### 6.4 The evolution of the Bondi energy-momentum

Finally, to obtain the equations of motion, we substitute the kinematic expression for into the Bondi evolution equation, the Bianchi identity, Eq. (2.52);

or its much more useful and attractive (real) form

Remark 12. The Bondi mass, , and the original mass of the Reissner–Nordström (Schwarzschild) unperturbed metric, , i.e., the harmonic of , differ by a quadratic term in the shear, the part of . This suggests that the observed mass of an object is partially determined by its time-dependent quadrupole moment.

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

This mass/energy loss equation contains the classical energy loss due to electric and magnetic dipole radiation and electric and magnetic quadrupole () radiation. (Note that agreement with the physical quadrupole radiation is recovered after making the aforementioned rescaling .) The gravitational energy loss is the conventional quadrupole loss by the identification (6.36) of with the gravitational quadrupole moment .

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

where

Finally, we can substitute in the from Eq. (6.49) to obtain Newton’s second law of motion:
with

#### Physical Content

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

• If the complex world line associated with the Maxwell center of charge coincides with the complex center of mass, i.e., if , the term
becomes the classical electrodynamic radiation reaction force.
• This result follows directly from the Einstein–Maxwell equations. There was no model building other than requiring that the two complex world lines coincide. Furthermore, there was no mass renormalization; the mass was simply the conventional Bondi mass as seen at infinity. The problem of the runaway solutions, though not solved here, is converted to the stability of the Einstein–Maxwell equations with the ‘coinciding’ condition on the two world lines. If the two world lines do not coincide, i.e., the Maxwell world line forms independent data, then there is no problem of unstable behavior. This suggests a resolution to the problem of the unstable solutions: one should treat the source as a structured object, not a point, and centers of mass and charge as independent quantities.
• The is the recoil force from momentum radiation.
• The can be interpreted as the gravitational radiation reaction.
• The first term in , i.e., , is identical to a term in the classical Lorentz–Dirac equations of motion. Again it is nice to see it appearing, but with the use of the mass loss equation it is in reality third order.