6.2 The complex center-of-mass world line

Before trying to determine the l = 1 harmonics of Eq. (6.28View Equation), several comments and repetitions (for emphasis) are in order:

  1. As previously noted, Eq. (6.28View Equation) is a function of both τ (via the ξi, ξij) and uret (via the ψ01i and Ψ). The extraction of the l = 1 part of ψ ∗10 must be taken on the constant τ cuts. In other words u ret must be eliminated by using Eq. (6.2View Equation).
  2. This elimination of uB (or uret) is done in the linear terms via the expansion:
    ( √ -- ) 2 √ -- ij η(uret) = η τ − ----ξiR(τ)Y10i + 2 ξR (τ)Y20ij (6.30) 2 √2-- [ ] ≈ η(τ) − ---η(s)′ ξiR(s)Y10i − 2ξiRj(s)Y20ij √ --2 --2-i 0 √-- ij 0 uret = τ − 2 ξ (τ)Y1i + 2 ξ (τ)Y2ij + ...

    In the nonlinear terms we can simply use

    u = τ. ret

  3. In the Clebsch–Gordon expansions of the harmonic products, though we need both the l = 1 and l = 2 terms in the calculation, we keep at the end only the l = 1 terms for the 0i ψ 1. (Note that there are no l = 0 terms since ψ∗10 is spin weight s = 1.)
  4. For completeness, we have included into the calculations Maxwell fields with both a complex dipole (electric and magnetic), i i i i D ℂ = qη = q(ηR + iηI) and complex quadrupole (electric and magnetic) fields kj kj kj Qℂ = Q E + iQ M.

We begin by focusing on the l = 1 portion of the right-hand-side of Eq. (6.28View Equation) in the complex center of mass frame. Assuming that ψ ∗0i = 0 1, we see that all remaining terms on this side of the equation are nonlinear, so we can simply make the replacement u → τ ret. On the left-hand side of the equation, extracting the l = 1 component of 0 ψ1 on a constant τ slice is more complicated though; using Eq. (6.30View Equation), we have that

√2- √ -- ψ01(uret) = ψ01(τ ) − ---ψ01(τ)′ξi(τ )Y 11i + 2ψ01(τ )′ξij(τ)Y 12ij. (6.31 ) 2
Using the Bianchi identity (2.53View Equation) and inserting the proper factor of √ -- 2 to account for the retarded Bondi time, we have that (to second order)
√ -- √ -- √ -- ψ0 ′= − --2∂ Ψ + --2-∂3¯σ0 + 2kϕ0ϕ¯0. 1 2 2 1 2

As ψ01′ enters Eq. (6.31View Equation) only in nonlinear terms, we only need to extract the linear portion of this Bianchi identity. Recalling (suppressing factors of c for the time being) that

√ -- 0 √ -- i′ 0 --2- ij′′ 0 0 i′′ 1 1- ij′′′ 1 ϕ1 = q + 2qη Y 1i + 6 Q ℂ Y 2ij, ϕ 2 = − 2q η Y 1i − 3Q ℂ Y2ij,

we readily determine that

√ -- √ -- √ -- √2kq ψ01′= 2ΨiY 11i + 3 2(Ψij − 24¯ξij)Y12ij − 2 2kq2¯ηi′′Y11i − -----Q¯ijℂ′′′Y21ij. (6.32 ) 3

Feeding (6.32View Equation) into Eq. (6.31View Equation) and performing the relevant Clebsch–Gordon expansions, we find:

√ -- √ -- 0 0i 2i j k 864 j ji 12 j ji 3456 2i lj lk ψ1(uret)|l=1 = ψ 1 − --2-Ψ ξ 𝜖jki +-5--ξ ¯ξ − -5-Ψ ξ + ---5----ξ ¯ξ 𝜖jki -- √ -- + √ 2ikq2ξk¯ηj′′𝜖 + 24-kq2ξjiη¯j′′ + 2-kqξj ¯Qji′′′+ 24-2ikqξljQ ¯lk ′′′𝜖 . (6.33 ) jki 5 5 ℂ 5 ℂ jki
Here, we have implicitly used the fact that, to our level of approximation, Ψij = − 24 ¯ξij.

We can now incorporate this into Eq. (6.28View Equation) to obtain the full complex center of mass equation as a function of τ. The l = 0,1 components of the mass aspect are replaced by the expressions

√ -- Ψ0 = − 2--2G-MB, Ψi = − 6G-P i, Ψij = − 24 ¯ξij; c2 c3

we insert k = 2Gc −4 and the appropriate factors of c elsewhere, at which point Eq. (6.28View Equation) can be re-expressed in a manner that determines the complex center of mass, with all terms being functions of τ:

√ -- √ -- √ -- 6 2G 6 2i 576G 6912 2i ψ01i= − --2---MB ξi + --3--GP kξj𝜖kji − ---3-P kξki +---------ξlj¯ξlk𝜖jki √c-- c 5c 5√ -- 2--2i 2 k j′′ 48G--2 jij′′ 4G- j ¯ji′′′ 16--2i- lj ¯lk′′′ − c6 Gq ξ η¯ 𝜖jki − 5c6 q ξ ¯η − 5c7qξ Q ℂ − 5c7 Gq ξ Q ℂ 𝜖jki. (6.34 )
Note that the linear term
√ -- 0i 6--2G- i 0 i ψ1 = − c2 MB ξ = 3Ψ ξ

coincides with the earlier results in the stationary case, Eq. (5.44View Equation).

Now, we recall our identification for the complex gravitational dipole,

√ -- ψ0i(τ) = − 6--2G-(Di + ic−1Ji), (6.35 ) 1 c2 (mass)
as well as the identification between the l = 2 harmonic coefficient of the UCF and the gravitational quadrupole:
√2G- √2G- ξij = ----4 QijG′r′av =----4 (QijM′′ass + iQijS′′pin). (6.36 ) 24c 24c
Feeding these into (6.34View Equation), we can separate out the real and imaginary parts via (6.35View Equation) to obtain expressions for the mass dipole and angular momentum, our primary results:
Di = M ξi − c−1P kξj𝜖 + 4G-P kQki′′ + -2G Qlj′′ Qlk′′ 𝜖 (mass) B R I jki 5c5 Mass 5c6 Spin Mass jki q2 ( k j′′ k j′′) Gq2 ( j′′ ji′′ j′′ ji′′ ) + --4 ξR ηI − ξIηR 𝜖jki + ---8- ηR Q Mass + ηI QSpin 3√c-- 15c + --2q(ξj Qji′′′+ ξjQji′′′) 15c5 R E I M √2Gq ( ) + ----9- QljM′a′ssQlkM′′′− QljS′p′inQlkE′′′ 𝜖jki (6.37 ) 45c
√ -- j 4G q2 ( j′′ j′′) 2q ( j ji′′′ j ji′′′) Ji = cMB ξiI + ξRP k𝜖jki +--4P kQkiS′p′in + --3 ξkRηR + ξkIηI 𝜖jki +---4- ξIQ E − ξRQ M 5c √ -3c ( 15c) Gq2--( j′′ ji′′ j′′ ji′′ ) --2Gq- lj′′ lk′′′ lj′′ lk′′′ + 15c7 ηR QSpin − ηI Q Mass + 45c8 QMassQ E + Q SpinQ M 𝜖jki. (6.38 )

Though these results are discussed at greater length later, we point out that Eqs. (6.37View Equation) and (6.38View Equation) already contains terms of obvious physical interest. Note that the first two items in J i are the spin,

−→ −→ S = cMB ξ I (6.39 )
(identified via the special case of the Kerr–Newman metric) and the orbital angular momentum
−→L = −→ξ × −→P . (6.40 ) R
The mass dipole Di (mass) has the conventional term −→ MB ξ R and a momentum-spin coupling term (which appears to be new):
−→ 1 −→ −→ D (mass) = MB ξ R + 2----P × S + ⋅⋅⋅ (6.41 ) cMB

We will see shortly that there is also a great deal of physical content to be found in the nonlinear terms of Eq. (6.34View Equation).


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