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)ˆlaζ,ζ¯+ ξij(τ)Y20ij + ... (261 ) ξ0(τ) 1 = -√--- − --ξi(τ)Y 01i + ξij(τ)Y20ij + ... 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 + ... (262 )
(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), √ -- uret = 2uB,
can be found by the following iteration process [27]; writing Equation (262View Equation) as
¯ τ = uret + F (τ,ζ,ζ), (263 )
with
√ -- 2 √ -- F (τ, ζ, ¯ζ) =---ξi(τ)Y10i(ζ, ¯ζ) − 2 ξij(τ)Y10ij(ζ,ζ¯) + ..., (264 ) 2
the iteration relationship, with the zeroth-order iterate, τ0 = uret, is
τ = u + F (τ ,ζ, ¯ζ). (265 ) n ret n−1
Though the second iterate easily becomes
¯ ( -- -) τ = T (uret,ζ,ζ ) = uret + F uret + F (uret,ζ,ζ),ζ,ζ ≈ uret + F + F ∂uretF. (266 )
UpdateJump To The Next Update Information 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). (267 ) 2
This relationship is important later.

We also have the linearized reality relations – easily found earlier or from Equation (267View Equation):

τ = s + iλ, (268 ) √ -- -- --2-i 0 √ --ij 0 λ = Λ(s,ζ,ζ ) = 2 ξI(s)Y1i − 2ξI (s)Y2ij, (269 ) ( √ -- √ -- ) τ = s + i --2ξi(s)Y 0− 2ξij(s)Y0 , (270 ) 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. (271 )

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

L (uB, ζ, ¯ζ) = ∂(τ)G(τ,ζ, ¯ζ) (272 ) i 1 ij 1 = ξ (τ)Y1i − 6 ξ (τ)Y1ij + ...
and
0 ¯ 2 ¯ σ (τ,ζ,ζ) = ∂(τ)G (τ,ζ,ζ), (273 ) = 24ξij(τ )Y2 + ..., 2ij
using uB = G, while the asymptotically shear-free NGC is given (again) by the null rotation
l∗a = la + bma + bma + bbna, (274 ) ∗a a a m = m + bn , n ∗a = na, −2 b = − L ∕r + O (r ).

From Equation (274View Equation), the transformed asymptotic Weyl tensor becomes, Equations (275View Equation) – (279View Equation),

ψ ∗0= ψ0− 4Lψ0 + 6L2ψ0 − 4L3 ψ0 + L4 ψ0, (275 ) 0 0 1 2 3 4 ψ ∗10= ψ01 − 3Lψ02 + 3L2ψ03 − L3 ψ04, (276 ) ψ ∗0= ψ0− 2Lψ0 + L2ψ0, (277 ) 2∗0 20 03 4 ψ 3 = ψ3 − Lψ 4, (278 ) ψ ∗0= ψ0. (279 ) 4 4
UpdateJump To The Next Update Information

The procedure is centered on Equation (276View Equation), where we search for and set to zero the l = 1 harmonic in ψ∗0 1 on an s = constant slice. This determines the complex center-of-mass world line and singles out a particular GCF referred to as the UCF,

X (τ, ζ, ¯ζ) = G (τ, ζ, ¯ζ),

with the real version,

X (s,ζ, ¯ζ) = G (s,ζ, ¯ζ), (280 ) 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, one for the center of charge, the other for the center of mass and the two associated UCFs. 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 σ0 and L are first order and, from Equation (52View Equation), that --0 ψ03 = ∂σ˙, Equation (276View Equation), to second order, is

∗0 0 2 0 ψ 1 = ψ1 − 3L[Ψ − ∂ ¯σ ], (281 )
where ψ02 has been replaced by the mass aspect, Equation (54View Equation), Ψ ≈ ψ02 + ∂2¯σ0.

Using the spherical harmonic expansions (see Equations (272View Equation) and (273View Equation)),

0 i 0 ij 0 Ψ = Ψ + Ψ Y1i + Ψ Y 2ij + ..., (282 ) ψ0 = ψ0iY 1+ ψ0ijY 1 + ..., (283 ) 1 1 1i 1∗0ij2ij ψ∗10 = ψ1∗0iY11i + ψ 1 Y21ij + ..., (284 ) ¯ i 1 ij 1 L(uB, ζ,ζ) = ξ (τ)Y1i − 6ξ (τ)Y2ij + ..., (285 ) σ0(uB, ζ, ¯ζ) = 24ξij(τ)Y22ij + ... (286 )
Remembering that 0 Ψ is zeroth order, Equation (281View Equation), becomes
∗0 0i 1 0ij 1 ψ 1 = ψ1 Y1i + ψ 1 Y2ij − 3[ξi(τ)Y 1− 6ξij(τ)Y 1][Ψ0 + ΨiY 0 + {Ψij − 24ξij(τ)}Y 0] 1i 2ij 1i 2ij
or
ψ ∗10= ψ01iY11i + ψ0i1jY21ij − 3ξi(τ)Y11iΨ0 − 3Ψi ξj(τ )Y11jY 01i (287 ) -ij − 3ξk (τ )[Ψij − 24ξ (τ)]Y 11kY 02ij ij 1 0 k ij 1 0 kl ij -ij 1 0 +18 ξ (τ)Y2ijΨ + Ψ 18ξ (τ)Y 2ijY1k + 18ξ (τ )[Ψ − 24ξ (τ)]Y 2klY2ij.
Note that the right-hand side of Equation (287View Equation) depends initially on both τ and u ret, with -- τ = T (uret,ζ,ζ ).

This equation, though complicated and unattractive, is our main source of information concerning the complex center-of-mass world line. Extracting this information, i.e., determining ψ ∗10i at constant values of s by expressing τ and uret as functions of s and setting it equal to zero, takes considerable effort.


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