Why would mass/energy increase between observers and charge remains same.
I guess some will leap simply saying: "who said mass changes?" Feynman, Lectures on Physics Vol I, ch15:

Secondly, under what situtations can we consider m changes with velocity?? Or is this a general statement?

Third: is it possible to define a radius or charge density (delta function?) for electron? I remember some phrase like "SR doesn't allow electron to have a radius", but don't know what it's based on.

The invariant mass of a moving object remains constant. The relativistic mass of a moving object changes. These concepts are defined in SR and in the previous FAQ entry.

GR also has some (several!) concepts of mass that are very similar, but much more involved. The most applicable to this situation is the ADM mass, energy, and momentum of a system. The ADM quantities can be defined in any asymptotically flat space-time, and match the behavior of their SR counterparts. (Some other candidates for "mass" in GR, such as the Komar mass, require a static system for their defintion, which does not allow them to address the issue of the "mass" of a moving system).

The ADM energy of the system increases with velocity, as does the ADM momentum. The ADM mass would generally be considered to be the invariant quantity of the ADM energy-momentum 4-vector, i.e. it would be an invariant quantity.

But this is not the end of the story.

Because it is really energy (in the form of the stress-energy tensor) which causes gravity, it is conceptually correct to think of gravity as being different from electromagnetism. In electromagnetism, the source of electric fields (charge) does not depend on velocity - it is invariant. In GR, the source of gravity (energy) DOES depend on velocity.

This is one of the things that makes GR harder than relativistic electrodynamics.

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Something that I should add - while the source term for gravity depends on energy, which depends on velocity, GR is still generally covariant. This means that if one moves at the same velocity as the mass, its gravity will look perfectly normal (i.e. what one is used to, spherically symmetric and approximately Newtonian for weak fields). The gravity of an object cannot be used to measure it's velocity. For another example, objects do not turn into black holes if they move too fast - the object is not a black hole in its own frame, so it's not a black hole in any frame.

In both Newtonian mechanics and SR, point particles would have an infinite self-energy, making them poorly behaved.

To resolve the problem of the self-energy of an electron takes quantum mechanics. Unfortunately quantum mechanics and GR are not fully integrated, so the topic of point particles in GR is not well defined.

Hmmm... Actually, I was expecting a QED-wise reply, maybe I should've posted this in QP forum.

For mass variance: what I've understood from your post and phys-faq, it seems OK to talk about relativistic mass as long as we have a flat space-time and can ignore GR, right?

Umm, what's the reason for that. Assuming that energy packets (photons) are responsible for electrodynamic interaction...

Speaking QP-wise, can we assign a radius/charge density to electron (again, would dirac delta do it?).

I don´t know much about this, but here it is:
Relativistic mass in flat spacetime is ok in the same sense that energy is ok (it´s actually the same). Rest mass has the advantage of being invariant.
About charge invariance: wikipedia says nobody knows why. I don´t know either.
There is a classical definition of the electron radius (where the self-energy equals it´s mass), but when you measure it, it´s a dirac delta. That causes those renormalisation problems, but again I don´t know anything about it.

The whole "relativistic mass" vs "invariant mass" debate is mostly a waste of time, and a source of of confusion. It is possible (at least for an isolated system) to simply view relativistic mass as energy. If you prefer to call energy by a different name, it doesn't really matter, you are changing the semantics, not the physics. It doesn't matter as long as everyone is assigning the same meaning to the words, at least - unfortunately, because there are a couple of different defintions of mass, it's quite possible for people to talk about the "mass" of a moving system and mean two different quantities even though they are using the same word.

Discussing non-isolated systems just makes the situation worse - our sole proponent of "relativistic mass" on the board loves to do this, but it usually just creates a giant muddle.

There isn't any clear well-documented, and widely accepted route from "relativistic mass" to "gravity" that I'm aware of. The route from "the stress energy tensor" to "gravity" is quite clear and well documented, however (though it may be difficult to grasp all the details, at least you know that there is a path that definitely gets you from here to there).

The fundamental difference between gravity and electromagnetism is that gravity couples to _everything_. Electromagnetism only couples to objects with charge.

The reason that electromagnetism is not affected by velocity is that charge is not affected by velocity.

"Energy" does not create electromagnetic fields, the source term is "charge", and it's a quantity that does not depend on velocity.

Since you seem to want to view everything in terms of QED, you might find the following useful:

There's more in the paper itself - the basic idea is you start out with a flat (SR-like) space-time, you then assume that you have a force that couples to _everything_, and you then find out that because this force couples to everything, that you cannot observe this flat background metric - the only quantities you can actually measure correspond to a curved metric. This curved metric is, if the theory is consistent, the same curved metric that GR predicts.

As far as your questions about the electron go - they, at least, would probably be best raised in the QM forum.