- A particularly interesting issue raised by our equations is that of the run-away (unstable)
behavior of the equations of motion for a charged particle (with or without an external field).
We saw in Eq. (6.54) that there was a driving term in the equation of motion depending on the
electric dipole moment (or the real center of charge). This driving term was totally independent
of the real center of mass and thus does not lead to the classical instability. However, if we
restrict the complex center of charge to be the same as the complex center of mass (a severe, but
very attractive restriction leading to ), then the innocuous driving dipole term becomes
the classical radiation reaction term – suggesting instability. (Note that in this coinciding case
there was no model building – aside from the coinciding lines – and no mass renormalization.)
A natural question then is: does this unstable behavior really remain? In other words, is it possible that the large number of extra terms in the gravitational radiation reaction or the momentum recoil force might stabilize the situation? Answering this question is extremely difficult. If the gravitational effects do not stabilize, then – at least in this special case – the Einstein–Maxwell equations are unstable, since the run-away behavior would lead to an infinite amount of electromagnetic dipole energy loss.

An alternative possible resolution to the classical run-away problem is simply to say that the classical electrodynamic model is wrong; and that one must treat the center of charge as different from the center of mass with its own dynamics.

- In our approximations, it was assumed that the complex world line yielded cuts of that were close to Bondi cuts. At the present we do not have any straightforward means of finding the world lines and their associated cuts of that are far from the Bondi cuts.
- As mentioned earlier, when the gravitational and electromagnetic world lines coincide we find the rather surprising result of the Dirac value for the gyromagnetic ratio. Though this appears to be a significant result, we unfortunately do not have any deeper understanding of it.
- Is it possible that the complex structures that we have been seeing and using are more than just a technical device to organize ideas, and that they have a deeper significance? One direction to explore this is via Penrose’s twistor and asymptotic twistor theory. It is known that much of the material described here is closely related to twistor theory; an example is the fact that asymptotic shear-free NGCs are really a special case of the Kerr theorem, an important application of twistor theory (see Appendix A). This connection is being further explored.
- With much of the kinematics and dynamics of ordinary classical mechanics sitting in our results, i.e., in classical GR, is it possible that ordinary particle quantization could play a role in understanding quantum gravity? Attempts along this line have been made [25, 15] but, so far, without much success.
- An interesting issue, not yet fully explored but potentially important, is what more can be said about the -space structures associated with the special regions (the -space ribbon of (4.40) that are related to the real cuts of null infinity. We touch on this briefly below.
- Another issue to be explored comes from the duality between the complex -space light-cones and the real shear-free but twisting NGCs in the real physical spacetime. From either one the other can be determined. It appears as if one might be able to reinterpret (almost) all the -space structures in terms of real structures associated with the optical parameters of the twisting NGCs and the real slicings associated with the ribbons. This reinterpretation would likely result in lost geometric simplicity and elegance – but perhaps would avoid the mysterious use of the complex -space for physical identifications.
- As a final remark, we want to point out that there is an issue that we have ignored: do the asymptotic solutions of the Einstein equations that we have discussed and used throughout this work really exist? By ‘really exist’ we mean the following: are there, in sufficiently general circumstances, Cauchy surfaces with physically-given data such that their evolution yields these asymptotic solutions? We have tacitly assumed throughout, with physical justification but no rigorous mathematical justification, that the full interior vacuum Einstein equations do lead to these asymptotic situations. However, there has been a great deal of deep and difficult analysis [24, 20, 21] showing, in fact, that large classes of solutions to the Cauchy problem with physically-relevant data do lead to the asymptotic behavior we have discussed. Recently there has been progress made on the same problem for the Einstein–Maxwell equations.

Living Rev. Relativity 15, (2012), 1
http://www.livingreviews.org/lrr-2012-1 |
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