6.3 The evolution of the complex center of mass
The evolution of the mass dipole and the angular momentum, defined from the , Eq. (6.34) and
Eqs. (6.37) with (6.38), is determined via the Bianchi identity
This relationship allows us the determine – kinematically – the Bondi momentum in terms of the dipole and
the complex world line.
By extracting the harmonic from Eq. (6.42), a process which involves several Clebsch–Gordon
expansions, we find
Using our various identifications for the complex gravitational dipole, the Bondi momentum, and the
complex gravitational quadrupole, Eq. (6.43) can be written as
or in terms of real and imaginary parts:
Eq. (6.46), 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 :
This results in an alternative flux law for angular momentum conservation,
whose terms appear to agree with the known angular momentum flux due to gravitational quadrupole and
electromagnetic dipole and quadrupole radiation .
As for the evolution equation for the mass dipole (6.45), we can obtain an expression for the Bondi
linear momentum by taking the derivative (with respect to retarded Bondi time) of Eq. (6.37) to eliminate
where , and are nonlinear terms representing dipole-dipole, dipole-quadrupole and
quadrupole-quadrupole coupling respectively,
- The first term of is the standard Newtonian kinematic expression for the linear momentum,
- The second term, , which is a contribution from the second derivative of the
electric dipole moment, , plays a special role for the case when the complex center of
mass coincides with the complex center of charge, . In this case, the second term is
exactly the contribution to the momentum that yields the classical radiation reaction force of
classical electrodynamics .
- Many of the remaining terms in , though apparently second order, are really of higher
order when the dynamics are considered. Others involve quadrupole interactions, which contain
high powers of .
- In the expression for we have already identified, in earlier discussions, the first two terms
and as the intrinsic spin angular momentum and the orbital angular
momentum. The further terms, a spin-spin, spin-quadrupole and quadrupole-quadrupole
interaction terms, are considerably smaller.
- As mentioned earlier, in Eq. (6.46) we see that there are five flux terms, the second is from
the gravitational quadrupole flux, the third and fifth are from the classical electromagnetic
dipole and electromagnetic quadrupole flux, while the fourth come from dipole-quadrupole
coupling. The Maxwell dipole part is identical to that derived from pure Maxwell theory .
We emphasize that this angular momentum flux law has little to do directly with the chosen
definition of angular momentum. The imaginary part of the Bianchi identity (6.42), with the
reality condition , is the angular momentum conservation law. How to identify the
different terms, i.e., identifying the time derivative of the angular momentum and the flux
terms, comes from different arguments. The identification of the Maxwell contribution to total
angular momentum and the flux contain certain arbitrary assignments: some terms on the
left-hand side of the equation, i.e., terms with a time derivative, could have been moved onto
the right-hand side and been called ‘flux’ terms. However, our assignments were governed by
the question of what terms appeared most naturally to be on different sides. The first term
appears to be a new prediction.
- The angular momentum conservation law can be considered as the evolution equation for the
imaginary part of the complex world line, i.e., . The evolution for the real part is
found from the Bondi energy-momentum loss equation.
- In the special case where the complex centers of mass and charge coincide, , we
have a rather attractive identification: since now the magnetic dipole moment is given
by and the spin by , we have that the gyromagnetic ratio is
leading to the Dirac value of , i.e., .