) | x ′ | r (cid:19) B (cid:21) | x ′ | − f µνq,a,p ( t ′ , x ′ ) (cid:21) , (2.7)Im (cid:20) (cid:18) | x | r (cid:19) B (cid:21) f µνq,a,p ( t, x ) = ✷ − (cid:20) Im (cid:20) (cid:18) | x ′ | r (cid:19) B (cid:21) ✷ ′ f µνq,a,p ( t ′ , x ′ ) (cid:21) + ✷ − (cid:20) (cid:20) B (cid:18) | x ′ | r (cid:19) B (cid:21) | x ′ | − (cid:0) ∂ | x ′ | f µνq,a,p ( t ′ , x ′ ) (cid:1) t ′ (cid:21) + ✷ − (cid:20) Im (cid:20) B ( B + 1) (cid:18) | x ′ | r (cid:19) B (cid:21) | x ′ | − f µνq,a,p ( t ′ , x ′ ) (cid:21) , (2.8)where (cid:0) ∂ | x ′ | f µνq,a,p ( t ′ , x ′ ) (cid:1) t ′ denotes the partial derivative of f µνq,a,p ( t ′ , x ′ ) with respect to | x ′ | ,ignoring the contribution from the variable t ′ . Each of the retarded integrals appearingin the above equations can be rewritten as the sum of two integrals, one over the region | x ′ | < R and the other over | x ′ | > R . Considering lemma A.1, the near zone integralsare convergent providing − q − < Re( B )+ q − a ≤
0, and − q − < Re( B )+ q +2 a − a >
0. (The near-zone integrals whose integrands include ∂ t ′ f µνq,a,p ( t ′ , x ′ ) converge if − q − < Re( B ) + q [for a ≤ − q − < Re( B ) + q + 2 a [for a > lternative method for matching ... B ) + a − < q −
2. (Since f µνq,a,p ( t, x ) is past-stationary and hence ∂ t f µνq,a,p ( t, x ) past-zero, for the far-zone integralsin which ∂ t ′ f µνq,a,p ( t ′ , x ′ ) appear to be convergent, the stronger condition Re( B )+ a < q − a ≤ − q − < Re( B ) < q − a ,and for a > − q − a − < Re( B ) < q − a , the retarded integrals are convergent.Therefore, irrespective of the value of a , there exists a vertical strip in the complexplane (whose width depends on the value of a ) in which both (2.7) and (2.8) hold.Consequently, in this region we can write (cid:18) | x | r (cid:19) B f µνq,a,p ( t, x ) = ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ✷ ′ f µνq,a,p ( t ′ , x ′ ) (cid:21) + ✷ − (cid:20) B (cid:18) | x ′ | r (cid:19) B | x ′ | − (cid:0) ∂ | x ′ | f µνq,a,p ( t ′ , x ′ ) (cid:1) t ′ (cid:21) + ✷ − (cid:20) B ( B + 1) (cid:18) | x ′ | r (cid:19) B | x ′ | − f µνq,a,p ( t ′ , x ′ ) (cid:21) . (2.9)It is straightforward to show that all the complex functions appearing in the aboveequation are also analytic in the aforementioned region. Hence, as a result of the identitytheorem, the equality between the analytic continuations of each side of (2.9) must holdin their common region of definition. Thus, considering that ( | x | /r ) B f µνq,a,p ( t, x ) isentire, we have (cid:18) | x | r (cid:19) B f µνq,a,p ( t, x ) = A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ✷ ′ f µνq,a,p ( t ′ , x ′ ) (cid:21) + A ✷ − (cid:20) B (cid:18) | x ′ | r (cid:19) B | x ′ | − (cid:0) ∂ | x ′ | f µνq,a,p ( t ′ , x ′ ) (cid:1) t ′ (cid:21) + A ✷ − (cid:20) B ( B + 1) (cid:18) | x ′ | r (cid:19) B | x ′ | − f µνq,a,p ( t ′ , x ′ ) (cid:21) , (2.10)which, after writing each retarded integral as the sum of the near-zone and far-zoneintegrals and summing over all values of q , a and p , reads (cid:18) | x | r (cid:19) B ¯ h µν ( t, x )= ∞ X q =0 ∞ X a = −∞ ∞ X p =0 A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ✷ ′ f µνq,a,p ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + ∞ X q =0 ∞ X a = −∞ ∞ X p =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ✷ ′ f µνq,a,p ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + ∞ X q =0 ∞ X a = −∞ ∞ X p =0 A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B B | x ′ | − (cid:0) ∂ | x ′ | f µνq,a,p ( t ′ , x ′ ) (cid:1) t ′ | x − x ′ | d x ′ (cid:21) + ∞ X q =0 ∞ X a = −∞ ∞ X p =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B B | x ′ | − (cid:0) ∂ | x ′ | f µνq,a,p ( t ′ , x ′ ) (cid:1) t ′ | x − x ′ | d x ′ (cid:21) lternative method for matching ... ∞ X q =0 ∞ X a = −∞ ∞ X p =0 A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B B ( B + 1) | x ′ | − f µνq,a,p ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + ∞ X q =0 ∞ X a = −∞ ∞ X p =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B B ( B + 1) | x ′ | − f µνq,a,p ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) . (2.11)Since there exists a common region in which all the integrals in each term onthe RHS of the above equation are analytic (depending on whether the region ofintegration is, this common region is either some right or left half-plane), using P ∞ q =0 P ∞ a = −∞ P ∞ p =0 f µνq,a,p ( t, x ) = ¯ h µν ( t, x ), one can write (cid:18) | x | r (cid:19) B ¯ h µν ( t, x ) = A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ✷ ′ ¯ h µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ✷ ′ ¯ h µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B B | x ′ | − (cid:0) ∂ | x ′ | ¯ h µν ( t ′ , x ′ ) (cid:1) t ′ | x − x ′ | d x ′ (cid:21) + A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B B | x ′ | − (cid:0) ∂ | x ′ | ¯ h µν ( t ′ , x ′ ) (cid:1) t ′ | x − x ′ | d x ′ (cid:21) + A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B B ( B + 1) | x ′ | − ¯ h µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B B ( B + 1) | x ′ | − ¯ h µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) . (2.12)As ¯ h µν ( t, x ) is smooth in the near zone, and its structure outside of that region is of theform P ∞ q =0 P ∞ a = −∞ P ∞ p =0 ˆ n Q | x | a (ln ( | x | /r )) p ˆ G µνQ,a,p ( t ), it can be readily shown that allthe analytic continuations appearing on the RHS of (2.12) are defined in some puncturedneighborhood of B = 0. Therefore, each of them possesses a Laurent expansion about B = 0. Further, it is obvious that the LHS of this equation has a Taylor expansionabout B = 0. Thus, since the coefficients of B n on both sides of the equation must beequal for each n , taking the coefficients of B into account, we reach¯ h µν ( t, x ) = FP B =0 A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ✷ ′ ¯ h µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + FP B =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ✷ ′ ¯ h µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + FP B =0 A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B B | x ′ | − (cid:0) ∂ | x ′ | ¯ h µν ( t ′ , x ′ ) (cid:1) t ′ | x − x ′ | d x ′ (cid:21) + FP B =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B B | x ′ | − (cid:0) ∂ | x ′ | ¯ h µν ( t ′ , x ′ ) (cid:1) t ′ | x − x ′ | d x ′ (cid:21) + FP B =0 A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B B ( B + 1) | x ′ | − ¯ h µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) lternative method for matching ...
12+ FP B =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B B ( B + 1) | x ′ | − ¯ h µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) . (2.13)Taking B and B ( B + 1) out of the integrals appearing respectively in the third and fifthterms on the RHS of (2.13), the integrals left after doing that are analytic at B = 0 dueto ¯ h µν ( t, x ) being smooth in the near zone. Thus, owing to the aforementioned complexcoefficients, the third and fifth terms on the RHS of (2.13) vanish. This, together with(1.31), yield¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) + FP B =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B B | x ′ | − (cid:0) ∂ | x ′ | ¯ h µν ( t ′ , x ′ ) (cid:1) t ′ | x − x ′ | d x ′ (cid:21) + FP B =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B B ( B + 1) | x ′ | − ¯ h µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) . (2.14)Since everywhere outside the near zone we have ¯ h µν ( t, x ) = M (cid:0) ¯ h µν (cid:1) ( t, x ), one canrewrite (2.14) as¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) + FP B =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B B | x ′ | − (cid:0) ∂ | x ′ | M (cid:0) ¯ h µν (cid:1) ( t ′ , x ′ ) (cid:1) t ′ | x − x ′ | d x ′ (cid:21) + FP B =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B B ( B + 1) | x ′ | − M (cid:0) ¯ h µν (cid:1) ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) . (2.15)If | x | < R , substituting the structure given in (1.35) for M (cid:0) ¯ h µν (cid:1) ( t, x ), taking thederivatives and using the 3-dimensional Taylor expansion, we get¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) − π ∞ X q =0 ∞ X a = −∞ ∞ X p =0 FP B =0 (2 aB + B ( B + 1)) · A Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ˆ n ′ I q | x ′ | a − × (cid:18) ln | x ′ | r (cid:19) p " ∞ X j =0 ( − j j ! x I ′ j ∂ ′ I ′ j ˆ G µνI q ,a,p ( t − | x ′ | c ) | x ′ | ! d x ′ − π ∞ X q =0 ∞ X a = −∞ ∞ X p =1 FP B =0 (2 pB ) · A Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ˆ n ′ I q | x ′ | a − × (cid:18) ln | x ′ | r (cid:19) p − " ∞ X j =0 ( − j j ! x I ′ j ∂ ′ I ′ j ˆ G µνI q ,a,p ( t − | x ′ | c ) | x ′ | ! d x ′ . (2.16)Now note that with the use of (A.10) and (A.1), we have ∞ X j =0 ( − j j ! x I ′ j ∂ ′ I ′ j ˆ G µνI q ,a,p ( t − | x ′ | c ) | x ′ | ! lternative method for matching ... ∞ X j =0 ( − j j ! x I ′ j [ j ] X k =0 (2 j − k + 1)!!(2 j − k + 1)!! δ { i ′ i ′ · · · δ i ′ k − i ′ k ˆ ∂ ′ i ′ k +1 ...i ′ j } ∆ ′ k ˆ G µνI q ,a,p ( t − | x ′ | c ) | x ′ | ! = ∞ X j =0 [ j ] X k =0 ( − j j ! (2 j − k + 1)!!(2 j − k + 1)!! x I ′ j δ { i ′ i ′ · · · δ i ′ k − i ′ k ˆ ∂ ′ i ′ k +1 ...i ′ j } (2 k ) ˆ G µνI q ,a,p ( t − | x ′ | c ) c k | x ′ | ! = ∞ X j =0 [ j ] X k =0 ( − j j ! (2 j − k + 1)!!(2 j − k + 1)!! j !2 k k ! ( j − k )! x I ′ j δ i ′ i ′ · · · δ i ′ k − i ′ k × ˆ ∂ ′ i ′ k +1 ...i ′ j (2 k ) ˆ G µνI q ,a,p ( t − | x ′ | c ) c k | x ′ | ! = ∞ X j =0 [ j ] X k =0 ( − j k k ! ( j − k )! (2 j − k + 1)!!(2 j − k + 1)!! | x | k x i ′ k +1 ...i ′ j ˆ ∂ ′ i ′ k +1 ...i ′ j (2 k ) ˆ G µνI q ,a,p ( t − | x ′ | c ) c k | x ′ | ! = ∞ X j =0 [ j ] X k =0 ( − j − k k k ! ( j − k )! (2 ( j − k ) + 1)!!(2 ( j − k ) + 2 k + 1)!! | x | k x I ′ J − k ˆ ∂ ′ I ′ J − k (2 k ) ˆ G µνI q ,a,p ( t − | x ′ | c ) c k | x ′ | ! = ∞ X ℓ =0 ∞ X k =0 ( − ℓ ℓ ! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k x I ′ ℓ ˆ ∂ ′ I ′ ℓ (2 k ) ˆ G µνI q ,a,p ( t − | x ′ | c ) c k | x ′ | ! , (2.17)where the third and sixth equalities have been obtained by arguments similar to thosefollowing (2.3). By means of (A.14), (2.17) takes the form ∞ X j =0 ( − j j ! x I ′ j ∂ ′ I ′ j ˆ G µνI q ,a,p ( t − | x ′ | c ) | x ′ | ! = ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! × | x | k ˆ x I ′ ℓ c k + ℓ − i " ˆ n ′ I ′ ℓ (2 k + ℓ − i ) ˆ G µνI q ,a,p ( t − | x ′ | c ) | x ′ | i +1 . (2.18)Combining (2.16) and (2.18) and using (A.3), we reach¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) − π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ∞ X q =0 ∞ X a = −∞ ∞ X p =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x I ′ ℓ c k + ℓ − i × Z ˆ n ′ I q ˆ n ′ I ′ ℓ dΩ ′ FP B =0 (2 aB + B ( B + 1)) ∂ p ∂B p (cid:20) r B A Z ∞R | x ′ | B + a − i − × (2 k + ℓ − i ) ˆ G µνI q ,a,p ( t − | x ′ | c ) d | x ′ | (cid:21) − π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ∞ X q =0 ∞ X a = −∞ ∞ X p =1 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x I ′ ℓ c k + ℓ − i × Z ˆ n ′ I q ˆ n ′ I ′ ℓ dΩ ′ FP B =0 (2 pB ) ∂ p − ∂B p − (cid:20) r B A Z ∞R | x ′ | B + a − i − lternative method for matching ... × (2 k + ℓ − i ) ˆ G µνI q ,a,p ( t − | x ′ | c ) d | x ′ | (cid:21) , (2.19)where we have also used the identity theorem. Assuming that the (1-variable) Taylorexpansion of ¯ h µν ( t, x ) and hence that of ˆ G µνI q ,a,p ( t ) have infinite radii of convergence aboutany arbitrary point t , we can writeˆ G µνI q ,a,p ( t − | x ′ | c ) = ∞ X j =0 ( − j c j j ! ( j ) ˆ G µνI q ,a,p ( t ) | x ′ | j , (2.20)and noting that ∂ t ˆ G µνI q ,a,p ( t − | x ′ | /c ) = (1) ˆ G µνI q ,a,p ( t − | x ′ | /c ), we findA Z ∞R | x ′ | B + a − i − k + ℓ − i ) ˆ G µνI q ,a,p ( t − | x ′ | c ) d | x ′ | = − ∞ X j =0 ( − j c j j ! R B + a − i + j B + a − i + j (2 k + ℓ − i + j ) ˆ G µνI q ,a,p ( t )= − A Z R | x ′ | B + a − i − k + ℓ − i ) ˆ G µνI q ,a,p ( t − | x ′ | c ) d | x ′ | . (2.21)As a result, one is allowed to make the replacement A R ∞R → − A R R in (2.19), andtherefore, after reversing the order of computations resulting in that equation, reaches¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) + 14 π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it h B | x ′ | − (cid:0) ∂ | x ′ | M (cid:0) ¯ h µν (cid:1) ( u ′ , x ′ ) (cid:1) u ′ + B ( B + 1) | x ′ | − M (cid:0) ¯ h µν (cid:1) ( u ′ , x ′ ) i d x ′ , (2.22)where u ′ = t − | x ′ | /c . In the case where R < | x | < ∞ , (2.15) can be rewritten as¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) + FP B =0 A (cid:20) − π Z R < | x ′ | < | x | (cid:18) | x ′ | r (cid:19) B B | x ′ | − (cid:0) ∂ | x ′ | M (cid:0) ¯ h µν (cid:1) ( t ′ , x ′ ) (cid:1) t ′ | x − x ′ | d x ′ (cid:21) + FP B =0 A (cid:20) − π Z R < | x ′ | < | x | (cid:18) | x ′ | r (cid:19) B B ( B + 1) | x ′ | − M (cid:0) ¯ h µν (cid:1) ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + FP B =0 A (cid:20) − π Z | x | < | x ′ | (cid:18) | x ′ | r (cid:19) B B | x ′ | − (cid:0) ∂ | x ′ | M (cid:0) ¯ h µν (cid:1) ( t ′ , x ′ ) (cid:1) t ′ | x − x ′ | d x ′ (cid:21) + FP B =0 A (cid:20) − π Z | x | < | x ′ | (cid:18) | x ′ | r (cid:19) B B ( B + 1) | x ′ | − M (cid:0) ¯ h µν (cid:1) ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) . (2.23)Factorizing B and B ( B + 1) out of the integrals in the second and third linesrespectively, we are left with the integrals which can be straightforwardly shown to lternative method for matching ... B = 0, and the second and third terms on the RHS of (2.23) are thereby,due to the complex coefficients factorized earlier, zero. Further, since the integralsappearing in the fourth and fifth terms on the RHS are over the region | x ′ | > | x | , byproceeding along the same lines as in the case where | x | < R , we can obtain¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) − π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ∞ X q =0 ∞ X a = −∞ ∞ X p =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x I ′ ℓ c k + ℓ − i × Z ˆ n ′ I q ˆ n ′ I ′ ℓ dΩ ′ FP B =0 (2 aB + B ( B + 1)) ∂ p ∂B p (cid:20) r B A Z ∞| x | | x ′ | B + a − i − × (2 k + ℓ − i ) ˆ G µνI q ,a,p ( t − | x ′ | c ) d | x ′ | (cid:21) − π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ∞ X q =0 ∞ X a = −∞ ∞ X p =1 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x I ′ ℓ c k + ℓ − i × Z ˆ n ′ I q ˆ n ′ I ′ ℓ dΩ ′ FP B =0 (2 pB ) ∂ p − ∂B p − (cid:20) r B A Z ∞| x | | x ′ | B + a − i − × (2 k + ℓ − i ) ˆ G µνI q ,a,p ( t − | x ′ | c ) d | x ′ | (cid:21) . (2.24)One can easily show that the R | x |R counterpart of the radial integral appearing inthe second and third terms on the RHS is entire. Therefore, due to the coefficients2 aB + B ( B + 1) and 2 pB , we are entitled to write the RHS of the above equation asin (2.19), and this means that even in the case under study one can obtain the sameequation as (2.22), thereby extending the domain of validity of that equation to be thewhole space (except for | x | → ∞ ).Now it is time to use the matching equation. Replacing M (cid:0) ¯ h µν (cid:1) by M ( h µν ) in(2.22), we get¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) + 14 π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it h B | x ′ | − (cid:16) ∂ | x ′ | M ( h µν )( u ′ , x ′ ) (cid:17) u ′ + B ( B + 1) | x ′ | − M ( h µν )( u ′ , x ′ ) i d x ′ , (2.25)and since we are able to demand that, as | x | →
0, the function R ′ µν ( t, x ) be so thatthe integrals R | x ′ | < R ( | x ′ | /r ) B (cid:2) ˆ x ′ L / | x ′ | ℓ + i +1 (cid:3) ∂ k + ℓ − it (cid:2) | x ′ | − (cid:0) ∂ | x ′ | R ′ µν ( u ′ , x ′ ) (cid:1) u ′ (cid:3) d x ′ and R | x ′ | < R ( | x ′ | /r ) B (cid:2) ˆ x ′ L / | x ′ | ℓ + i +1 (cid:3) ∂ k + ℓ − it [ | x ′ | − R ′ µν ( u ′ , x ′ )] d x ′ are analytic in some lternative method for matching ... B = 0 (including B = 0), we can write¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) + 14 π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it h B | x ′ | − (cid:0) ∂ | x ′ | M ( h µν ) ( u ′ , x ′ ) (cid:1) u ′ + B ( B + 1) | x ′ | − M ( h µν ) ( u ′ , x ′ ) i d x ′ . (2.26)The integrals R R < | x ′ | ( | x ′ | /r ) B (cid:2) ˆ x ′ L / | x ′ | ℓ + i +1 (cid:3) ∂ k + ℓ − it (cid:2) | x ′ | − ∂ | x ′ | M ( h µν AS ) ( x ′ ) (cid:3) d x ′ and R R < | x ′ | ( | x ′ | /r ) B (cid:2) ˆ x ′ L / | x ′ | ℓ + i +1 (cid:3) ∂ k + ℓ − it [ | x ′ | − M ( h µν AS ) ( x ′ )] d x ′ are identically zero if k > i < ℓ . In the case where k = 0 and i = ℓ , due to themaximal power of | x | in the structure of M ( h µν AS ) ( x ) being −
1, these integralsare analytic in some neighborhood of B = 0 (including B = 0). Moreover,the integrals R R < | x ′ | ( | x ′ | /r ) B (cid:2) ˆ x ′ L / | x ′ | ℓ + i +1 (cid:3) ∂ k + ℓ − it (cid:2) | x ′ | − (cid:0) ∂ | x ′ | M ( h µν PZ ) ( u ′ , x ′ ) (cid:1) u ′ (cid:3) d x ′ and R R < | x ′ | ( | x ′ | /r ) B (cid:2) ˆ x ′ L / | x ′ | ℓ + i +1 (cid:3) ∂ k + ℓ − it [ | x ′ | − M ( h µν PZ ) ( u ′ , x ′ )] d x ′ are analytic insome neighborhood of B = 0 (including B = 0) due to M ( h µν PZ ) ( t, x ) being past-zero.Therefore, owing to the coefficients B and B ( B + 1), we haveFP B =0 A Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it h B | x ′ | − (cid:0) ∂ | x ′ | M ( h µν ) ( u ′ , x ′ ) (cid:1) u ′ + B ( B + 1) | x ′ | − M ( h µν ) ( u ′ , x ′ ) i d x ′ = 0 . (2.27)Adding this vanishing expression to its counterpart on the RHS of (2.26), it becomes¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) + 14 π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it h B | x ′ | − (cid:0) ∂ | x ′ | M ( h µν ) ( u ′ , x ′ ) (cid:1) u ′ + B ( B + 1) | x ′ | − M ( h µν ) ( u ′ , x ′ ) i d x ′ + 14 π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 A Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it h B | x ′ | − (cid:0) ∂ | x ′ | M ( h µν ) ( u ′ , x ′ ) (cid:1) u ′ + B ( B + 1) | x ′ | − M ( h µν ) ( u ′ , x ′ ) i d x ′ . (2.28) lternative method for matching ... ✷ (cid:20) (cid:18) | x | r (cid:19) B M ( h µν ) ( t, x ) (cid:21) = (cid:18) | x | r (cid:19) B ✷ M ( h µν ) ( t, x ) + (cid:18) | x | r (cid:19) B h B | x | − ∂ | x | M ( h µν ) ( t, x )+ B ( B + 1) | x | − M ( h µν ) ( t, x ) i , (2.29)which, after using (1.22), can be rearranged as (cid:18) | x | r (cid:19) B h B | x | − ∂ | x | M ( h µν ) ( t, x ) + B ( B + 1) | x | − M ( h µν ) ( t, x ) i = ∆ (cid:20) (cid:18) | x | r (cid:19) B M ( h µν ) ( t, x ) (cid:21) − c (cid:18) | x | r (cid:19) B ∂ t M ( h µν ) ( t, x ) − (cid:18) | x | r (cid:19) B Λ µν ( M ( h )) ( t, x ) , (2.30)and, defining u = t − | x | /c , it is apparent that we can write (cid:18) | x | r (cid:19) B h B | x | − (cid:0) ∂ | x | M ( h µν ) ( u, x ) (cid:1) u + B ( B + 1) | x | − M ( h µν ) ( u, x ) i = (cid:18) ∆ (cid:20) (cid:18) | x | r (cid:19) B M ( h µν ) ( u, x ) (cid:21)(cid:19) u − c (cid:18) | x | r (cid:19) B ∂ u M ( h µν ) ( u, x ) − (cid:18) | x | r (cid:19) B Λ µν ( M ( h )) ( u, x ) . (2.31)With the use of the above equation, (2.28) can be rewritten as¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) − π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × (cid:20) FP B =0 A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) d x ′ + FP B =0 A Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) d x ′ (cid:21) + 14 π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × ∂ k + ℓ − it (cid:20) FP B =0 A Z | x ′ | < R ˆ x ′ L | x ′ | ℓ + i +1 (cid:26)(cid:18) ∆ ′ (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21)(cid:19) u ′ − c (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) (cid:27) d x ′ lternative method for matching ...
18+ FP B =0 A Z R < | x ′ | ˆ x ′ L | x ′ | ℓ + i +1 (cid:26)(cid:18) ∆ ′ (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21)(cid:19) u ′ − c (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) (cid:27) d x ′ (cid:21) . (2.32)It is not difficult to show that (cid:18) ∆ ′ (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21)(cid:19) u ′ − c (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ )= ∆ ′ (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21) + 2 c ∂ ′ j (cid:20) (cid:18) | x ′ | r (cid:19) B | x ′ | − x ′ j ∂ u ′ M ( h µν ) ( u ′ , x ′ ) (cid:21) − c (cid:18) | x ′ | r (cid:19) B | x ′ | − ∂ u ′ M ( h µν ) ( u ′ , x ′ ) , (2.33)and hence, introducing D i = ( { x ′ ∈ R | | x ′ | < R} , i = 1, { x ′ ∈ R | R < | x ′ |} , i = 2, (2.34)by means of the Gauss’ and Green’s theorems we can writeA Z D i ˆ x ′ L | x ′ | ℓ + i +1 (cid:26)(cid:18) ∆ ′ (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21)(cid:19) u ′ − c (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) (cid:27) d x ′ = Z D i (cid:18) | x ′ | r (cid:19) B (cid:20) ∆ ′ (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) M ( h µν ) ( u ′ , x ′ ) d x ′ − c Z D i (cid:18) | x ′ | r (cid:19) B | x ′ | − x ′ j (cid:20) ∂ ′ j (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ − c Z D i (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ + Z ∂D i ˆ x ′ L | x ′ | ℓ + i +1 ∂ ′ k (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21) d σ ′ ( i ) k − Z ∂D i (cid:20) ∂ ′ k (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) d σ ′ ( i ) k + 2 c Z ∂D i x ′ k ˆ x ′ L | x ′ | ℓ + i +2 (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d σ ′ ( i ) k , (2.35)providing we restrict ourselves to the original domain of analyticity of the LHS, namelyin either some right or left half-plane depending on whether i is. For i = 1 we haveA Z | x ′ | < R ˆ x ′ L | x ′ | ℓ + i +1 (cid:26)(cid:18) ∆ ′ (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21)(cid:19) u ′ − c (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) (cid:27) d x ′ lternative method for matching ... Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B (cid:20) ∆ ′ (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) M ( h µν ) ( u ′ , x ′ ) d x ′ − c Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B | x ′ | − x ′ j (cid:20) ∂ ′ j (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ − c Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ + Z | x ′ | = R ˆ x ′ L | x ′ | ℓ + i +1 ∂ ′ k (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21) d σ ′ (1) k − Z | x ′ | = R (cid:20) ∂ ′ k (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) d σ ′ (1) k + 2 c Z | x ′ | = R x ′ k ˆ x ′ L | x ′ | ℓ + i +2 (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d σ ′ (1) k . (2.36)As a result of the identity theorem, one can equate the LHS of the above equation andthe analytic continuation of its RHS wherever they are both defined. This, togetherwith the fact that the surface integrals are entire, yieldA Z | x ′ | < R ˆ x ′ L | x ′ | ℓ + i +1 (cid:26)(cid:18) ∆ ′ (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21)(cid:19) u ′ − c (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) (cid:27) d x ′ = A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B (cid:20) ∆ ′ (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) M ( h µν ) ( u ′ , x ′ ) d x ′ − c A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B | x ′ | − x ′ j (cid:20) ∂ ′ j (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ − c A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ + Z | x ′ | = R ˆ x ′ L | x ′ | ℓ + i +1 ∂ ′ k (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21) d σ ′ (1) k − Z | x ′ | = R (cid:20) ∂ ′ k (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) d σ ′ (1) k + 2 c Z | x ′ | = R x ′ k ˆ x ′ L | x ′ | ℓ + i +2 (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d σ ′ (1) k . (2.37)For i = 2 we getA Z R < | x ′ | ˆ x ′ L | x ′ | ℓ + i +1 (cid:26)(cid:18) ∆ ′ (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21)(cid:19) u ′ − c (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) (cid:27) d x ′ = Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B (cid:20) ∆ ′ (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) M ( h µν ) ( u ′ , x ′ ) d x ′ lternative method for matching ... − c Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B | x ′ | − x ′ j (cid:20) ∂ ′ j (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ − c Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ + Z | x ′ | = R ˆ x ′ L | x ′ | ℓ + i +1 ∂ ′ k (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21) d σ ′ (2) k − Z | x ′ | = R (cid:20) ∂ ′ k (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) d σ ′ (2) k + 2 c Z | x ′ | = R x ′ k ˆ x ′ L | x ′ | ℓ + i +2 (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d σ ′ (2) k + Z | x ′ |→∞ ˆ x ′ L | x ′ | ℓ + i +1 ∂ ′ k (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21) d σ ′ (2) k − Z | x ′ |→∞ (cid:20) ∂ ′ k (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) d σ ′ (2) k + 2 c Z | x ′ |→∞ x ′ k ˆ x ′ L | x ′ | ℓ + i +2 (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d σ ′ (2) k . (2.38)Taking Re( B ) to be a sufficiently large negative number, the last three surface integralson the RHS of the above equation vanish. Again, as a consequence of the identitytheorem, the sum of the analytic continuations of the remaining terms on the RHS of(2.38) and the LHS of it must be equal wherever they are all defined. Thus, consideringthat the surface integrals are entire functions, we reachA Z R < | x ′ | ˆ x ′ L | x ′ | ℓ + i +1 (cid:26)(cid:18) ∆ ′ (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21)(cid:19) u ′ − c (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) (cid:27) d x ′ = A Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B (cid:20) ∆ ′ (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) M ( h µν ) ( u ′ , x ′ ) d x ′ − c A Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B | x ′ | − x ′ j (cid:20) ∂ ′ j (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ − c A Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ + Z | x ′ | = R ˆ x ′ L | x ′ | ℓ + i +1 ∂ ′ k (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21) d σ ′ (2) k − Z | x ′ | = R (cid:20) ∂ ′ k (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) d σ ′ (2) k + 2 c Z | x ′ | = R x ′ k ˆ x ′ L | x ′ | ℓ + i +2 (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d σ ′ (2) k . (2.39) lternative method for matching ... Z D i ˆ x ′ L | x ′ | ℓ + i +1 (cid:26)(cid:18) ∆ ′ (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21)(cid:19) u ′ − c (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) (cid:27) d x ′ = A Z D i (cid:18) | x ′ | r (cid:19) B (cid:20) ∆ ′ (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) M ( h µν ) ( u ′ , x ′ ) d x ′ − c A Z D i (cid:18) | x ′ | r (cid:19) B | x ′ | − x ′ j (cid:20) ∂ ′ j (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ − c A Z D i (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ + Z | x ′ | = R ˆ x ′ L | x ′ | ℓ + i +1 ∂ ′ k (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21) d σ ′ ( i ) k − Z | x ′ | = R (cid:20) ∂ ′ k (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) d σ ′ ( i ) k + 2 c Z | x ′ | = R x ′ k ˆ x ′ L | x ′ | ℓ + i +2 (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d σ ′ ( i ) k . (2.40)Irrespective of whether i is, it can be easily shown that all the terms appearing in theabove equation are analytic in some punctured neighborhood of B = 0. Therefore, eachof them has a Laurent expansion around B = 0 (the last three terms on the RHS areanalytic at B = 0 and each of them thereby possesses a Taylor expansion about thispoint). Since the coefficients of B n on both sides of (2.40) must be equal for each n ,considering the coefficients of B , we end withFP B =0 A Z D i ˆ x ′ L | x ′ | ℓ + i +1 (cid:26)(cid:18) ∆ ′ (cid:20) (cid:18) | x ′ | r (cid:19) B M ( h µν ) ( u ′ , x ′ ) (cid:21)(cid:19) u ′ − c (cid:18) | x ′ | r (cid:19) B ∂ u ′ M ( h µν ) ( u ′ , x ′ ) (cid:27) d x ′ = FP B =0 A Z D i (cid:18) | x ′ | r (cid:19) B (cid:20) ∆ ′ (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) M ( h µν ) ( u ′ , x ′ ) d x ′ − c FP B =0 A Z D i (cid:18) | x ′ | r (cid:19) B | x ′ | − x ′ j (cid:20) ∂ ′ j (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ − c FP B =0 A Z D i (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d x ′ + Z | x ′ | = R ˆ x ′ L | x ′ | ℓ + i +1 ∂ ′ k M ( h µν ) ( u ′ , x ′ ) d σ ′ ( i ) k − Z | x ′ | = R (cid:20) ∂ ′ k (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) M ( h µν ) ( u ′ , x ′ ) d σ ′ ( i ) k lternative method for matching ...
22+ 2 c Z | x ′ | = R x ′ k ˆ x ′ L | x ′ | ℓ + i +2 ∂ u ′ M ( h µν ) ( u ′ , x ′ ) d σ ′ ( i ) k . (2.41)Thus, after summing (2.41) over i = 1 and i = 2, applying ∂ u ′ M ( h µν ) ( u ′ , x ′ ) = ∂ t M ( h µν ) ( u ′ , x ′ ) and combining the result with (2.32), we get¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) − π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) d x ′ + 14 π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B (cid:20) ∆ ′ (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ k + ℓ − it M ( h µν ) ( u ′ , x ′ ) d x ′ − π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B | x ′ | − x ′ j (cid:20) ∂ ′ j (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ k + ℓ − i +1 t M ( h µν ) ( u ′ , x ′ ) d x ′ − π ∞ X ℓ =0 ∞ X k =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ k + ℓ − i +1 t M ( h µν ) ( u ′ , x ′ ) d x ′ . (2.42)where we have written R R instead of the sum of R | x ′ | < R and R R < | x ′ | for the sake of brevity,and the surface integrals have not appeared due to ~ d σ ′ (1) being the opposite of ~ d σ ′ (2) at each point on the surface | x ′ | = R . Using (A.11), the third term on the RHS of theabove equation takes the form " = − π ∞ X ℓ =1 ∞ X k =0 ℓ X i =1 ℓ ! ( ℓ + i )!2 i ( i − ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ k + ℓ − i +1 t M ( h µν ) ( u ′ , x ′ ) d x ′ . (2.43)Further, by means of (A.2) and (A.8), one can rewrite the fourth term as " = − π ∞ X ℓ =1 ∞ X k =0 ℓ − ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ +2 ∂ k + ℓ +1 t M ( h µν ) ( u ′ , x ′ ) d x ′ lternative method for matching ... − π ∞ X ℓ =1 ∞ X k =0 ℓ X i =1 ℓ − ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ k + ℓ − i +1 t M ( h µν ) ( u ′ , x ′ ) d x ′ + 14 π ∞ X k =0 k )!! (2 k + 1)!! | x | k c k +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B | x ′ | − ∂ k +1 t M ( h µν ) ( u ′ , x ′ ) d x ′ + 14 π ∞ X ℓ =1 ∞ X k =0 ℓ + 1) ℓ ! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ +2 ∂ k + ℓ +1 t M ( h µν ) ( u ′ , x ′ ) d x ′ + 14 π ∞ X ℓ =1 ∞ X k =0 ℓ X i =1 ℓ ! ( ℓ + i + 1)!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ k + ℓ − i +1 t M ( h µν ) ( u ′ , x ′ ) d x ′ . (2.44)The fifth term on the RHS can also be rewritten as follows: " = − π ∞ X k =0 k )!! (2 k + 1)!! | x | k c k +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B | x ′ | − ∂ k +1 t M ( h µν ) ( u ′ , x ′ ) d x ′ − π ∞ X ℓ =1 ∞ X k =0 ℓ ! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ +2 ∂ k + ℓ +1 t M ( h µν ) ( u ′ , x ′ ) d x ′ − π ∞ X ℓ =1 ∞ X k =0 ℓ X i =1 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +2 ∂ k + ℓ − i +1 t M ( h µν ) ( u ′ , x ′ ) d x ′ . (2.45)Substituting (2.43)-(2.45) into (2.42), one finds that the sum of the last three terms onthe RHS is zero. Hence, we have¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) d x ′ . (2.46) lternative method for matching ... | x ′ | = R , which were brought aboutafter using Gauss’ and Green’s theorems. The first term on the RHS of (2.46) is obviouslya particular solution to (1.31) and the second term a solution to the homogeneousd’Alembertian equation (to see this, we just need to apply d’Alembertian operator tothe second term). Therefore, ¯ h µν ( t, x ) given in (2.46) is indeed a solution to (1.31), andit is legitimate provided that it fulfills the harmonic gauge condition.
3. Harmonic gauge condition
By applying ∂ µ to both sides of (2.46), we find ∂ µ ¯ h µν ( t, x ) ⋆ = 16 πGc (cid:26) ∂ µ FP B =0 A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + ∂ µ FP B =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21)(cid:27) − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × (cid:20) ∂ FP B =0 A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ ν ( M ( h )) ( u ′ , x ′ ) d x ′ + ∂ FP B =0 A Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ ν ( M ( h )) ( u ′ , x ′ ) d x ′ (cid:21) − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! ∂ j (cid:0) | x | k x L (cid:1) c k + ℓ − i × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ , (3.1)where to write the last term on the RHS, we have also used (A.8). We begin byexamining the first term on the RHS of the above equation. In the original domain ofanalyticity of R D i ( | x ′ | /r ) B [¯ τ µν ( t ′ , x ′ ) / | x − x ′ | ] d x ′ , where D i is as in (2.34), one can,by virtue of the Gauss’ theorem and the conservation equation, write ∂ µ A (cid:20) − π Z D i (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) = B (cid:20) − π Z D i (cid:18) | x ′ | r (cid:19) B | x ′ | − n ′ j ¯ τ jν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + 14 π Z ∂D i (cid:18) | x ′ | r (cid:19) B ¯ τ jν ( t ′ , x ′ ) | x − x ′ | d σ ′ ( i ) j . (3.2)Now, by reasoning analogous to what led us from (2.35) to (2.41), we get ∂ µ FP B =0 A (cid:20) − π Z D i (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) lternative method for matching ...
25= Residue B =0 A (cid:20) − π Z D i (cid:18) | x ′ | r (cid:19) B | x ′ | − n ′ j ¯ τ jν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + 14 π Z ∂D i ¯ τ jν ( t ′ , x ′ ) | x − x ′ | d σ ′ ( i ) j , (3.3)and hence, taking into account that ~ d σ ′ (1) = − ~ d σ ′ (2) at each point on the surface | x ′ | = R , the first term on the RHS of (3.1) takes the form " = 16 πGc Residue B =0 A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B | x ′ | − n ′ j ¯ τ jν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) + 16 πGc Residue B =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B | x ′ | − n ′ j ¯ τ jν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) = 16 πGc Residue B =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B | x ′ | − n ′ j ¯ τ jν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) = 16 πGc Residue B =0 A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B | x ′ | − n ′ j Λ jν (cid:0) ¯ h (cid:1) ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21) , (3.4)where Λ µν (cid:0) ¯ h (cid:1) is nothing but ¯Λ µν , and to write the second and third equalitieswe have respectively used the facts that the integral appearing in the first line is,because of the smoothness of ¯ τ µν ( t, x ) in the near zone, analytic at B = 0, and( − ¯ g ) ¯ T µν vanishes outside the near zone. Noting that outside the near zone we haveΛ µν (cid:0) ¯ h (cid:1) = Λ µν (cid:0) M (cid:0) ¯ h (cid:1)(cid:1) , due to the structure of Λ µν (cid:0) M (cid:0) ¯ h (cid:1)(cid:1) ( t, x ) being similar to thestructure of M (cid:0) ¯ h (cid:1) ( t, x ) in this region, in a manner analogous to the one employed toobtain (2.22), we can rewrite (3.4) as " = 14 π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × Residue B =0 A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 | x ′ | − n ′ j × ∂ k + ℓ − it Λ jν (cid:0) M (cid:0) ¯ h (cid:1)(cid:1) ( u ′ , x ′ ) d x ′ . (3.5)Replacing M (cid:0) ¯ h µν (cid:1) by M ( h µν ) with the use of the matching equation, by argumentssimilar to the ones following (2.25)-(2.27), we reach " ⋆ = 14 π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × Residue B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 | x ′ | − n ′ j × ∂ k + ℓ − it Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ . (3.6)Now we examine the second term on the RHS of (3.1). In the original domain ofanalyticity of R D i ( | x ′ | /r ) B (cid:2) ˆ x ′ L / | x ′ | ℓ + i +1 (cid:3) ∂ k + ℓ − it Λ ν ( M ( h )) ( u ′ , x ′ ) d x ′ , where D i is lternative method for matching ... ∂ A Z D i (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ ν ( M ( h )) ( u ′ , x ′ ) d x ′ = − ∂ k + ℓ − it Z D i (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 (cid:0) ∂ ′ j Λ jν ( M ( h )) ( u ′ , x ′ ) (cid:1) u ′ d x ′ , (3.7)where we have also used the conservation equation. Considering ∂ ′ j Λ jν ( M ( h )) ( u ′ , x ′ ) = − c | x ′ | − x ′ j ∂ t Λ jν ( M ( h )) ( u ′ , x ′ )+ (cid:0) ∂ ′ j Λ jν ( M ( h )) ( u ′ , x ′ ) (cid:1) u ′ , (3.8)by using the Gauss’ theorem, we get ∂ A Z D i (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ ν ( M ( h )) ( u ′ , x ′ ) d x ′ = B Z D i (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 | x ′ | − n ′ j ∂ k + ℓ − it Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ + Z D i (cid:18) | x ′ | r (cid:19) B (cid:20) ∂ ′ j (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ k + ℓ − it Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ − c Z D i (cid:18) | x ′ | r (cid:19) B x ′ j ˆ x ′ L | x ′ | ℓ + i +2 ∂ k + ℓ − i +1 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ − Z ∂D i (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ jν ( M ( h )) ( u ′ , x ′ ) d σ ′ ( i ) j . (3.9)Proceeding with the arguments analogous to those employed to derive (2.41), we reach ∂ FP B =0 A Z D i (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ ν ( M ( h )) ( u ′ , x ′ ) d x ′ = Residue B =0 A Z D i (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 | x ′ | − n ′ j ∂ k + ℓ − it Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ + FP B =0 A Z D i (cid:18) | x ′ | r (cid:19) B (cid:20) ∂ ′ j (cid:18) ˆ x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ k + ℓ − it Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ − c FP B =0 A Z D i (cid:18) | x ′ | r (cid:19) B x ′ j ˆ x ′ L | x ′ | ℓ + i +2 ∂ k + ℓ − i +1 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ − Z ∂D i ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ jν ( M ( h )) ( u ′ , x ′ ) d σ ′ ( i ) j . (3.10)This, together with the fact that ~ d σ ′ (1) is the opposite of ~ d σ ′ (2) anywhere on the surface | x ′ | = R , enable us to rewrite the second term on the RHS of (3.1) as " ⋆ = − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × Residue B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 | x ′ | − n ′ j ∂ k + ℓ − it Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ lternative method for matching ... − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B (cid:20) ∂ ′ j (cid:18) x ′ L | x ′ | ℓ + i +1 (cid:19) (cid:21) ∂ k + ℓ − it Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ + 14 π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B x ′ jL | x ′ | ℓ + i +2 ∂ k + ℓ − i +1 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ . (3.11)where to write the second and third terms, we have also used (A.8). By means of (A.2),(A.8) and (A.15) one can rewrite the second term on the RHS of (3.11) as " = − π ∞ X k =0 ∞ X ℓ =0 (2 ℓ + 1)!2 ℓ ( ℓ !) (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 3)!! | x | k ˆ x jL c k +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ +2 ∂ k +1 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ + 14 π ∞ X k =0 ∞ X ℓ =0 (2 ℓ + 1)!2 ℓ ( ℓ !) (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ jL | x ′ | ℓ +3 ∂ kt Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =1 ℓ ! ( ℓ + i )!2 i ( i − ℓ − i + 1)! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 3)!! | x | k ˆ x jL c k + ℓ − i +2 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − i +2 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ + 14 π ∞ X k =0 ∞ X ℓ =0 ℓ X i =1 ℓ ! ( ℓ + i )!2 i ( i − ℓ − i + 1)! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ jL | x ′ | ℓ + i +2 ∂ k + ℓ − i +1 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ . (3.12)Further, with the use of (A.15), the third term on the RHS of (3.11) takes the form " = 14 π ∞ X k =0 ∞ X ℓ =0 ℓ ! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ jL | x ′ | ℓ +2 ∂ k + ℓ +1 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ + 14 π ∞ X k =0 ∞ X ℓ =0 ℓ ! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 3)!! | x | k ˆ x jL c k + ℓ +2 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ +1 ∂ k + ℓ +2 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ lternative method for matching ...
28+ 14 π ∞ X k =0 ∞ X ℓ =0 (2 ℓ + 1)!2 ℓ ( ℓ !) (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 3)!! | x | k ˆ x jL c k +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ +2 ∂ k +1 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ + 14 π ∞ X k =0 ∞ X ℓ =0 ℓ X i =1 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ jL | x ′ | ℓ + i +2 ∂ k + ℓ − i +1 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ + 14 π ∞ X k =0 ∞ X ℓ =0 ℓ X i =1 ℓ ! ( ℓ + i + 1)!2 i i ! ( ℓ − i + 1)! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 3)!! | x | k ˆ x jL c k + ℓ − i +2 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − i +2 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ . (3.13)Finally, we examine the third term on the RHS of (3.1). Using (A.2), (A.8) and theformula for ˆ x ′ L x jL obtained through making the change x ↔ x ′ in (A.15), one can reach " = − π ∞ X k =0 ∞ X ℓ =0 ℓ ! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 3)!! | x | k ˆ x jL c k + ℓ +2 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ +1 ∂ k + ℓ +2 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ − π ∞ X k =0 ∞ X ℓ =0 ℓ ! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ jL | x ′ | ℓ +2 ∂ k + ℓ +1 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ − π ∞ X k =0 ∞ X ℓ =0 (2 ℓ + 1)!2 ℓ ( ℓ !) (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ jL | x ′ | ℓ +3 ∂ kt Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =1 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 3)!! | x | k ˆ x jL c k + ℓ − i +2 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − i +2 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =1 ℓ ! ( ℓ + i + 1)!2 i i ! ( ℓ − i + 1)! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i +1 × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ jL | x ′ | ℓ + i +2 ∂ k + ℓ − i +1 t Λ jν ( M ( h )) ( u ′ , x ′ ) d x ′ . (3.14)Now, substituting (3.12) and (3.13) into (3.11), and then substituting (3.6), (3.11) and(3.14) into (3.1), it is clear that ∂ µ ¯ h µν ( t, x ) = 0, or in other words, ¯ h µν ( t, x ) given by lternative method for matching ... B µνL ( t ), it isalso worth noting that, despite the appearance of R and r , ¯ h µν ( t, x ) can be shown tobe independent of these two constants. While the proof of the independence from R isstraightforward, the r -independence proof is more technical and we devote appendix Bto that.
4. Determination of the moments ˆ B µνL ( t )To solve the PN-to-PM matching problem completely, we must determine the momentsˆ B µνL ( t ). Let us first rewrite (2.46) as¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) + ∞ X k =0 ∞ X ℓ =0 c k (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L ∂ kt (cid:26) − πℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L × ℓ X i =0 ( ℓ + i )!2 i i ! ( ℓ − i )! 1 | x ′ | ℓ + i +1 ∂ ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) c ℓ − i d x ′ (cid:27) . (4.1)Comparing the above equation with (1.28), it can be seen that the second term on theRHS has been already written in the appropriate form. However, one needs to rewritethe first term. Assuming that the (1-variable) Taylor expansion of ¯ τ µν ( t, x ) about anyarbitrary point t has an infinite radius of convergence, we can write16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) = 16 πGc ∞ X k =0 c k ∂ kt FP B =0 A (cid:20) − π Z R (cid:18) | x ′ | r (cid:19) B | x − x ′ | k − (2 k )! ¯ τ µν ( t, x ′ ) d x ′ (cid:21) + 4 Gc FP B =0 A Z | x ′ | < | x | (cid:18) | x ′ | r (cid:19) B ∞ X k =0 | x − x ′ | k (2 k + 1)! ∂ k +1 t ¯ τ µν ( t, x ′ ) c k +1 d x ′ + 4 Gc FP B =0 A Z | x | < | x ′ | (cid:18) | x ′ | r (cid:19) B ∞ X k =0 | x − x ′ | k (2 k + 1)! ∂ k +1 t ¯ τ µν ( t, x ′ ) c k +1 d x ′ . (4.2)In the above equation, the first term on the RHS is in the appropriate form, however,the other two terms have to be rewritten. In order to do so, we rewrite the k sum bymeans of the Taylor expansion for functions of three variables. We have ∞ X k =0 | x − x ′ | k (2 k + 1)! ∂ k +1 t c k +1 = ∞ X k =0 ∞ X j =0 ( − j j ! (2 k + 1)! x ′ J ∂ J | x | k ∂ k +1 t c k +1 , | x ′ | < | x | , ∞ X k =0 ∞ X j =0 ( − j j ! (2 k + 1)! x J ∂ ′ J | x ′ | k ∂ k +1 t c k +1 , | x | < | x ′ | , (4.3)where the expressions appearing on the RHS have been obtained through the expansionsabout x and x ′ respectively. Since both of these expressions are of the same general lternative method for matching ... x ↔ x ′ to obtain the result for the other one. With the use of (A.10)-(A.13), we find ∞ X k =0 ∞ X j =0 ( − j j ! (2 k + 1)! x ′ J ∂ J | x | k ∂ k +1 t c k +1 = ∞ X ℓ =0 ∞ X m =0 ∞ X k =0 ( − ℓ ℓ ! (2 ℓ + 1)!! | x ′ | m ˆ x ′ L (2 m )!! (2 ℓ + 2 m + 1)!! × | x | k ˆ x L (2 k )!! (2 ℓ + 2 k + 1)!! ∂ k +2 ℓ +2 m +1 t c k +2 ℓ +2 m +1 . (4.4)As stated above, by making the change x ↔ x ′ , the expression for | x | < | x ′ | can also berewritten as ∞ X k =0 ∞ X j =0 ( − j j ! (2 k + 1)! x J ∂ ′ J | x ′ | k ∂ k +1 t c k +1 = ∞ X ℓ =0 ∞ X m =0 ∞ X k =0 ( − ℓ ℓ ! (2 ℓ + 1)!! | x | m ˆ x L (2 m )!! (2 ℓ + 2 m + 1)!! × | x ′ | k ˆ x ′ L (2 k )!! (2 ℓ + 2 k + 1)!! ∂ k +2 ℓ +2 m +1 t c k +2 ℓ +2 m +1 . (4.5)Changing the index of summation m to k in the above equation, and vice versa, onefinds out that the RHS of (4.4) is the same as that of (4.5). Therefore, regardless ofwhether | x ′ | is greater or less than | x | , we have ∞ X k =0 | x − x ′ | k (2 k + 1)! ∂ k +1 t c k +1 = ∞ X k =0 ∞ X ℓ =0 c k (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L × ∂ kt (cid:20) ( − ℓ ℓ ! ∞ X m =0 | x ′ | m ˆ x ′ L (2 m )!! (2 ℓ + 2 m + 1)!! ∂ ℓ +2 m +1 t c ℓ +2 m +1 (cid:21) . (4.6)Using the above equation, (4.2) takes the form16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) = 16 πGc ∞ X k =0 c k ∂ kt FP B =0 A (cid:20) − π Z R (cid:18) | x ′ | r (cid:19) B | x − x ′ | k − (2 k )! ¯ τ µν ( t, x ′ ) d x ′ (cid:21) + ∞ X k =0 ∞ X ℓ =0 c k (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L ∂ kt (cid:26) Gc ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B × ∞ X m =0 | x ′ | m ˆ x ′ L (2 m )!! (2 ℓ + 2 m + 1)!! ∂ ℓ +2 m +1 t ¯ τ µν ( t, x ′ ) c ℓ +2 m +1 d x ′ (cid:27) , (4.7)and hence, ¯ h µν ( t, x ) can be rewritten as¯ h µν ( t, x ) ⋆ = 16 πGc ∞ X k =0 c k ∂ kt FP B =0 A (cid:20) − π Z R (cid:18) | x ′ | r (cid:19) B | x − x ′ | k − (2 k )! ¯ τ µν ( t, x ′ ) d x ′ (cid:21) + ∞ X k =0 ∞ X ℓ =0 c k (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L ∂ kt (cid:26) Gc ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B × ∞ X m =0 | x ′ | m ˆ x ′ L (2 m )!! (2 ℓ + 2 m + 1)!! ∂ ℓ +2 m +1 t ¯ τ µν ( t, x ′ ) c ℓ +2 m +1 d x ′ (cid:27) lternative method for matching ... ∞ X k =0 ∞ X ℓ =0 c k (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L ∂ kt (cid:26) − πℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L × ℓ X i =0 ( ℓ + i )!2 i i ! ( ℓ − i )! 1 | x ′ | ℓ + i +1 ∂ ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) c ℓ − i d x ′ (cid:27) . (4.8)Comparing (1.28) and (4.8), we get ∞ X k =0 ∞ X ℓ =0 c k (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L ∂ kt ˆ B µνL ( t )= ∞ X k =0 ∞ X ℓ =0 c k (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L ∂ kt (cid:26) Gc ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B × ∞ X m =0 | x ′ | m ˆ x ′ L (2 m )!! (2 ℓ + 2 m + 1)!! ∂ ℓ +2 m +1 t ¯ τ µν ( t, x ′ ) c ℓ +2 m +1 d x ′ (cid:27) + ∞ X k =0 ∞ X ℓ =0 c k (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L ∂ kt (cid:26) − πℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L × ℓ X i =0 ( ℓ + i )!2 i i ! ( ℓ − i )! 1 | x ′ | ℓ + i +1 ∂ ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) c ℓ − i d x ′ (cid:27) . (4.9)We demand (4.9) hold for all x . Thus, for any k and ℓ we must have ∂ kt ˆ B µνL ( t ) = ∂ kt (cid:26) Gc ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B × ∞ X m =0 | x ′ | m ˆ x ′ L (2 m )!! (2 ℓ + 2 m + 1)!! ∂ ℓ +2 m +1 t ¯ τ µν ( t, x ′ ) c ℓ +2 m +1 d x ′ (cid:27) + ∂ kt (cid:26) − πℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L × ℓ X i =0 ( ℓ + i )!2 i i ! ( ℓ − i )! 1 | x ′ | ℓ + i +1 ∂ ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) c ℓ − i d x ′ (cid:27) . (4.10)Taking k = 0, we findˆ B µνL ( t )= 4 Gc ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ∞ X m =0 | x ′ | m ˆ x ′ L (2 m )!! (2 ℓ + 2 m + 1)!! ∂ ℓ +2 m +1 t ¯ τ µν ( t, x ′ ) c ℓ +2 m +1 d x ′ − πℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L ℓ X i =0 ( ℓ + i )!2 i i ! ( ℓ − i )! 1 | x ′ | ℓ + i +1 ∂ ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) c ℓ − i d x ′ . (4.11)The first term on the RHS of the above equation is past-zero since ¯ τ µν ( t, x ′ ) is past-stationary, and hence, due to 2 ℓ + 2 m + 1 ≥ ∂ ℓ +2 m +1 t ¯ τ µν ( t, x ′ ) past-zero. WritingΛ µν ( M ( h )) ( u ′ , x ′ ) as Λ µν AS ( M ( h )) ( x ′ )+Λ µν PZ ( M ( h )) ( u ′ , x ′ ), one can easily deduce that,since ∂ ℓ − it Λ µν AS ( M ( h )) ( x ) = 0 if i < ℓ , and if i = ℓ , we haveA Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 Λ µν AS ( M ( h )) ( x ′ ) d x ′ lternative method for matching ... − A Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 Λ µν AS ( M ( h )) ( x ′ ) d x ′ , (4.12)which is itself due to the particular structure of Λ µν AS ( M ( h )) ( x ), the always-stationaryconstituting part of the second term is actually zero. Furthermore, it is evident thatthe remaining constituting part of the second term is past-zero. In light of theseconsiderations, ˆ B µνL ( t ) is past-zero and can be rewritten asˆ B µνL ( t )= 4 Gc ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ∞ X m =0 | x ′ | m ˆ x ′ L (2 m )!! (2 ℓ + 2 m + 1)!! ∂ ℓ +2 m +1 t ¯ τ µν ( t, x ′ ) c ℓ +2 m +1 d x ′ − πℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L ℓ X i =0 ( ℓ + i )!2 i i ! ( ℓ − i )! 1 | x ′ | ℓ + i +1 ∂ ℓ − it Λ µν PZ ( M ( h )) ( u ′ , x ′ ) c ℓ − i d x ′ . (4.13)
5. Closed form of ¯ h µν ( t, x )To better compare the result given at the end of section 2 with that of [15], we nextobtain the closed form of the sum of the last two terms on the RHS of its equivalence,that is (4.8). To do so, we first need to find the closed form of ˆ B µνL ( t ). The first termon the RHS of (4.13) can be written as " = 1 c ℓ +1 (2 ℓ + 1)!! ∂ ℓ +1 t (cid:26) Gc ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L × (cid:20) ∞ X m =0 (2 ℓ + 1)!!2 m m ! (2 ℓ + 2 m + 1)!! (cid:18) | x ′ | c (cid:19) m ∂ mt ¯ τ µν ( t, x ′ ) (cid:21) d x ′ (cid:27) . (5.1)Since we have assumed that the Taylor expansion of ¯ τ µν ( t, x ) has an infinite radius ofconvergence about any arbitrary point t (this assumption first was stated above (4.2)),it is apparent that we can write ∞ X m =0 (2 ℓ + 1)!!2 m m ! (2 ℓ + 2 m + 1)!! (cid:18) | x ′ | c (cid:19) m ∂ mt ¯ τ µν ( t, x ′ ) = Z − f ( z )¯ τ µν ( t + z | x ′ | c , x ′ ) d z, (5.2)provided that f ( z ) is an even function of z fulfilling Z − f ( z ) z m d z = (2 ℓ + 1)!! (2 m )!2 m m ! (2 ℓ + 2 m + 1)!! . (5.3)We have(2 m )!2 m m ! (2 ℓ + 2 m + 1)!! = (2 m )! (2 ℓ + 2 m + 2)!!2 m m ! (2 ℓ + 2 m + 2)! = 2 ℓ + m +1 (2 m )! ( ℓ + m + 1)!2 m m ! (2 ℓ + 2 m + 2)!= 2 ℓ +1 (2 m )! m ! ( ℓ + m + 1)!(2 ℓ + 2 m + 2)! . (5.4)Taking into account (A.17) and (A.18), (5.4) takes the form(2 m )!2 m m ! (2 ℓ + 2 m + 1)!! = 2 ℓ +1 (cid:18) m Γ( m + ) √ π (cid:19) (cid:18) √ π ℓ + m +1 Γ( ℓ + m + ) (cid:19) lternative method for matching ...
33= 12 ℓ +1 Γ( m + )Γ( ℓ + m + ) = 12 ℓ +1 ℓ ! Γ( ℓ + 1)Γ( m + )Γ( ℓ + m + ) , (5.5)and by using (A.23), one reaches(2 m )!2 m m ! (2 ℓ + 2 m + 1)!! = 12 ℓ +1 ℓ ! Z t m − (1 − t ) ℓ d t = 22 ℓ +1 ℓ ! Z (cid:0) z (cid:1) m − (cid:0) − z (cid:1) ℓ z d z = 22 ℓ +1 ℓ ! Z z m (cid:0) − z (cid:1) ℓ d z = 12 ℓ +1 ℓ ! Z − z m (cid:0) − z (cid:1) ℓ d z. (5.6)Combining (5.3) and (5.6) and then comparing the LHS and the RHS of the resultantequation, f ( z ) can be found as f ( z ) = δ ℓ ( z ) = (2 ℓ + 1)!!2 ℓ +1 ℓ ! (cid:0) − z (cid:1) ℓ , (5.7)and hence, (5.1) becomes " = 1 c ℓ +1 (2 ℓ + 1)!! ∂ ℓ +1 t (cid:26) Gc ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L × Z − δ ℓ ( z )¯ τ µν ( t + z | x ′ | c , x ′ ) d z d x ′ (cid:27) = 1 c ℓ +1 (2 ℓ + 1)!! ∂ ℓ +1 t (cid:26) Gc ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L × Z − δ ℓ ( z )¯ τ µν ( t − z | x ′ | c , x ′ ) d z d x ′ (cid:27) , (5.8)where to obtain the final form we changed the integration variable from z to − z andthen used the fact that δ ℓ ( z ) is an even function of z . (5.8) is called the closed form of(5.1). Before examining the second term on the RHS of (4.13), first note that ∂∂z h P ( n ) ℓ ( z ) ∂ mt w ( t − z | x ′ | c , x ′ ) i = P ( n +1) ℓ ( z ) ∂ mt w ( t − z | x ′ | c , x ′ ) − | x ′ | c P ( n ) ℓ ( z ) ∂ m +1 t w ( t − z | x ′ | c , x ′ ) , (5.9)where w ( t, x ) is a general function and P ℓ ( z ) denotes the ℓ th-degree Legendrepolynomial. Integrating both sides of (5.9) with respect to z , we reach Z ba P ( n ) ℓ ( z ) ∂ m +1 t w ( t − z | x ′ | c , x ′ ) d z = − c | x ′ | P ( n ) ℓ ( z ) ∂ mt w ( t − z | x ′ | c , x ′ ) (cid:12)(cid:12)(cid:12) z = bz = a + c | x ′ | Z ba P ( n +1) ℓ ( z ) ∂ mt w ( t − z | x ′ | c , x ′ ) d z. (5.10)Since the LHS of the above equation and the integral appearing in the last term on itsRHS are of the same general form, one can write Z ba P ( n ) ℓ ( z ) ∂ m +1 t w ( t − z | x ′ | c , x ′ ) d z = − c | x ′ | P ( n ) ℓ ( z ) ∂ mt w ( t − z | x ′ | c , x ′ ) (cid:12)(cid:12)(cid:12) z = bz = a − (cid:18) c | x ′ | (cid:19) P ( n +1) ℓ ( z ) ∂ m − t w ( t − z | x ′ | c , x ′ ) (cid:12)(cid:12)(cid:12) z = bz = a lternative method for matching ... (cid:18) c | x ′ | (cid:19) Z ba P ( n +2) ℓ ( z ) ∂ m − t w ( t − z | x ′ | c , x ′ ) d z. (5.11)By repeating this process of substitution into the last term on the RHS, k times in all,we get Z ba P ( n ) ℓ ( z ) ∂ m +1 t w ( t − z | x ′ | c , x ′ ) d z = − k − X i =0 (cid:18) c | x ′ | (cid:19) i +1 P ( n + i ) ℓ ( z ) ∂ m − it w ( t − z | x ′ | c , x ′ ) (cid:12)(cid:12)(cid:12) z = bz = a + (cid:18) c | x ′ | (cid:19) k Z ba P ( n + k ) ℓ ( z ) ∂ m +1 − kt w ( t − z | x ′ | c , x ′ ) d z, (5.12)which, by taking m = ℓ , n = 0 and k = ℓ + 1 and noting that P ( j ) ℓ ( z ) is identically zeroif j > ℓ (due to P ℓ ( z ) being a polynomial of degree ℓ ), takes the form Z ba P ℓ ( z ) ∂ ℓ +1 t w ( t − z | x ′ | c , x ′ ) d z = − ℓ X i =0 (cid:18) c | x ′ | (cid:19) i +1 P ( i ) ℓ ( z ) ∂ ℓ − it w ( t − z | x ′ | c , x ′ ) (cid:12)(cid:12)(cid:12) z = bz = a . (5.13)Furthermore, providing that w ( t, x ) is a past-zero function, after choosing a = 1 and b → ∞ , we find Z ∞ P ℓ ( z ) ∂ ℓ +1 t w ( t − z | x ′ | c , x ′ ) d z = ℓ X i =0 (cid:18) c | x ′ | (cid:19) i +1 P ( i ) ℓ (1) ∂ ℓ − it w ( t − | x ′ | c , x ′ ) , (5.14)which by virtue of (A.27) becomes Z ∞ P ℓ ( z ) ∂ ℓ +1 t w ( t − z | x ′ | c , x ′ ) d z = c ℓ +1 | x ′ | ℓ ℓ X i =0 ( ℓ + i )!2 i i ! ( ℓ − i )! 1 | x ′ | ℓ + i +1 ∂ ℓ − it w ( t − | x ′ | c , x ′ ) c ℓ − i , (5.15)or equivalently, ℓ X i =0 ( ℓ + i )!2 i i ! ( ℓ − i )! 1 | x ′ | ℓ + i +1 ∂ ℓ − it w ( t − | x ′ | c , x ′ ) c ℓ − i = 1 c ℓ +1 ∂ ℓ +1 t (cid:20) | x ′ | ℓ Z ∞ P ℓ ( z ) w ( t − z | x ′ | c , x ′ ) d z (cid:21) = 1 c ℓ +1 ∂ ℓ +1 t (cid:20) ( − ℓ ℓ ℓ ! | x ′ | ℓ Z ∞ h d ℓ d z ℓ (cid:0) − z (cid:1) ℓ i w ( t − z | x ′ | c , x ′ ) d z (cid:21) , (5.16)where to obtain the last equality we have used the Rodrigues’ formula (A.25).Integrating by parts ℓ times and noting that we have assumed w ( t, x ) is a past-zerofunction and the n th derivative of ( z − ℓ vanishes at z = 1 for n < ℓ , the integral inthe last line of (5.16) can be expressed as Z ∞ h d ℓ d z ℓ (cid:0) − z (cid:1) ℓ i w ( t − z | x ′ | c , x ′ ) d z = (cid:18) | x ′ | c (cid:19) ℓ Z ∞ (cid:0) − z (cid:1) ℓ ∂ ℓt w ( t − z | x ′ | c , x ′ ) d z. (5.17) lternative method for matching ... ℓ X i =0 ( ℓ + i )!2 i i ! ( ℓ − i )! 1 | x ′ | ℓ + i +1 ∂ ℓ − it w ( t − | x ′ | c , x ′ ) c ℓ − i = 1 c ℓ +1 ∂ ℓ +1 t Z ∞ (1 − z ) ℓ ℓ ℓ ! w ( t − z | x ′ | c , x ′ ) d z = 1 c ℓ +1 (2 ℓ + 1)!! ∂ ℓ +1 t (cid:26) ( − ℓ Z ∞ δ ℓ ( z ) w ( t − z | x ′ | c , x ′ ) d z (cid:27) , (5.18)and by means of (5.18), one finds that the closed form of the second term on the RHSof (4.13) reads " = 1 c ℓ +1 (2 ℓ + 1)!! ∂ ℓ +1 t (cid:26) π ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L × Z ∞ γ ℓ ( z )Λ µν PZ ( M ( h )) ( t − z | x ′ | c , x ′ ) d z d x ′ (cid:27) , (5.19)where γ ℓ ( z ) = − δ ℓ ( z ). However, in order to use the closed form (5.19) in computations,we must rewrite it in terms of Λ µν ( M ( h )) = Λ µν AS ( M ( h )) + Λ µν PZ ( M ( h )). Such a closedform is obtainable. In order to derive it, first note that, although R ∞ γ ℓ ( z ) d z is divergentand hence meaningless, FP j = ℓ A R ∞ γ j ( z ) d z , where j ∈ C and γ j ( z ) = ( − j +1 j Γ(2 j + 2)[Γ( j + 1)] (cid:0) z − (cid:1) j , (5.20)in which Γ denotes the gamma function defined in appendix A, is well-defined, andby being well-defined we mean that R ∞ γ j ( z ) d z can be analytically continued downto some neighborhood of any point with j = ℓ ∈ N . To see this, first we need toshow that there exists a region in the complex plane where R ∞ γ j ( z ) d z is defined,and further, is analytic (if instead of (5.20), we had considered the former expressionfor γ j ( z ), i.e., ( − j +1 [(2 j + 1)!! / j j !] ( z − j , which is identical to the newly defined γ j ( z ) for j ∈ N , R ∞ γ j ( z ) d z would have been defined nowhere in the complex planedue to the divergence for j ∈ N and (2 j + 1)!! and j ! being ill-defined for j ∈ C − N ).Making the change t = 1 /z , we reach Z ∞ γ j ( z ) d z = ( − j +1 j +1 Γ(2 j + 2)[Γ( j + 1)] Z (1 − t ) j t − j − d t = ( − j +1 j +1 Γ(2 j + 2)[Γ( j + 1)] Z t ( − j − ) − (1 − t ) ( j +1) − d t. (5.21)Taking j to be in the strip − < Re( j ) < − , the integral appearing on the RHS of thelast equality in the above equation is nothing but B ( − j − , j + 1) ( B denotes the betafunction defined in appendix A). With the use of (A.23), (5.21) can be rewritten as Z ∞ γ j ( z ) d z = ( − j +1 j +1 Γ(2 j + 2)[Γ( j + 1)] Γ( − j − )Γ( j + 1)Γ( ) lternative method for matching ...
36= ( − j +1 j +1 √ π Γ(2 j + 2)Γ( j + 1) Γ( − j −
12 ) . (5.22)Since the gamma function is everywhere-analytic and nowhere-zero in its originaldomain of definition, from (5.22) it can be deduced that R ∞ γ j ( z ) d z is analytic inthe aforementioned vertical strip. By means of (A.21), one can rewrite (5.22) as Z ∞ γ j ( z ) d z = ( − j +1 j +1 √ π " Γ(2 j + 2 + 2 n ) Q n − i =0 (2 j + 2 + i ) Γ( j + 1 + n ) Q n − m =0 ( j + 1 + m ) − × " Γ( − j − + n ) Q n − k =0 (cid:0) − j − + k (cid:1) = ( − j +1 j +1 √ π " Γ(2 j + 2 + 2 n ) (cid:2)Q n − i =0 (2 j + 3 + 2 i ) (cid:3) (cid:2)Q n − i =0 (2 j + 2 + 2 i ) (cid:3) × " n Γ( j + 1 + n ) Q n − m =0 (2 j + 2 + 2 m ) − " n Γ( − j − + n )( − n Q n − k =0 (2 j + 1 − k ) = ( − j +1+ n j +1 √ π Q n − m = − n (2 j + 3 + 2 m ) Γ(2 j + 2 + 2 n )Γ( j + 1 + n ) Γ( − j −
12 + n ) . (5.23)By virtue of the identity theorem, the equality between the analytic continuations of R ∞ γ j ( z ) d z and the RHS of the last equality in (5.23) must hold wherever they areboth defined. Therefore, we haveA Z ∞ γ j ( z ) d z = ( − j +1+ n j +1 √ π Q n − m = − n (2 j + 3 + 2 m ) AΓ(2 j + 2 + 2 n )AΓ( j + 1 + n ) AΓ( − j −
12 + n ) , (5.24)which in the strip − n − < Re( j ) < − + n readsA Z ∞ γ j ( z ) d z = ( − j +1+ n j +1 √ π Q n − m = − n (2 j + 3 + 2 m ) Γ(2 j + 2 + 2 n )Γ( j + 1 + n ) Γ( − j −
12 + n ) . (5.25)As it can be seen, in this strip, which is n units wider from each side in comparison withthe original striplike domain of analyticity of R ∞ γ j ( z ) d z , A R ∞ γ j ( z ) d z has 2 n simplepoles at the points with j half-integer (namely at j = − m − with − n ≤ m ≤ n − j integer. Therefore,taking n = ℓ + 1 (which is also compatible with the implicit requirement n ≥ j = ℓ A R ∞ γ j ( z ) d z is well-defined as claimed earlier, and moreover, since A R ∞ γ j ( z ) d z is analytic at j = ℓ , we findFP j = ℓ A Z ∞ γ j ( z ) d z = A Z ∞ γ j ( z ) d z (cid:12)(cid:12)(cid:12) j = ℓ = ( − ℓ +2 ℓ +1 √ π Q ℓm = − ℓ − (2 ℓ + 3 + 2 m ) Γ(4 ℓ + 4)Γ(2 ℓ + 2) Γ( 12 )= 12 ℓ +1 (4 ℓ + 3)!! (4 ℓ + 3)!(2 ℓ + 1)! = (4 ℓ + 2)!!2 ℓ +1 (2 ℓ + 1)! = (4 ℓ + 2)!!(4 ℓ + 2)!! = 1 . (5.26) lternative method for matching ... B =0 A R R ( | x ′ | /r ) B ˆ x ′ L R ∞ γ ℓ ( z )Λ µν AS ( M ( h )) ( x ′ ) d z d x ′ is notdefined, it is now obvious FP B =0 A R R ( | x ′ | /r ) B ˆ x ′ L FP j = ℓ A R ∞ γ j ( z )Λ µν AS ( M ( h )) ( x ′ ) d z d x ′ is well-defined, and in factFP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L FP j = ℓ A Z ∞ γ j ( z )Λ µν AS ( M ( h )) ( x ′ ) d z d x ′ = FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L Λ µν AS ( M ( h )) ( x ′ ) FP j = ℓ A Z ∞ γ j ( z ) d z d x ′ = FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L Λ µν AS ( M ( h )) ( x ′ ) d x ′ = FP B =0 A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L Λ µν AS ( M ( h )) ( x ′ ) d x ′ + FP B =0 A Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B ˆ x ′ L Λ µν AS ( M ( h )) ( x ′ ) d x ′ = FP B =0 A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L Λ µν AS ( M ( h )) ( x ′ ) d x ′ − FP B =0 A Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L Λ µν AS ( M ( h )) ( x ′ ) d x ′ = 0 , (5.27)where we could write the fourth equality due to the particular structure ofΛ µν AS ( M ( h )) ( x ). Further, it can be proven that R ∞ γ j ( z )Λ µν PZ ( M ( h )) ( t − z | x ′ | /c, x ′ ) d z is analytic in the half-plane Re( j ) > − µν PZ ( M ( h )) ( t, x ) being a past-zerofunction. Hence, we haveFP j = ℓ A Z ∞ γ j ( z )Λ µν PZ ( M ( h )) ( t − z | x ′ | c , x ′ ) d z = FP j = ℓ Z ∞ γ j ( z )Λ µν PZ ( M ( h )) ( t − z | x ′ | c , x ′ ) d z = Z ∞ γ ℓ ( z )Λ µν PZ ( M ( h )) ( t − z | x ′ | c , x ′ ) d z. (5.28)With the use of (5.27) and (5.28), we getFP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L Z ∞ γ ℓ ( z )Λ µν PZ ( M ( h )) ( t − z | x ′ | c , x ′ ) d z d x ′ = FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L FP j = ℓ A Z ∞ γ j ( z )Λ µν ( M ( h )) ( t − z | x ′ | c , x ′ ) d z d x ′ . (5.29)Thus, the closed form (5.19) takes the form " = 1 c ℓ +1 (2 ℓ + 1)!! ∂ ℓ +1 t (cid:26) π ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L × FP j = ℓ A Z ∞ γ j ( z )Λ µν ( M ( h )) ( t − z | x ′ | c , x ′ ) d z d x ′ (cid:27) . (5.30) lternative method for matching ... B µνL ( t )as ˆ B µνL ( t ) = 1 c ℓ +1 (2 ℓ + 1)!! ∂ ℓ +1 t (cid:26) Gc ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L × Z − δ ℓ ( z )¯ τ µν ( t − z | x ′ | c , x ′ ) d z d x ′ + 14 π ( − ℓ ℓ ! FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L × FP j = ℓ A Z ∞ γ j ( z )Λ µν ( M ( h )) ( t − z | x ′ | c , x ′ ) d z d x ′ (cid:27) . (5.31)Now note that we can write ∞ X k =0 ∞ X ℓ =0 c k (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L ∂ kt ˆ B µνL ( t )= ∞ X ℓ =0 ˆ ∂ L ˆ U µνL ( t − | x | c ) + ˆ V µνL ( t + | x | c ) | x | ! , (5.32)which is due to the LHS being a solution to the homogeneous d’Alembertian equationwhose most general solution is given as on the RHS [5]. In order to determineˆ U µνL ( t − | x | /c ) and ˆ V µνL ( t + | x | /c ) corresponding to the LHS, the following two propertiesmust be taken into account:(i) Retardation effects are small in the near zone.(ii) The solution is smooth in the near zone.The property (i) allows us to use the Taylor expansion. Using it as well as (A.12) and(A.13), we get ∞ X ℓ =0 ˆ ∂ L ˆ U µνL ( t − | x | c ) + ˆ V µνL ( t + | x | c ) | x | ! = ∞ X k =0 h (2 k ) ˆ U µν ( t ) + (2 k ) ˆ V µν ( t ) i c k (2 k )! | x | k − + ∞ X k =0 ∞ X ℓ =1 h (2 k ) ˆ U µνL ( t ) + (2 k ) ˆ V µνL ( t ) i c k (2 k )! (2 k −
1) (2 k − · · · (2 k − ℓ + 1) ˆ n L | x | k − ℓ − − ∞ X k =0 ∞ X ℓ =0 h (2 k +2 ℓ +1) ˆ U µνL ( t ) − (2 k +2 ℓ +1) ˆ V µνL ( t ) i c k +2 ℓ +1 (2 k + 2 ℓ + 1)!! | x | k ˆ x L (2 k )!! . (5.33)Obviously, the property (ii) requires that each of the first two terms on the RHS of theabove equation vanishes. Therefore, we must have ˆ V µνL ( t ) = − ˆ U µνL ( t ), and hence,ˆ B µνL ( t ) = 1 c ℓ +1 (2 ℓ + 1)!! ∂ ℓ +1 t h − U µνL ( t ) i . (5.34) lternative method for matching ... U µνL ( t ) = − Gc ( − ℓ ℓ ! ˆ A µνL ( t )2 + ˆ C µνL ( t ) , (5.35)where ˆ C µνL ( t ) is a constant STF tensor, andˆ A µνL ( t ) = ˆ F µνL ( t ) + ˆ R µνL ( t ) , (5.36)ˆ F µνL ( t ) = FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L Z − δ ℓ ( z )¯ τ µν ( t − z | x ′ | c , x ′ ) d z d x ′ , (5.37)ˆ R µνL ( t ) = FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L FP j = ℓ A Z ∞ γ j ( z ) M ( τ µν ) ( t − z | x ′ | c , x ′ ) d z d x ′ , (5.38)where in (5.38), by M ( τ µν ) we mean [ c / πG ] Λ µν ( M ( h )). All in all, ¯ h µν ( t, x ) givenby (2.46) can be written as¯ h µν ( t, x ) ⋆ = 16 πGc ∞ X k =0 c k ∂ kt FP B =0 A (cid:20) − π Z R (cid:18) | x ′ | r (cid:19) B | x − x ′ | k − (2 k )! ¯ τ µν ( t, x ′ ) d x ′ (cid:21) − Gc ∞ X ℓ =0 ( − ℓ ℓ ! ˆ ∂ L ˆ A µνL ( t − | x | c ) − ˆ A µνL ( t + | x | c )2 | x | ! , (5.39)which is in full agreement with the result derived in [15]. As a bonus, since the secondterm on the RHS of (4.13), whose closed form is given by (5.30), corresponds only tothe second term on the RHS of (4.1), and vice versa, we can also write¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) − Gc ∞ X ℓ =0 ( − ℓ ℓ ! ˆ ∂ L ˆ R µνL ( t − | x | c ) − ˆ R µνL ( t + | x | c )2 | x | ! , (5.40)which is nothing but the result stated in [18].
6. Summary
Starting with a trivial equality, by virtue of a specific process of analytic continuation,we derived an expression for the post-Newtonian approximation of h µν ( t, x ). We thendemonstrated that this approximate solution to the Einstein field equation satisfies theharmonic gauge condition. Finally, we obtained the closed form of the stated solution,mainly by means of the properties of the gamma and beta functions, and in this way,verified the compatibility of the result of this paper with that of the 2002 paper byPoujade and Blanchet. Acknowledgments
The author would like to express his profound gratitude toward Nahid Ahmadi forhelpful discussions. lternative method for matching ... Appendix A. A collection of useful definitions, lemmas and formulae
A.1. Basic formulae ∆ F ( t − | x | c ) | x | ! = 1 c F (2) ( t − | x | c ) | x | . (A.1) ∂ k x L = ( , ℓ = 0, δ k { i x i ...i ℓ } , ℓ ≥
1. (A.2) ∂∂B (cid:18) | x | r (cid:19) B = (cid:18) | x | r (cid:19) B ln | x | r . (A.3) f ( t + a ) = ∞ X j =0 j ! a j f ( j ) ( t ) . ( ) (A.4) f ( x + a ) = ∞ X j =0 j ! a J ∂ J f ( x ) . ( ) (A.5) A.2. Symmetric-trace free Cartesian tensorsDefinition A.1.
A Cartesian tensor is called symmetric-trace-free if its any singlecontraction vanishes. We denote (the components of) a symmetric-trace-free Cartesiantensor of rank q by ˆ T Q or T . Definition A.2.
The symmetric-trace-free part of a general tensor T Q is defined asˆ T Q = [ q ] X m =0 a qm δ ( i i · · · δ i m − i m T ( i m +1 ...i q )) a a ...a m a m , (A.6)where a qm = ( − m q !( q − m )! (2 q − m − q − m )!! . (A.7)Two general Cartesian tensors F L and G L satisfy the following formula:ˆ F L ˆ G L = ˆ F L G L . (A.8)Further, two important lemmas containing symmetric-trace-free tensors (whose proofsare given in [16]) are as follows. Lemma A.1.
When | x | >
0, the integral R | x ′ | < R (cid:2) ˆ n ′ Q | x ′ | a (ln | x ′ | ) p g ( t, x ′ ) / | x − x ′ | (cid:3) d x ′ with 0 < R < ∞ , q ∈ N , a ∈ R , p ∈ R ≥ and g ( t, x ) a bounded function in R convergesif a > − q − Lemma A.2.
When | x | < ∞ , the integral R R < | x ′ | (cid:2) ˆ n ′ Q | x ′ | a (ln | x ′ | ) p g ( t, x ′ ) / | x − x ′ | (cid:3) d x ′ with 0 < R < ∞ , q ∈ N , a ∈ R , p ∈ R ≥ and g ( t, x ) a bounded function in R convergesif a < q − lternative method for matching ... n L = [ ℓ ] X k =0 (2 ℓ − k + 1)!!(2 ℓ − k + 1)!! δ { i i · · · δ i k − i k ˆ n i k +1 ...i ℓ } , (A.9) ∂ L = [ ℓ ] X k =0 (2 ℓ − k + 1)!!(2 ℓ − k + 1)!! δ { i i · · · δ i k − i k ˆ ∂ i k +1 ...i ℓ } ∆ k , (A.10)∆ (cid:2) ˆ n Q | x | a +2 (cid:3) = ( a − q + 2) ( a + q + 3) ˆ n Q | x | a , (A.11)ˆ ∂ L | x | λ = λ ( λ − · · · ( λ − ℓ + 2) ˆ n L | x | λ − ℓ , (A.12)ˆ ∂ L | x | j = 0 if j = 0 , , , ..., ℓ − , (A.13)ˆ ∂ L F ( t − | x | c ) | x | ! = ( − ℓ ˆ n L ℓ X j =0 ( ℓ + j )!2 j j ! ( ℓ − j )! F ( ℓ − j ) ( t − | x | c ) c ℓ − j | x | j +1 , (A.14)and last but not least,ˆ x L x ′ jL = ˆ x L [ ℓ +12 ] X k =0 (2 ( ℓ + 1) − k + 1)!!(2 ( ℓ + 1) − k + 1)!! δ { i i · · · δ i k − i k ˆ x ′ i k +1 ...i ℓ j } | x ′ | k = ˆ x L (cid:20) (2 ℓ + 3)!!(2 ℓ + 3)!! ˆ x ′ jL + (2 ℓ − ℓ + 1)!! δ j { i ˆ x ′ i ...i ℓ } | x ′ | (cid:21) = ˆ x L ˆ x ′ jL + ℓ ℓ + 1 ˆ x jL − ˆ x ′ L − | x ′ | . (A.15) A.3. Gamma and beta functionsDefinition A.3.
The gamma function is defined asΓ( x ) = Z ∞ t x − e − t d t for Re( x ) > . (A.16)One can showΓ( n ) = ( n − n ∈ Z ≥ , (A.17)Γ( n + 12 ) = (2 n )!4 n n ! √ π for n ∈ Z ≥ . (A.18)Moreover, by integrating by parts, one obtainsΓ( x + 1) = Z ∞ t x e − t d t = − t x e − t (cid:12)(cid:12)(cid:12) ∞ + Z ∞ xt x − e − t d t = x Γ( x ) for Re( x ) > , (A.19)or equivalently,Γ( x ) = Γ( x + 1) x . (A.20)Repeatedly using the above equation, n times in all, one getsΓ( x ) = Γ( x + n ) x ( x + 1) · · · ( x + n −
1) = Γ( x + n ) Q n − k =0 ( x + k ) . (A.21) lternative method for matching ... Definition A.4.
The beta function is defined as B ( x, y ) = Z t x − (1 − t ) y − d t for Re( x ) > y ) > . (A.22)It can be shown B ( x, y ) = Γ( x )Γ( y )Γ( x + y ) . (A.23) A.4. Other mathematical relationsKirchhoff ’s no-incoming radiation condition.
A general function f ( t, x ) fulfills thiscondition if [19] lim | x |→∞ t + | x | c =const (cid:20) ∂∂ | x | ( | x | f ( t, x )) + 1 c ∂∂t ( | x | f ( t, x )) (cid:21) = 0 . (A.24) Rodrigues’ formula.
It is a formula for Legendre polynomials given by P ℓ ( z ) = 12 ℓ ℓ ! d ℓ d z ℓ (cid:0) z − (cid:1) ℓ . (A.25)By means of the Rodrigues’ formula and the Leibniz’s formula for the n th derivative ofa product of more than two factors, which reads( f f · · · f m ) ( n ) = X k ,k ,...,k m k + k + ··· + k m = n n ! k ! k ! · · · k m ! m Y j =1 f ( k j ) j , (A.26)we can show P ( i ) ℓ (1) = ( ℓ + i )!2 i i ! ( ℓ − i )! , i ≤ ℓ ,0 , i > ℓ . (A.27)The expression for i > ℓ comes from the fact that P ℓ ( z ) is a ℓ th-degree polynomial . Inorder to obtain the expression for i ≤ ℓ , first we note that P ( i ) ℓ ( z ) = 12 ℓ ℓ ! d ℓ + i d z ℓ + i (cid:0) z − (cid:1) ℓ . (A.28)Taking n = ℓ + i , m = ℓ and f = f = ... = f m = z − P ( i ) ℓ ( z ) = 12 ℓ ℓ ! X k ,k ,...,k ℓ k + k + ··· + k ℓ = ℓ + i ( ℓ + i )! k ! k ! · · · k ℓ ! ℓ Y j =1 (cid:0) z − (cid:1) ( k j ) . (A.29)From the above equation, it is obvious that only the terms with all k j ’s fulfilling1 ≤ k j ≤ P ( i ) ℓ (1). This, together with the fact that the number of k j ’s is ℓ , lead us to deduce from the constraint P ℓj =1 k j = ℓ + i that in each nonzero term of thesum, exactly ( ℓ − i ) first derivatives and i second derivatives appear. Therefore, for allthese terms 1 / ( k ! k ! · · · k ℓ !) equals 1 / i . Further, it is clear that Q ℓj =1 ( z − ( k j ) | z =1 lternative method for matching ... ℓ , and hence, each of the nonzero terms contributingto the sum has the value of 2 ℓ − i ( ℓ + i )!. This makes it possible to evaluate the sumby multiplying the number of its nonzero terms by 2 ℓ − i ( ℓ + i )!. The problem of findingthat number is equivalent to the problem of counting the number of ways in which onecan put i indistinguishable objects in ℓ distinguishable boxes ( i ≤ ℓ ). The answer is (cid:0) ℓi (cid:1) ,and therefore, P ( i ) ℓ (1) = 12 ℓ ℓ ! (cid:20) ℓ i (cid:18) ℓi (cid:19) ( ℓ + i )! (cid:21) = ( ℓ + i )!2 i i ! ( ℓ − i )! . (A.30) Appendix B. Independence of ¯ h µν ( t, x ) from r To prove the independence from r , we need to show that the RHS of (2.46) equalsthe same expression with r replaced by another arbitrary constant r . Making thesubstitution ( | x ′ | /r ) B = ( | x ′ | /r ) B ( r /r ) B and using the Taylor expansion of ( r /r ) B ,we get¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) + 16 πGc FP B =0 (cid:26)(cid:18) ∞ X j =1 j ! (cid:18) ln r r (cid:19) j B j (cid:19) · A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21)(cid:27) − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) d x ′ − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 (cid:26)(cid:18) ∞ X j =1 j ! (cid:18) ln r r (cid:19) j B j (cid:19) · A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 × ∂ k + ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) d x ′ (cid:27) , (B.1)which, after using¯ τ µν ( t, x ) = c πG ¯Λ µν ( t, x ) = c πG Λ µν (cid:0) ¯ h (cid:1) ( t, x ) = c πG Λ µν (cid:0) M (cid:0) ¯ h (cid:1)(cid:1) ( t, x ) , (B.2)whose domain of validity is outside the near zone, takes the form¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i lternative method for matching ... × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) d x ′ + 16 πGc FP B =0 (cid:26)(cid:18) ∞ X j =1 j ! (cid:18) ln r r (cid:19) j B j (cid:19) · A (cid:20) − π Z | x ′ | < R (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21)(cid:27) + FP B =0 (cid:26)(cid:18) ∞ X j =1 j ! (cid:18) ln r r (cid:19) j B j (cid:19) · A (cid:20) − π Z R < | x ′ | (cid:18) | x ′ | r (cid:19) B Λ µν (cid:0) M (cid:0) ¯ h (cid:1)(cid:1) ( t ′ , x ′ ) | x − x ′ | d x ′ (cid:21)(cid:27) − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 (cid:26)(cid:18) ∞ X j =1 j ! (cid:18) ln r r (cid:19) j B j (cid:19) · A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 × ∂ k + ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) d x ′ (cid:27) , (B.3)The third term on the RHS of the above equation is zero due to the near-zone integralappearing in it being analytic at B = 0 and having the coefficients B k with k ≥ " = 14 π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 (cid:26)(cid:18) ∞ X j =1 j ! (cid:18) ln r r (cid:19) j B j (cid:19) · A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 × ∂ k + ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) d x ′ (cid:27) . (B.4)Thus, we have¯ h µν ( t, x ) ⋆ = 16 πGc FP B =0 A ✷ − (cid:20) (cid:18) | x ′ | r (cid:19) B ¯ τ µν ( t ′ , x ′ ) (cid:21) − π ∞ X k =0 ∞ X ℓ =0 ℓ X i =0 ℓ ! ( ℓ + i )!2 i i ! ( ℓ − i )! (2 ℓ + 1)!!(2 k )!! (2 ℓ + 2 k + 1)!! | x | k ˆ x L c k + ℓ − i × FP B =0 A Z R (cid:18) | x ′ | r (cid:19) B ˆ x ′ L | x ′ | ℓ + i +1 ∂ k + ℓ − it Λ µν ( M ( h )) ( u ′ , x ′ ) d x ′ , (B.5)which means, as stated earlier, ¯ h µν ( t, x ) is independent of r . References [1] Einstein A 1916
Sitzungsber. K. Preuss. Akad. Wiss.
Sitzungsber. K. Preuss. Akad. Wiss.
Phil. Trans. R. Soc. Lond. A Rev. Mod. Phys. Phil. Trans. R. Soc. Lond. A lternative method for matching ... [6] Kerlick G D 1980 Gen. Relativ. Gravit. Gen. Relativ. Gravit. Theory of Space Time and Gravitation
Living Rev. Relativity Bulletin de la S. M. F. Ann. Inst. Henri Poincar´e ( Phys. Th´eor. ) Phys. Rev. D Phys. Rev. D Class. Quantum Grav. Phys. Rev. D Gravitational Waves vol 1
Theory and Experiments (New York: OxfordUniversity Press) p 257[18] Blanchet L Faye G and Nissanke S 2005