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 [8]:

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 [43].

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 and find:

where , and are nonlinear terms representing dipole-dipole, dipole-quadrupole and quadrupole-quadrupole coupling respectively,

Remark 11. In the calculation leading to Eq. (6.49), nonlinear terms with (or its derivatives) were replaced by the linear expression . Additionally, we have neglected the time derivatives of the Bondi mass, ; this is because, as we shall see momentarily, these derivatives are themselves second order quantities and hence give a vanishing contribution to Eq. (6.49) at our level of approximation.

Physical Content

• 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 [43].
• 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 [43]. 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., .