icarus wrote: No. Why would you chose either the proton or electron rest frames for a magnetic moment calc.? Does either of those frames have any relevance whatsoever to the laboratory/observer frame that will be experimentally verifying/dispatching your notion?
According to Einstein's Special Relativity they are all equivalent; also to the center-of-mass reference frame (I note that you avoided my challenge about positronium: what a pity). All that is required is that what you observe in one inertial reference frame can be transformed into the other refrence frame by means of the Lorentz transformation: You cannot transform a magnetic moment so that it flips when going from one inertial reference frame to another. I just used this argument to prove how futile and stupid the argument is that you must
choose a reference frame in which the center-of-mass is stationary.
In fact there is no reason why you cannot choose the origin of your inertial refreence frame to be at the charge-centre of the two charges. In terms of Dodds wrong calculation this will give ZERO magnetic moment.
In fact, relative to an external observer (needed to measure the presence/absence of the hypothesised mag. moment) both frames would be rotating,
Really!!! If you choose one of these reference frames as the stationary, inertial refrence frame, as allowed by Einstein's first postulate, it is only the other refrence frame that is rotating: Not both!
both are thus non-inertial and violate the SR principle, that you seem to want to include in this simple gedanken case and that does nothing but confuse .... unless you have a deeper insight.
Both cannot be non-inertial unless you have a unique stationary reference frame which is NOT possible. You can measure movement along any inertial reference frame and this reference frame need not be unique.
Another point, magnetic moments, and angular momentum or any moment calcs are not typically frame invariant ... you have to choose wisely to get sensible/useful results.
This remark can lead to a long drawn out discussion. Suffice to say that I am of the opinion that you should have used "fudging" instead of 'wise choice".
And if you must know, I keep digging with you here because I'm kind of fascinated by the solitary charge having no electric field hypothesis though (i.e. put away the tinfoil beanie) ... it is one I've toyed with in the past for other reasons, but how would it ever be proven or not?
Eureka, we are finally touching brains!! I am excited and I am not writing this to be sarcastic. The fact is that there are two possibilities (either there is a field or there is not a field) which cannot be proved either way by experiment. This is so since to determine whether there is an electric-field energy around a solitary charge you must use a test charge; and then you do not have a solitary charge anymore. In fact the electric-field lines for two cahrges do NOT correspond to the spherically symmetric field-lines around a solitary charge as is wrongly illustrated in elementary textbooks.
So a wise physicist would look for circumstantial evidence to decide which of bthe two possibilities is more self-consistent than the other.
I would reason that the assumption which leads to infinities in calculations must be the wrong one! And the assumption which consistently lead to infinities is (guess what) that there actually is an electric-field energy around a solitary charge. In fact Coulomb's law has only been proved BETWEEN charges not around a solitary charge.
And is it even needed as a concept if the only tools we use to describe EM are the interactions between charges .... does it change any calculable results what the solitary state is?
Yes it is needed since such a field is used to calculate that there is a magnetic field around a single moving charge and thus leads to the wrong conclusion that Bohr's atomic model has a magnetic moment.
Another case which cannot be proved by experiments is whether there are free charge-carriers withi a metal when there is no electric-field within the metal. To measure free charge carriers you must apply an electric-field so you can never know whther the charge carrirers are generated by the applied electric field or not. In my book I show that the assumtion that there are no free charge carriers when there is no electric field (ignoring tempertaure effects if course) is afr more self-consistent than to assume that there are free charge carriers even when there is not an electric field.