Relativistic mass and Electromagnetism in particle accelerators?

Question

Dear Physics Stack Exchange,

I've been rather troubled as of late on trying to see the problems or issues inherent in crank scientist or layman views on physics topics about special relativity. One of those that I have come across is a disagreement about the interpretation that particle accelerators give proof that relativistic mass (relativistic momentum) exists and thusly supports special relativity. Some have claimed that you could get the same results of particles going close to a certain speed but no faster given the speed of the interacting field is finite. So when an electron goes faster it begins to lessen the interaction or exchange of momentum or something similar between the electron and thusly the field or particles that are influencing its motion because its going closer to the speed of the interacting field. I've seen this time and time again as a response so I would love/prefer a mathematical response. I'm a college freshman now in multivariable calculus so present with discretion. Thank you for your time or effort in what ever manner possible.

Sincerely, a freshman college student.

Edit: I know that relativistic mass is a contentious subject and really I'm talking about relativistic momentum. I already know that the rest mass in special relativity is an invariant and is technically the real mass but that is not what's of contention.

Answer 1

Some have claimed that you could get the same results of particles going close to a certain speed but no faster given the speed of the interacting field is finite. So when an electron goes faster it begins to lessen the interaction or exchange of momentum or something similar between the electron and thusly the field or particles that are influencing its motion because its going closer to the speed of the interacting field.

First, it is highly doubtful that this method could quantitatively reproduce the observed behavior for any device. In other words, it is one thing to wave your hands and talk about lessening interactions at higher speeds, but it is a whole different thing to actually write down an equation that can quantitatively predict exactly how much the interaction lessens as a function of velocity in a way that agrees with known data. Physics is a quantitative science, so it is not just enough to say “what goes up must come down”, but you also need to say quantitatively when and where it comes down.

Second, there are some accelerators where this argument doesn’t even apply in principle. For instance, in a cyclotron the field in the dees is static. So the speed of the interacting field is zero (as much as it even makes sense to talk about the speed of a field as opposed to the speed of waves in the field)