6.1 A brief summary – Before continuing

Very briefly, for the purpose of organizing the many strands so far developed, we summarize our procedure for finding the complex center of mass. We begin with the gravitational radiation data, the Bondi shear, 0 ¯ σ (uB,ζ, ζ) and solve the good-cut equation,
∂2Z = σ0(Z,ζ, ¯ζ),

with solution a uB = Z (z ,ζ, ¯ζ) and the four complex parameters a z defining the solution space. Next we consider an arbitrary complex world line in the solution space, za = ξa(τ) = (ξ0(τ),ξi(τ)), so that uB = Z(ξa(τ),ζ, ¯ζ) = G(τ,ζ,ζ¯), a GCF, which can be expanded in spherical harmonics as

uB = G (τ,ζ, ¯ζ) = ξa(τ)ˆla(ζ,ζ¯) + ξij(τ)Y02ij + ... (6.1 ) ξ0(τ) 1 = -√--- − --ξi(τ)Y 01i + ξij(τ)Y 02ij + ... 2 2
Assuming slow motion and the gauge condition ξ0(τ ) = τ (see Section 4), we have
-τ-- 1-i 0 ij 0 uB = √2-− 2ξ (τ)Y1i + ξ (τ)Y2ij + ... (6.2 )
(Though the world line is arbitrary, the quadrupole term, ξij(τ ), and higher harmonics, are determined by both the Bondi shear and the world line.)

The inverse function,

τ = T(uret,ζ, ¯ζ), (6.3 ) √ -- uret = 2uB,
can be found by the following iteration process [40]: writing Eq. (6.2View Equation) as
¯ τ = uret + F (τ,ζ,ζ), (6.4 )
with
√ -- 2 √ -- F (τ, ζ, ¯ζ) =---ξi(τ)Y10i(ζ, ¯ζ) − 2 ξij(τ)Y10ij(ζ,ζ¯) + ..., (6.5 ) 2
the iteration relationship, with the zeroth-order iterate, τ0 = uret, is
¯ τn = uret + F (τn−1,ζ,ζ). (6.6 )
To second order, this is
( -- -) τ = T(uret,ζ, ¯ζ) = uret + F uret + F (uret,ζ,ζ ),ζ, ζ ≈ uret + F + F ∂uretF,

but for most of our calculations, all that is needed is the first iterate, given by

√ -- 2 √ -- τ = T(uret,ζ, ¯ζ) = uret +---ξi(uret)Y10i(ζ, ¯ζ) − 2ξij(uret)Y 01ij(ζ, ¯ζ). (6.7 ) 2
This relationship is, in principle, an important one.

We also have the linearized reality relations – easily found earlier or from Eq. (6.7View Equation):

τ = s + iλ, √ -- (6.8 ) -- --2-i 0 √ --ij 0 λ = Λ(s,ζ,ζ ) = 2 ξI(s)Y1i − 2ξI (s)Y2ij, (6.9 ) ( √ -- ) τ = s + i --2ξi(s)Y 0− √2-ξij(s)Y0 , (6.10 ) 2 I 1i I 2ij √ -- (R) √ -- -- √ --(R) --2-i 0 √ --ij 0 uret = 2GR (s,ζ,ζ) = 2uB = s − 2 ξR (s )Y 1i + 2ξR(s)Y2ij. (6.11 )

The associated angle field, L, and the Bondi shear, 0 σ, are given parametrically by

¯ ¯ L (uB, ζ,ζ) = ∂(τ)G(τ,ζ, ζ) (6.12 ) = ξi(τ)Y11i − 6 ξij(τ)Y21ij + ...
and
0 2 σ (uB,ζ, ¯ζ) = ∂(τ)G (τ,ζ, ¯ζ), (6.13 ) = 24ξij(τ)Y 2 + ..., 2ij
using the inverse to uB = G (τ,ζ,ζ¯), Eq. (6.7View Equation). The asymptotically shear-free NGC is given by performing the null rotation
-- - - l∗a = la + bma + bma + bbna, (6.14 ) m ∗a = ma + bna, ∗a a n = n , b = − L ∕r + O (r −2).

As stated in Eqs. (4.6View Equation) – (4.10View Equation), under (6.14View Equation) the transformed asymptotic Weyl tensor becomes

ψ ∗0= ψ0− 4Lψ0 + 6L2ψ0 − 4L3 ψ0 + L4 ψ0, (6.15 ) 0 0 1 2 3 4 ψ ∗10= ψ01 − 3Lψ02 + 3L2ψ03 − L3 ψ04, (6.16 ) ψ ∗0= ψ0− 2Lψ0 + L2ψ0, (6.17 ) 2∗0 20 03 4 ψ 3 = ψ3 − Lψ 4, (6.18 ) ψ ∗0= ψ0. (6.19 ) 4 4

The procedure for finding the complex center of mass is centered on Eq. (6.16View Equation), where we search for and set to zero the l = 1 harmonic in ψ ∗0 1 on a τ = constant slice. This determines the complex center-of-mass world line and singles out a particular GCF referred to as the UCF,

uB = X (τ,ζ, ¯ζ) = G (τ,ζ, ¯ζ), (6.20 )
with the real version,
u (R) = X (s,ζ,ζ¯) = G (s, ζ, ¯ζ), (6.21 ) ret R R
for the gravitational field in the general asymptotically-flat case.

For the case of the Einstein–Maxwell fields, in general there will be two complex world lines and two associated UCFs, one for the center of charge, the other for the center of mass. For later use we note that the gravitational world line will be denoted by a ξ, while the electromagnetic world line by a η. Later we consider the special case when the two world lines and the two UCFs coincide, i.e., ξa = ηa.

From the assumption that 0 σ and L are first order and, from Eqs. (2.48View Equation) and (2.49View Equation) (e.g., ψ03 = ∂˙σ0), Eq. (6.16View Equation), to second order, is

∗0 0 2 0 ψ 1 = ψ 1 − 3L(Ψ − ∂ ¯σ ), (6.22 )
where ψ0 2 has been replaced by the mass aspect (2.50View Equation): Ψ ≈ ψ0 + ∂2¯σ0 2.

Using the spherical harmonic expansions (see Eqs. (6.12View Equation) and (6.13View Equation)),

Ψ = Ψ0 + ΨiY 10i + ΨijY 02ij + ..., (6.23 ) 0 0i 1 0ij 1 ψ1 = ψ1 Y1i + ψ1 Y 2ij + ..., (6.24 ) ψ∗0 = ψ ∗0iY 1+ ψ ∗0ijY 1 + ..., (6.25 ) 1 1 1i 1 2ij L(uB, ζ, ¯ζ) = ξi(τ)Y11i − 6ξij(τ)Y21ij + ..., (6.26 ) 0 ¯ ij 2 σ (uB, ζ,ζ) = 24ξ (τ)Y2ij + ... (6.27 )
and remembering that Ψ0 is zeroth order, Eq. (6.22View Equation), becomes
∗0 0 i 1 ij 1 0 i 0 ij -ij 0 ψ1 = ψ1 − 3[ξ (τ )Y1i − 6ξ (τ)Y2ij][Ψ + Ψ Y1i + {Ψ − 24ξ (τ )}Y 2ij]

or, re-arranging and performing the relevant Clebsch–Gordon expansions,

√ -- 0 ∗0 0 i 1 3--2i k j 1 108- ik k 1 18-k ik ¯ik 1 ψ1 = ψ 1 + 3Ψ ξ Y1i + 2 ξ Ψ 𝜖kjiY 1i − 5 ξ Ψ Y1i − 5 ξ (Ψ − 24ξ )Y1i (6.28 ) 216√2i- − -------ξkj(Ψkl − 24ξ¯kl)𝜖jliY11i + l ≥ 2 harmonic contributions . (6.29 ) 5
Note that though Eq. (6.28View Equation) depends initially on both τ and uret, with -- τ = T (uret,ζ,ζ), we will eventually replace all the uB (or uret) by their expressions in terms of τ, using Eq. (6.2View Equation). The transformation equation is then a function only of τ and (¯ ζ, ζ), at least to the given order in our perturbative framework.

This equation, though complicated and unattractive, is our main source of information concerning the complex center-of-mass world line. The information is extracted in the following way: Considering only the l = 1 harmonics at constant τ in Eq. (6.28View Equation), we set the l = 1 harmonics of ψ ∗0 1 (with constant τ) to zero (i.e., ∗0i ψ1 = 0). The three resulting relations are used to determine the three spatial components, k ξ (τ), of a ξ (τ ) (with 0 ξ = τ). This fixes the complex center of mass in terms of 0i 0 ψ 1 , Ψ, i Ψ, and other data which is readily interpreted physically. Alternatively it allow us to express ψ0i1 in terms of the ξa(τ).

Extracting this information takes a bit of effort.


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