6.3 The evolution of the complex center of mass

The evolution of the mass dipole and the angular momentum, defined from the 0i ψ1, Eq. (6.34View Equation) and Eqs. (6.37View Equation) with (6.38View Equation), is determined via the Bianchi identity
ψΛ™01 = − ∂ ψ02 + 2σ0ψ03 + 2k Ο•01¯Ο•02. (6.42 )
This relationship allows us the determine – kinematically – the Bondi momentum in terms of the dipole and the complex world line.

By extracting the l = 1 harmonic from Eq. (6.42View Equation), a process which involves several Clebsch–Gordon expansions, we find

√ -- √ -- √ -- 0i′ √ -- i 1728--2i-kj¯kl′ 4--2G- 2 i′′ 4--2i 2 j′k′′ ψ1 = 2cΨ + 5 ξ ξ − c5 q ¯η − c6 Gq η ¯η πœ–kji √ -- + 8G-q¯ηj′′Qji′′− 8G-qηj′ ¯Qji′′′ − 8-2iGQkj ′′Q¯kl′′′πœ– . (6.43 ) 5c7 β„‚ 5c7 β„‚ 15c8 β„‚ β„‚ lji
Using our various identifications for the complex gravitational dipole, the Bondi momentum, and the complex gravitational quadrupole, Eq. (6.43View Equation) can be written as
( )′ 12i 2q2 2i Di(mass) + ic−1Ji = P i − --6GQkjG′ra′vQ¯klG′r′′avπœ–jli +---3 ¯ηi′′ +--4q2ηj′¯ηk′′πœ–kji √-5c √ -- 3c 3c 2--2- j′′ ji′′ 2--2- j′ ¯ji′′′ -4i-- kj′′¯ kl′′′ − 15c5 q¯η Q β„‚ + 15c5qη Qβ„‚ + 45c6Q β„‚ Q β„‚ πœ–lji, (6.44 )
or in terms of real and imaginary parts:
2q2 2q2 ( )′ 12G ( ) ′ Di′(mass) = Pi + --3ηiR′′+ --4- ηjR′ηkI′ πœ–kji − --6-- QkjM′a′ssQklS′p′in πœ–jli √ --3c 3c 5c 2--2q-( j′ ji′′′ j′′ ji′′ j′ ji′′′ j′′ ji′′) -4--( kj′′ kl′′) ′ + 15c5 ηRQ E − ηR Q E + ηI QM − ηI QM + 45c6 Q E Q M πœ–lji, (6.45 )
2q2 2q2( ) 12G ( ) J i′ = − ---ηiI′′+ ---- ηj′RηkR′′ + ηjI′ ηkI′′ πœ–kji − ----- QkjM′a′ssQklM′a′′ss + QkSjp′′inQkSlp′′i′n πœ–jli 3c√2-- 3c3 5c5 2 2q ( j′ ji′′ j′ ji′′)′ 4 ( kj′′ kl′′′ kj′′ kl′′′) + 15c4-- ηI Q E − ηRQ M + 45c5- Q E Q E + QM Q M πœ–lji. (6.46 )

Eq. (6.46View Equation), which is the conservation of angular momentum, has several things to note. As there are two terms appearing as total derivatives (the first and fourth), it might be more natural to include them in an alternative definition of angular momentum [8Jump To The Next Citation Point]:

2 √ -- J i= Ji + 2q-ηi′− 2--2q-(ηj′Qji′′− ηj′Qji′′). (6.47 ) T 3c2 I 15c4 I E R M
This results in an alternative flux law for angular momentum conservation,
JTi′= (Flux )iT, 2 ( ) ( ) (Flux)i = 2q-- ηj′ηk′′+ ηj′ηk′′ πœ–kji − 12G-- Qkj′′Qkl ′′′ + Qkj′′ Qkl′′′ πœ–jli (6.48 ) T 3c3 R( R I I 5c)5 Mass Mass Spin Spin -4--- kj′′ kl′′′ kj′′ kl′′′ + 45c5 QE Q E + Q M Q M πœ–lji,
whose terms appear to agree with the known angular momentum flux due to gravitational quadrupole and electromagnetic dipole and quadrupole radiation [43Jump To The Next Citation Point].

As for the evolution equation for the mass dipole (6.45View Equation), we can obtain an expression for the Bondi linear momentum by taking the derivative (with respect to retarded Bondi time) of Eq. (6.37View Equation) to eliminate i D (mass) and find:

2 P i = MB ξiR′− 2q--ηi′R′+ 𝔓i1 + 𝔓i2 + 𝔓i3, (6.49 ) 3c3
where 𝔓i1, 𝔓i2 and 𝔓i3 are nonlinear terms representing dipole-dipole, dipole-quadrupole and quadrupole-quadrupole coupling respectively,
2 𝔓i = -q- (3ηk′′ξj − ηj′′ξk − 2ηk′ηj′)′πœ– − MB--(ξk′′ξj)′πœ– , 1 3c4 R I I r √ R- I jki c R I jki 4G ( ) ′ 2 2q ( j′′ j′ ji′′′ j′′ ji′′ j′ ji′′′) 𝔓i2 = ---5MB ξkR′QkiM′a′ss + ----5- ηR Qji′′ − ηRQ E + ηI Q M − ηI QM 5c√ -- 15c --2q-( j ji′′′ j ji′′′)′ Gq2-( j′′ ji′′ j′′ ji′′)′ + 15c5 ξRQ E + ξIQ M + 15c8 ηI Q Spin − 7ηR Q Mass , 2G ( )′ 4 ( ) ′ √2Gq ( )′ 𝔓i3 = --- QkjMa′′ssQklSp′′in πœ–jli + ----- QkjE′′QklM′′ πœ–jli + ------ QljM′a′ssQlkM′′′− QljS′p′inQlkE′′′ πœ–jki. c6 45c6 45c9

Remark 11. In the calculation leading to Eq. (6.49View Equation), nonlinear terms with i P (or its derivatives) were replaced by the linear expression P i ≈ MB ξiR′− 23c−3q2ηi′R′. Additionally, we have neglected the time derivatives of the Bondi mass, M B′; this is because, as we shall see momentarily, these derivatives are themselves second order quantities and hence give a vanishing contribution to Eq. (6.49View Equation) at our level of approximation.

Physical Content


  Go to previous page Go up Go to next page