As for "transverse mass is different from longitudinal mass."?

That one refers to the fact that electrostatic energy behaves as having some sort of electromagnetic mass, which can increase the normal mechanical mass of the bodies, right? Now, that one is quite confusing as it suddenly discuss electromagnetic 'forces', that depending on if they are 'transverse, at an right angle to the motion induced, or 'longitudinal', that is following the induced motion then presents us with different mass. Intuitively it seems true that a force acting transverse, at a right angle to the induced motion, will introduce a higher resistance than a 'force' applied in the direction of the motion 'longitudinally'. But now we are discussing 'forces' and different types of 'resistance' it seems to me, and then wonder what effects they might have on a 'rest mass' motion?

"Since a lot of relativistic effects are keyed off of speed, you can imagine then that applying forces along the direction of motion (which change speed) would have significantly different effects than forces perpendicular to motion (which don't change speed)."

For a clearer understanding of how I like to see it you might look at

MATTER, MASS AND ELECTROMAGNETIC MASS And to get to how Einstein saw it I believe

Transverse Mass in Einstein’s Electrodynamics to express it nicely. One can get quite a clear impression of the ideas from those two without needing to dive too deep in the math. Have to admit that the wiki

Mass in special relativity fail to make the distinctions between them clear to me. But reading those others I refer to I find me begetting a succinct description of the difference between the concepts rest mass and electromagnetic mass.

==

Got to admit though that the concept of 'rest mass' is more ambiguous than it seems at a first look, well, to me it is. It's pretty simple to see that electromagnetic forces are involved in all descriptions of matter that I know of, I can't really see how to exclude it to get to some state of matter not involving those? But it's still quite simple to differ between radiation and matter, just take a look at your telly. If you read "MATTER, MASS AND ELECTROMAGNETIC MASS" you can see that the author brings out some thoughts of his own at the end, wherein he questions Einsteins ideas, or maybe how they are interpreted? Those ones I'll leave to the author to defend

( Although it seems that some suggest that there is a difference between 'energy' and 'mass' making the equivalence slightly misaligned according to

The Equivalence of Mass and Energy. And

The Inertia of Energy have this to say about the Mass/Energy-equivalence.

"The exact proportionality between the extra inertia and the extra energy of a moving particle naturally suggests that the energy itself has contributed the inertia, and this in turn suggests that all of the particle’s inertia (including its rest inertia m0) corresponds to some form of energy. This leads to the hypothesis of a very general and important relation, E = mc2, which signifies a fundamental equivalence between energy and inertial mass. From this we might imagine that all inertial mass is potentially convertible to energy, although it's worth noting that this does not follow rigorously from the principles of special relativity. It is just a hypothesis suggested by special relativity (as it is also suggested by Maxwell's equations)." )

It all seems to fall down to if there would be any matter if those different 'forces' were taken away, and looked at it that way, I don't expect it to be. But not all of them are electromagnetic, right

So wanting to justify matter from that point of view you will need to find a way of transforming all those 'forces' into some electromagnetic equivalences it seems to me? Don't expect that to be do-able myself. We define electromagnetism to the 'electrons' orbitals and interaction with 'photons', virtual or not, am I right? And what they do seems to be bringing in are some 'kinetic energy' into the atoms 'equilibrium' that then gets transformed?

Awhh