6.3 Results

6.3.1 Preliminaries

Before describing in details our results, it appears to be worthwhile to very roughly survey the results and describe the logical steps taken to reach them. Virtually everything, as we said earlier, follows from the equation for 0i ψ1, i.e., Equation (292View Equation).

In the final results, we will include, from Section 3, the Maxwell field. Though we do consider the case where the two complex world lines (the complex center-of-mass and center-of-charge lines) differ from each other, some discussion will be directed to the special case of coinciding world lines.

  1. The first step is to decompose the ψ01i into its real and imaginary parts, identifying the real center of mass and the total angular momentum (as seen from infinity) described in the given Bondi coordinate system. In a different Bondi system they would undergo a specific transformation. These results are the analogues of Equation (115View Equation). It should be emphasized that there are alternative definitions, [59], but using our approximations all should reduce to our expression.
  2. The second step is to look at the evolution equation for ψ01i, i.e., insert our ψ01 into the Bianchi identity, (57View Equation), now including the Maxwell field. We obtain, from the l = 1 harmonic, the evolution of 0i ψ 1. After again decomposing it into the real and imaginary parts we find the kinematic description of the Bondi linear momentum, i i′ P = M ξ + ... (i.e., the usual kinematic expression P = M v plus additional terms) and the evolution (conservation law) for the angular momentum including a flux expression, i.e., Ji′ = (F lux)i.
  3. The third step is to reinsert the kinematic expression for the Bondi mass into the evolution equation for 0 ψ2, i.e., Equation (56View Equation). From the reality condition on the mass aspect, Equation (55View Equation) or Equation (53View Equation), only the real part is relevant. It leads to the evolution equation for the real part of the complex world line, a second-order ODE, that can be identified with Newton’s second law, F i = M ξi′′, with F i being the recoil and radiation reaction forces. The l = 0 harmonic term is the energy/mass loss equation of Bondi.

Before continuing we note that the l = 2 coefficients in ψ0 1 and ψ0 2, i.e., ψ0ij 1 and ψ0ij 2, appear frequently in second-order expressions, e.g., in Equation (292View Equation). Thus, knowing them, in terms of the free data, to first order is sufficient. By going to the linearized Bianchi identities (with the linearized Maxwell field) and the expression for the Bondi shear σ0,

˙0 2-0 ψ 2 = − ∂ ˙σ , (294 ) ψ˙01 = − ∂ψ02 + 2kq ¯ϕ02, (295 ) 0 ¯ ij 2 σ (τ,ζ,ζ) = 24ξ (τ )Y2ij + ..., (296 ) ϕ01 = q, (297 ) -0 1 --ij′′′ ϕ2 = − 2qηi′′(uret)Y11i −--Q ℂ Y 12ij, (298 ) 3
we easily find
0ij -ij ψ2 = − 24ξ (uret), √ -- (299 ) 0ij′ √ --ij 2 −3 -ij′′′ ψ 1 = − 72 2ξ (uret) − ---c kq Qℂ , 3
where a constant of integration was set to zero via initial conditions. These expressions are frequently used in the following. In future expressions we will restore explicitly ‘c’, via the derivative, (′) → c−1(′) and replace the gravitational coupling constant by k = 2Gc −4.

6.3.2 The real center of mass and the angular momentum

Returning to the basic relation, Equation (292View Equation), using Equations (299View Equation) we obtain

√ -- 0j 0 j 3(12)2-ij i 18--2- −1 0 ji i ψ1 = 3Ψ ξ + 5 ξ ξ + 5 c Ψ ξ v (300 ) 2 -- √ -- -- − 108-Ψiξij + 3(12)-ξiRξij + 4--2c−7Gq Qijℂ′′′ξiR 5( 5 √ -- 5 √ -- ) 3 (12 )3 2 -im 3 2 +i -c− 1Ψ0 ξkξi′ −-------- ξmRkξ + -----ξiΨk 𝜖ikj 2 5 2 ( 3√ -- -- -- ) +i 3(12)---2ξmk ξim − 32Gc −7qQim ′′′ξmk 𝜖ikj. 5 5 ℂ R
By replacing 0 Ψ and i Ψ, in terms of the Bondi mass and linear momentum, then decomposing the individual terms, e.g., ξi = ξiR + iξiI,Qijℂ = QijE + iQiMj, into their real and imaginary parts, the full expression is decomposed as
√ -- 6 2G ψ01i≡ − ---2--(Di(mass) + ic−1Ji) (301 ) √c-- 6--2G- i i = − c2 (MB ξR + iMB ξI) + ...
The physical identifications – first from the real part, are, initially, a tentative definition of the mass dipole moment,
(T)j j { 1 } #j D (mass) = MB ξR + c− 1 MB ξkR′ξiI + --MB ξkRξiI′ 𝜖kij + 𝒟(mass), (302 ) √ -- √2-- #j 54 2 − 1 i′ ij 6 2 −1 ji i′ 𝒟 (mass) = − -----c MB ξRξR − ----c MB ξI ξI (303 ) 5√ -- 5 − 36--2G −1c2(ξiξij+ ξiξij) 5 R R II 36√2-- √2-- − -----G −1c2ξiRξiRj− ---c−5qξiRQijE ′′′ { 5 2 15 } 2(12)-- −1 2 im mk 8--−5 mk im ′′′ − 5 G c ξI ξR − 15c qξR Q M 𝜖ikj √ -- + 36--2c− 4q2ηi′′ξij+ c− 4q2ξiηk′′𝜖 , 5 R R I R ikj
and – from the imaginary part, again, a tentative definition of the total angular momentum,
{ } (T)j j k′i 1 i k′ #j J = MBc ξI + MB ξRξR + 2MB ξIξI 𝜖ikj + 𝒥 , (304 ) √ -- √ -- #j 3(12)--2c3 i ij i ij 54--2- i′ij 𝒥 = − 5 G (ξIξR − 2ξR ξI ) − 5 MB ξRξI (305 ) √ -- √ -- + 6--2MB ξjiξi′ + --2c−4qξi Qij′′′− -8-c−4qξmkQim ′′′𝜖ikj. 5√ -- R I 15 R M 15 R E 36 2 −3 2 i′′ ij −3 2 i k′′ + -----c q ηR ξI + c q ξR ηR 𝜖ikj. 5

The reason for referring to these identifications as tentative is the following:

If there were no Maxwell field present, then the terms involving the electromagnetic dipole, q ηi, and quadruple, Qim ℂ, would not appear and these identifications, D (T )j (mass) and J(T)i, would then be considered to be firm; however, if a Maxwell field is present, we will see later that the identifications must be modified. Extra Maxwell terms are ‘automatically’ added to the above expressions when the conservation laws are considered.

As an important point we must mention a short cut that we have already taken in the interests of simplifying the presentation. When the linearized Equation (300View Equation) is substituted into the linearized Bianchi identity, ˙0 0 ψ1 = − ∂ψ 2, we obtain the linear expression for the momentum, i c3-0i P ≡ − 6G ψ2, in terms of the linearized expression for ˙ψ01i, namely,

Pi = M ξi′− 2c−3q2ηi′′= M vj − 2c−3q2ηi′′. (306 ) B R 3 R B R 3 R
This expression is then ‘fed’ into the full nonlinear Equation (300View Equation) leading to the relations Equations (302View Equation) and (304View Equation).

Considering now only the pure gravitational case, there are several comments and observations to be made.

  1. Equations (302View Equation) and (304View Equation) have been split into two types of terms: terms that contain only dipole information and terms that contain quadrupole information. The dipole terms are explicitly given, while the quadrupole terms are hidden in the 𝒟#i and 𝒥 #i.
  2. In Ji we identify Si as the intrinsic or spin angular momentum,
    i i S = MBc ξI. (307)
    This identification comes from the Kerr or Kerr–Newman metric [37Jump To The Next Citation Point]. The second term,
    MB ξk′ξi𝜖ikj = (−→r × −→P )j, (308) R R
    is the orbital angular momentum. The third term,
    1 1 -MB ξiIξkI′𝜖ikj = ------2SiSk′𝜖ikj, 2 2MBc

    though very small, represents a spin-spin contribution to the total angular momentum.

  3. In the mass dipole expression (T)i D (mass), the first term is the classical Newton mass dipole, while the next two are dynamical spin contributions.

In the following Sections 6.3.3 and 6.3.4 further physical results (with more comments and observations) will be found from the dynamic equations (asymptotic Bianchi identities) when they are applied to D (T(ma)iss) and J(T)i.

6.3.3 The evolution of the complex center of mass

The evolution of the mass dipole and the angular momentum, defined from the 0i ψ 1, Equation (301View Equation), is determined by the Bianchi identity,

˙0 0 0 0 0¯0 ψ 1 = − ∂ ψ2 + 2σ ψ 3 + 2k ϕ1ϕ2. (309 )
In the analysis of this relationship, the asymptotic Maxwell equations
ϕ˙0 = − ∂ ϕ0, (310 ) 10 20 0 0 ϕ˙0 = − ∂ ϕ1 + σ ϕ2,
and their solution, from Section 3, Equation (123View Equation) (needed only to first order),
0 i 1 − 1 ij′ 1 ϕ0 = 2qη (uret)Y1i + c Qℂ Y2ij√ +-..., (311 ) √ -- 2 ϕ01 = q + 2c−1qηi′(uret)Y 01i + ---c−2Qijℂ′′Y 02ij + ..., 6 ϕ0= − 2c−2qηi′′(uret)Y −1− 1c−3Qij′′′Y− 1+ ..., 2 1i 3 ℂ 2ij
must be used. By extracting the l = 1 harmonic from Equation (309View Equation), we find
√-- √ -- i2(12)3 2 -kj′ √ -- -- √ -- -- ψ01i′= 2cΨi + ----------ξklξ 𝜖lji − 8 2Gc −5q2ηi′′ + i4 2Gc− 6q2ηj′′ηm ′𝜖mji (312 ) 5 √-- 8- − 7 -k′′ ki′′ 8- −7 m ′-im′′′ 8--2- −8 mj ′′-lm′′′ + 5 Gc qη Q ℂ − 5Gc qη Q ℂ − i 15 Gc Qℂ Qℂ 𝜖lji.
Using Equation (301View Equation),
√ -- 6 2G ψ0i1 ≡ − ---2--(Di(mass) + ic−1Ji), c

with the (real)

2 MB = − -√c---Ψ0, (313 ) 2 2G
3 P i = −-c- Ψi, (314 ) 6G
we obtain, (1) from the real part, the kinematic expression for the (real) linear momentum and, (2) from the imaginary part, the conservation or flux law for angular momentum.

(1) Linear Momentum:

i i′ 2-− 3 2 i′′ i P = D (mass) − 3c q ηR + Π , (315 ) 4 (12 )2c2 2c −4q2 Πi = − ---------{ξkIlξkRj′− ξkRlξkIj′}𝜖lji −-------{ηjR′′ηlI′− ηjI′′ηlR′}𝜖lji √ 5- G 3 2 2 −5 k′′ ki′′ k′′ ki′′ m ′ im′′′ m ′ im′′′ + -15--c q{ηR Q E + ηI Q M − ηR Q E + ηI Q M } 4 + ---c−6{QmjM ′′QmlE ′′′− QmjE ′′QmlM ′′′}𝜖lji. 45

Using

{ } ′ (T)i′ i′ −1 k′j 1- k j′ #i′ D (mass) = MB ξR − c MB ξRξI + 2MB ξRξI 𝜖jki + 𝒟 (mass),

we get the kinematic expression for the linear momentum,

i i′ 2- −3 2 i′′ − 1 k′ j′ k′′j P = MB ξR − 3 c q ηR − c {MB ξR ξI + MB ξR ξI (316 ) 1 + -MB (ξkR ξj′I )′} 𝜖jki + 𝒟#(im′ass) + Πi, 2 2 2 −4 2 i 4(12)-c- kl kj′ kl kj′ 2c--q-- j′′ l ′ j′′ l ′ Π = − 5 G {ξI ξR − ξR ξI }𝜖lji − 3 {η R ηI − ηI ηR }𝜖lji 2√2-- + ----c−5q{ηkR′′ QkiE ′′+ ηkI′′ QkiM ′′− ηmR′QiEm′′′+ ηmI′QiMm′′′} 15 + 4-c−6{Qmj ′′Qml ′′′− Qmj ′′Qml ′′′}𝜖 , 45 M E E M lji
or
2 j′ Pi = MB ξiR′− -c−3q2ηiR′′+ c−1MB ξR ξkI′ 𝜖ijk + Ξi, (317 ) 3 Ξi = − c−1{MB ξk′′ξj + 1-MB (ξkξj′)′}𝜖jki + 𝒟#i ′ + Πi, R I 2 R I (mass)
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(2) Angular Momentum Flux:

Ji′ = (F lux )i, (318 ) J i = J(T)i + 2q2c− 2ηi′, (319 ) 3 I (T)i j k′ i 1- i k′ #i J = MBc ξI + {MB ξR ξR + 2MB ξIξI }𝜖ikj + 𝒥 , (320 ) 2 3 (F lux )i = − 4(12)-c-{ξkRlξkRj′+ ξkIl ξkIj′}𝜖lji (321 ) 5 G 2-2 −3 j′′l′ j′′l′ − 3q c {ηR ηR + ηI ηI} 𝜖lji 4 −5 mj′′ ml′′′ mj′′ ml′′′ + --c {Q E Q E + Q M Q M }𝜖lji 45√ -- 2--2- −4 k′′ ki′′ k′′ ki′′ − 15 qc {ηR Q M − ηI Q E } 2√2-- + ----qc−4{ηmI ′QimE ′′′− ηmR ′QiMm′′′}. 15

There are a variety of comments to be made about the physical content contained in these relations:

6.3.4 The evolution of the Bondi energy-momentum

Finally, to obtain the equations of motion, we substitute the kinematic expression for P i into the Bondi evolution equation, the Bianchi identity, Equation (56View Equation);

˙0 0 0 0 0¯0 ψ 2 = − ∂ ψ3 + σ ψ 4 + kϕ 2ϕ2, (323 )
or its much more useful and attractive (real) equivalent expression
˙ -˙ − 4 0-0 Ψ = ˙σσ + 2Gc ϕ 2ϕ2, (324 )
with
-- Ψ ≡ ψ02 + ∂2σ-+ σσ˙= Ψ. (325 )

Remark: The Bondi mass, 2 MB = − 2√c2G-Ψ0, and the original mass of the Reissner–Nordström (Schwarzschild) unperturbed metric, M = − -√c2--ψ00 RN 2 2G 2, i.e., the l = 0 harmonic of ψ0 2, differ by a quadratic term in the shear, the l = 0 part of -- σ ˙σ. This suggests that the observed mass of an object is partially determined by its time-dependent quadrupole moment.

Looking only at the l = 0 and l = 1 spherical harmonics and switching to the ‘′’ derivatives with the −1 c inserted, we first obtain the Bondi mass loss theorem:

′ 2(12)2c ( ij ij ij ij) 2q2( i′′i′′ i′′ i′′) --1---( ij′′′ ij′′′ ij′′′ ij′′′) M B = − 5G vRvR + v I vI − 3c5 η RηR + ηI ηI − 180c7 Q E Q E + QM Q M .

If we identify ξij with the gravitational quadrupole moment ij Q Grav via

ij ij ij ---G----( ij′′ ij′′) GQijG′r′av ξ = (ξR + iξI ) = 12 √2c4 Q Mass + iQ Spin = 12 √2c4 ,

and the electric and magnetic dipole moments by

Di = q(ηi + iηi) = Di + iDi , ℂ R I E M

the mass loss theorem becomes

M ′ = − -G- (Qij′′′Qij′′′ + Qij ′′′Qij ′′′ ) (326 ) B 5c7 Mass Mass Spin Spin -2- ( i′′ i′′ i′′ i′′) --1---( ij′′′ ij′′′ ij′′′ ij′′′) − 3c5 D ED E + D M D M − 180c7 Q E Q E + Q M Q M .

The mass/energy loss equation contains the classical energy loss due to electric and magnetic dipole radiation and electric and magnetic quadrupole (QijE,QijM) radiation. The gravitational energy loss is the conventional quadrupole loss by the above identification of ξij with the gravitational quadrupole moment ij Q Grav.

The momentum loss equation, from the l = 1 part of Equation (324View Equation), becomes

P k′ = Frkecoil, (327 ) 2G ( ) 2 Fkrecoil ≡ ---6- QljS′p′′inQijM′a′′ss − QlMja′′′ssQijS′′p′in 𝜖ilk − --4Dl′M DiE′𝜖ilk 15c ( ) 3(c ) + -1--- Dj ′Djk ′′′+ Dj ′Djk′′′ + --1--- Dlj′′′Dij′′′− Dlj′′′Dij′′′ 𝜖ilk. 15c5 E E M M 540c6 E M M E
Finally, substituting the Pi from Equation (317View Equation), we have Newton’s second law of motion:
i′′ i MB ξR = F , (328 )
with
F i = − M ′ξi′− c−1M (ξj′ξk′)′𝜖 + 2c− 3q2ηi′′′+ F i − Ξi′. (329 ) B R B R I ijk 3 R recoil
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There are several things to observe and comment on concerning Equations (328View Equation) and (329View Equation):


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