From Paul Camp: As a sometime relativity theorist, I have to say that the idea of a velocity dependent mass makes my teeth itch. And one of my colleagues (also a relativist) thinks that is exactly what you did, in equation 14.41.
I don't think so. I think what you are trying to do is distinguish between inertia and mass, giving an operational definition for inertia which pretty much isn't done elsewhere. You're doing this in order to get to the relativistic definitions of momentum and energy. This leads to a very unfortunate equation where you seem to be using the idea of velocity dependent mass. I know that you aren't, but that is the specific point my colleague can't get past.
Personally, I'd advocate throwing the whole idea of inertia over the side and speaking only of things that can be measured. For us relativity guys, everything measurable needs to be in terms of invariant quantities and covariant relationships. What you can measure are spacetime events. Differentiating with respect to proper time gives you a 4-velocity. Multiplying by mass gives you a 4-momentum, the same quantity that emerges from Noether's theorem BUT only if the mass you multiply is the rest mass.
Again the issue arises in the context of constructing a plausibility argument for relativistic modifications of Newtonian quantities. My colleague is wrong. It is not an incorrect argument. But for me it is a backwards argument.
What we should do is define momentum from the beginning as what it is, and consider appropriate models in limiting situations like v << c. That makes the problem go away.
But if you want to stick with the approach you're using, and it has some things to recommend it, at the very least use different notations for inertia and mass. Don't call them both m. Even with subscripts.
You can take either of these comments and do with them what you will. It is your judgment. I'd like to use the book but I have a colleague who thinks Knight hung the Moon because it looks familiar. He's also a relativity theorist and is using his claim that you are using velocity dependent mass as a "major blunder."
From Paul Camp: As a sometime relativity theorist, I have to say that the idea of a velocity dependent mass makes my teeth itch. And one of my colleagues (also a relativist) thinks that is exactly what you did, in equation 14.41.
I don't think so. I think what you are trying to do is distinguish between inertia and mass, giving an operational definition for inertia which pretty much isn't done elsewhere. You're doing this in order to get to the relativistic definitions of momentum and energy. This leads to a very unfortunate equation where you seem to be using the idea of velocity dependent mass. I know that you aren't, but that is the specific point my colleague can't get past.
Personally, I'd advocate throwing the whole idea of inertia over the side and speaking only of things that can be measured. For us relativity guys, everything measurable needs to be in terms of invariant quantities and covariant relationships. What you can measure are spacetime events. Differentiating with respect to proper time gives you a 4-velocity. Multiplying by mass gives you a 4-momentum, the same quantity that emerges from Noether's theorem BUT only if the mass you multiply is the rest mass.
Again the issue arises in the context of constructing a plausibility argument for relativistic modifications of Newtonian quantities. My colleague is wrong. It is not an incorrect argument. But for me it is a backwards argument.
What we should do is define momentum from the beginning as what it is, and consider appropriate models in limiting situations like v << c. That makes the problem go away.
But if you want to stick with the approach you're using, and it has some things to recommend it, at the very least use different notations for inertia and mass. Don't call them both m. Even with subscripts.
You can take either of these comments and do with them what you will. It is your judgment. I'd like to use the book but I have a colleague who thinks Knight hung the Moon because it looks familiar. He's also a relativity theorist and is using his claim that you are using velocity dependent mass as a "major blunder."