A thought occurred to me when reading the Encarta article on relativity:

Now, it occurs to me that the velocity of the electron, being relative to the "original" reference frame of the radioactive sample, can also be considered to be the (relative) velocity of the radioactive sample, the laboratory, the earth, the solar system, etc. - if one only considers things from the electron's frame of reference.

So, one would expect to see a corresponding, velocity-induced mass increase in the original sample, the laboratory, the earth, solar system, etc. - because of the relative velocity of those things from the point of view of the electron's frame of reference.

Yes, certainly- from the electron's frame of reference. Of course, us mere mortals, who are traveling in "the laboratory, the earth, solar system, etc." would see no such change.

However, the frames of reference are not divorced. The mass effect on the electron is measurable from our frame of reference, and is also an effect on the electron itself (increasing the charge/mass ratio as measured against a static magnetic field {static WRT the original isotope}). Thus, we would expect that our relativistic mass increase would also be noticeable to us - would affect, for instance, the charge/mass ratio of every charged particle in our reference frame.

The increased inertia of a high speed particle is consequent to the fact that time is observed to pass slower in the frame of the particle - so the effective inertia is increased because a greater force is required to change its momentum in a given interval of time as measured in the slowed time frame of the particle - although it is common to speak of increased mass, what has increased is the particles opposition to a velocity changing force. We do not notice any change because some electron has acquired a relativistic velocity wrt to the earth. Our clocks are unaffected by the experiment. Einstein did opine that the higher value of the particles inertia would result in a higher gravitational mass, i.e., the effective inertial mass and gravitatinal mass are always equivalent

Possibly. One thing you may be overlooking is that a static electric or magnetic field in one frame transforms into part electric and part magnetic field in a different frame. Furthermore, if the electron is accelerated by a magnetic field (or by an electric field in its "own perspecive"), then its rest frame is an accelerated frame in which electromagnetic waves are induced in the lab frame. Futhermore, we usually deal in charge densities in the macroscopic lab frame, which of course transform nontrivially.

What I recommend that you do is to devise an experiment in the lab frame that would demonstrate a charge to mass ratio statically. Then, transform to the electron's frame and see what it looks like. Don't forget to transform all fields, densities, etc.

Thanks for all the replies. I think that I have got this now (but there's a follow-up question at the end).

What was throwing me was that mass is a quality which is experienced at v=0. Thus, I had thought that the electron's mass was "really" increasing (i.e., in its own reference frame), and thus that (e.g.,) two such electrons, coincident and sharing a velocity and direction, would therefore experience a quadrupled gravitic attraction toward each other. Put another way, I was assessing relativistic mass as a change in the "proper" mass, rather than a change in the mass effect of the particle wrt the original isotope.

I also neglected to take into account the fact that the magnetic field, while static wrt the lab and isotope, is moving at .5c wrt the electron - hence, the mass/charge ratio should properly be calculated from the perspective of the lab's reference frame and not from the electron's.

Now, if you have made it through all that waffle, here is a puzzler:

I am passing a solar system at a relative velocity of .707c ([itex]\gamma=0.5[/itex]). This causes me to see that the mass/velocity/oribital distance does not cohere with Kepler's/Newton's laws of planetary motion. Yet, relativity tells us that the laws of physics should appear the same for every observer. This seems to be a conflict.

This is one of my questions as well. Off of the top of my head, I would say that the answer lies in GR, but I do not know GR well enough at this point to attack the question.

Kepler's laws are static weak field approximation to gravitation, I believe. When a system is travelling at 0.5 c, at least the static part of the approximation breaks down. But also, even though you have recognized rather appropriately the distinction between proper and relativistic mass, the source of gravitation is not found strictly in mass, but rather in stress-energy, which includes kinetic energy. Therefore, I would venture to say that the weak field part of the approximation breaks down as well.

In other words, I would say that we would discard Kepler's laws in this case, and not count them among the "laws of physics."

Alternatively, we could transform to the rest frame of the center of mass of the system and see that Kepler's laws do hold to a high degree of approximation. We can perform this transformation in our own rest frame. Then, we would interpret the strange behavior as a consequence of the relativistic effects.

Because this is a mass/gravity-related question, one naturally wants to resort to GR. I am not sure that this is appropriate, however, since I would guess that GR is incapable of producing sufficient mathematical rationale to explain the apparent discrepancy. Further, it is also inadequate to take refuge in length contraction, since this exacerbates the problem, or time dilation, since it cannot address the question.

Further, since the relationship between mass, orbital distance, and orbital period is resolveable in Newtonian mechanics, then I would say that this is exactly the sort of "law" which Einstein meant to include under the explanatory umbrella of SR.

Of course, one wants to say that what is meant by "the laws of physics appearing identical to all observers" means "when one takes the theory of special relativity into account," but somehow I feel that this compromises the original ideals which produced SR.

Let me know if you come across any resolution for this.

What do you mean by "mathematical rationale?" I believe that GR demonstrates the mathematical toolkit necessary to address the issue, so I will choose this side. I do not have much time lately, though, so all I can do is basically come here and generalize.

I thought that these two features were the issue, not suggestions to resolve it. If not for them, what is the issue?

Newtonian mechanics does not apply to velocities near c nor to arbitrarily large gravitational fields. It is true that Einstein first toyed with the idea of retarded potential to incorporate gravity into SR, but he left the notion of "laws of physics" vague enough that Newtonian Gravitation need not be included as such. Newton's Law of Universal Gravitation (circa 1600's) has existed in a state of subtle falsification for a century at least (i.e. the consistent anomoly in the orbit of Mercury). It will be interesting to survey the results of the GPB, which incidently launched last week (or possibly the week before).

By postulating that the laws of physics should be independent of the particular lab frame chosen requires that, if we are to observe this non-Keplerian behavior in the 0.5 c solar system, then observers at rest in this solar system should also observe non-Keplerian behavior in our laboratory (in principle, of course). This axiom was posed in order to dismast the notion of an absolute rest frame (against which the evidence was surmounting at the time of Einstein's proposal). If, for instance, observers at rest within the 0.5 c solar system observed Keplerian behavior in our lab, then the first axiom would indeed be falsified. (Another for instance: I would say that the anisotropy of the CBR also seems to demonstrate a possible falsification, but, I know almost nothing about it.) The first axiom does not suggest what are to be the laws of physics; what it does is essentially give a criterion for which mathematical formulations of physical relationships can be declared "laws of physics." There are many demonstrations that support the relativity of the lab frame, so I believe that the first axiom is reasonable.

What is wrong with that? The theory is logically consistent in its region of validity.