Before describing in details our results, it appears to be worthwhile to very roughly survey the results and
describe the logical steps taken to reach them. Virtually everything, as we said earlier, follows from the
equation for , i.e., Equation (292).
In the final results, we will include, from Section 3, the Maxwell field. Though we do consider the case
where the two complex world lines (the complex center-of-mass and center-of-charge lines)
differ from each other, some discussion will be directed to the special case of coinciding world
- The first step is to decompose the into its real and imaginary parts, identifying the
real center of mass and the total angular momentum (as seen from infinity) described in the
given Bondi coordinate system. In a different Bondi system they would undergo a specific
transformation. These results are the analogues of Equation (115). It should be emphasized
that there are alternative definitions, , but using our approximations all should reduce to
- The second step is to look at the evolution equation for , i.e., insert our into the
Bianchi identity, (57), now including the Maxwell field. We obtain, from the harmonic,
the evolution of . After again decomposing it into the real and imaginary parts we find
the kinematic description of the Bondi linear momentum, (i.e., the usual
kinematic expression plus additional terms) and the evolution (conservation law)
for the angular momentum including a flux expression, i.e., .
- The third step is to reinsert the kinematic expression for the Bondi mass into the evolution
equation for , i.e., Equation (56). From the reality condition on the mass aspect,
Equation (55) or Equation (53), only the real part is relevant. It leads to the evolution equation
for the real part of the complex world line, a second-order ODE, that can be identified with
Newton’s second law, , with being the recoil and radiation reaction forces.
The harmonic term is the energy/mass loss equation of Bondi.
Before continuing we note that the coefficients in and , i.e., and , appear
frequently in second-order expressions, e.g., in Equation (292). Thus, knowing them, in terms of the free
data, to first order is sufficient. By going to the linearized Bianchi identities (with the linearized Maxwell
field) and the expression for the Bondi shear ,
we easily find
where a constant of integration was set to zero via initial conditions. These expressions are frequently used
in the following. In future expressions we will restore explicitly ‘’, via the derivative, and
replace the gravitational coupling constant by .
6.3.2 The real center of mass and the angular momentum
Returning to the basic relation, Equation (292), using Equations (299) we obtain
By replacing and , in terms of the Bondi mass and linear momentum, then decomposing the
individual terms, e.g., , into their real and imaginary parts, the full
expression is decomposed as
The physical identifications – first from the real part, are, initially, a tentative definition of the mass dipole
and – from the imaginary part, again, a tentative definition of the total angular momentum,
The reason for referring to these identifications as tentative is the following:
If there were no Maxwell field present, then the terms involving the electromagnetic dipole, , and
quadruple, , would not appear and these identifications, and , would then be
considered to be firm; however, if a Maxwell field is present, we will see later that the identifications must
be modified. Extra Maxwell terms are ‘automatically’ added to the above expressions when the conservation
laws are considered.
As an important point we must mention a short cut that we have already taken in the interests of
simplifying the presentation. When the linearized Equation (300) is substituted into the linearized Bianchi
identity, , we obtain the linear expression for the momentum, , in terms of the
linearized expression for , namely,
This expression is then ‘fed’ into the full nonlinear Equation (300) leading to the relations Equations (302)
Considering now only the pure gravitational case, there are several comments and observations to be
- Equations (302) and (304) have been split into two types of terms: terms that contain only
dipole information and terms that contain quadrupole information. The dipole terms are
explicitly given, while the quadrupole terms are hidden in the and .
- In we identify as the intrinsic or spin angular momentum,
This identification comes from the Kerr or Kerr–Newman metric . The second term,
is the orbital angular momentum. The third term,
though very small, represents a spin-spin contribution to the total angular momentum.
- In the mass dipole expression , the first term is the classical Newton mass dipole, while the
next two are dynamical spin contributions.
In the following Sections 6.3.3 and 6.3.4 further physical results (with more comments and
observations) will be found from the dynamic equations (asymptotic Bianchi identities) when they are
applied to and .
6.3.3 The evolution of the complex center of mass
The evolution of the mass dipole and the angular momentum, defined from the , Equation (301), is
determined by the Bianchi identity,
In the analysis of this relationship, the asymptotic Maxwell equations
and their solution, from Section 3, Equation (123) (needed only to first order),
must be used. By extracting the harmonic from Equation (309), we find
Using Equation (301),
with the (real)
we obtain, (1) from the real part, the kinematic expression for the (real) linear momentum and, (2) from
the imaginary part, the conservation or flux law for angular momentum.
(1) Linear Momentum:
we get the kinematic expression for the linear momentum,
(2) Angular Momentum Flux:
There are a variety of comments to be made about the physical content contained in these
- 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 .
- The third term, , is the classical Mathisson–Papapetrou spin-velocity
contribution to the linear momentum. If the evolution equation (angular momentum
conservation) is used, this term becomes third order.
- 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 . Though it is nice to see the Mathisson–Papapetrou term, it should be
treated more as a suggestive result rather than a physical prediction.
Aside: Later, in the discussion of the Bondi energy loss theorem, we will see that we can relate ,
i.e., the shear term, to the gravitational quadrupole by
- In the expression for we have already identified, in the earlier discussion, the first two terms,
and as the intrinsic spin angular momentum and the orbital angular
momentum. The third term, , a spin-spin interaction term,
considerably smaller, can be interpreted as a spin-precession contribution to the total
angular momentum. An interesting contribution to the total angular momentum comes from
the term, , i.e., a contribution from the time-varying magnetic
- As mentioned earlier, our identification of as the total angular momentum in the absence of a
Maxwell field agrees with most other identifications (assuming our approximations). Very strong
support of this view, with the Maxwell terms added in, comes from the flux law. In Equation (321) we
see that there are five flux terms, the first is from the gravitational quadrupole flux, the second and
third are from the classical electromagnetic dipole and electromagnetic quadrupole flux, while the
fourth and fifth come from electromagnetic-gravitational 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, Equation (309), is the 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
explicit time derivatives, thereby being assigned to the time derivative of the angular
- 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., .
6.3.4 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, Equation (56);
or its much more useful and attractive (real) equivalent expression
Remark: 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.
Looking only at the and spherical harmonics and switching to the ‘’ derivatives with
the inserted, we first obtain the Bondi mass loss theorem:
If we identify with the gravitational quadrupole moment via
and the electric and magnetic dipole moments by
the mass loss theorem becomes
The mass/energy loss equation contains the classical energy loss due to electric and magnetic dipole
radiation and electric and magnetic quadrupole () radiation. The gravitational energy loss is the
conventional quadrupole loss by the above identification of with the gravitational quadrupole moment
The momentum loss equation, from the part of Equation (324), becomes
Finally, substituting the from Equation (317), we have Newton’s second law of motion:
There are several things to observe and comment on concerning Equations (328) and (329):
- 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
- 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.