6.4 The evolution of the Bondi energy-momentum

Finally, to obtain the equations of motion, we substitute the kinematic expression for i P into the Bondi evolution equation, the Bianchi identity, Eq. (2.52View Equation);
˙ψ0 = − ∂ψ0 + σ0 ψ0 + kϕ0ϕ¯0 , 2 3 4 2 2

or its much more useful and attractive (real) form

√ -- √ -- Ψ′ = --2σ0 ′σ0′ + --2kϕ02¯ϕ02. (6.51 ) c c

Remark 12. The Bondi mass, -√c2- 0 MB = − 2 2G Ψ, and the original mass of the Reissner–Nordström (Schwarzschild) unperturbed metric, 2 MRN = − -c√--ψ002 2 2G, i.e., the l = 0 harmonic of ψ02, differ by a quadratic term in the shear, the l = 0 part of ˙- σσ. This suggests that the observed mass of an object is partially determined by its time-dependent quadrupole moment.

Upon extracting the l = 0 harmonic portion of Eq. (6.51View Equation) as well as inserting our various physical identifications for the objects involved, we obtain the Bondi mass loss theorem:

( ) 2 M ′= − G-- Qjk′′′Qjk ′′′ + Qjk ′′′Qjk ′′′ − 4q--(ηk′′ηk′′+ ηk′′ηk ′′) B 5c7 Mass Mass Spin Spin 3c5 E E M M -4---( jk′′′ jk′′′ jk′′′ jk′′′) − 45c7 Q E QE + Q M Q M . (6.52 )
This mass/energy loss equation contains the classical energy loss due to electric and magnetic dipole radiation and electric and magnetic quadrupole (ij ij Q E,Q M) radiation. (Note that agreement with the physical quadrupole radiation is recovered after making the aforementioned rescaling √-- Qijℂ → 2 2Qijℂ.) The gravitational energy loss is the conventional quadrupole loss by the identification (6.36View Equation) of ξij with the gravitational quadrupole moment Qij Grav.

The momentum loss equation, from the l = 1 part of Eq. (6.51View Equation), is then identified with the recoil force due to momentum radiation:

Pi′ = F irecoil,

where

( ) 2( ) Firecoil = -2G-- Qjl′′′Qkj ′′′− Qjl′′′ Qkj′′′ 𝜖kli + 2q--ηj′′ηkI′′− ηj′′ηkR′′𝜖kji 15c6√ -- Spin Mass Mass Spin 3c4 R I 4 2q ( j′′ ji′′′ j′′ ji′′′) 4 ( jl′′′ kj′′′ jl′′′ kj′′′) − ----5- ηR Q E + ηI Q M − -----6 QM Q E − QE Q M . (6.53 ) 15c 135c
Finally, we can substitute in the i P from Eq. (6.49View Equation) to obtain Newton’s second law of motion:
i′′ i MB ξR = F , (6.54 )
with
2 Fi = − M ′BξiR′+ -c−3q2ηiR′′′+ Friecoil − Ξi′, (6.55 ) 3 Ξ = 𝔓1 + 𝔓2 + 𝔓3 (6.56 )

Physical Content

There are several things to observe and comment on concerning Eqs. (6.54View Equation) and (6.55View Equation):


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