--- On Tue, 6/23/09, John Denker <jsd@av8n.com> wrote:
"The four-momentum p i.e. m d(x)/d(tau) is conserved. The naive hypothetical
momentum-like quantity m d(x)/d(t) is not conserved"
I take it, m means the same thing in both sentences, namely, the invariant(rest) mass. If so, the second sentence is as true as the first one, except that in this case it advocates for the relativistic mass. If you replace m here with m*gamma, then m*gamma*v conserves in both meanings of the word - as conservation in time and Lorentz-invariance (when
performed correctly). Unfortunately, the term "conserves" is frequently used to denote both. There is no need to be ashamed of the fact that relativistic mass is m (rel) = m (inv)*gamma. At least, this is no more shameful than m(inv) = sqr[m(rel)^2 - p^2/c^2]. "Bottom line: Using velocity-dependent definitions for mass is a mistake"
It depends on how one uses it. The concept of relativistc mass cannot be held responsible for erroneous statements about it made by some of its opponents.
"-- For circular motion, as in the aforementioned cyclotron, the
3-acceleration (based on the coordinate time, t) differs from the
proper acceleration by
two factors of gamma.
-- For straight-line motion, the 3-acceleration differs from the proper
acceleration by three factors of gamma.
-- Meanwhile the total energy (rest energy plus kinetic energy) differs
from the rest energy by one factor of gamma.
These are well-known results."
They are, indeed. Except that only one dependence (m(rest)*gamma) is the relativistic mass.
"If you want to account for all three of these phenomena using velocity-dependent mass, it must be defined as rest mass times gamma, rest mass times gamma^2, and rest mass times gamma^3 all simultaneously."
Again, only one of them is relativistic mass. The existance of longitudinal mass is not
a valid argument against relativistic mass. Using such logics is similar to arguing only for the isotropic dielectric constant on the grounds that actual permittivity is generally a tensor quantity. I can aslo reverse the argument and direct it against the rest mass, saying that it is, at best, a secondary notion actually based on the relativistic mass.Indeed, the rest mass of a system of non-interacting particles is merely the sum of their relativistic masses. Saying instead that it is the "rest energy/c^2" is only decorative replacement, since the rest energy is again nothing else but the sum of the individual relativistic energies. As an extreme example, annihilation of positronium produces two photons, which, according to the "true religion", can only be named as massless particles. Then where does the rest mass of the system of these masless particles come from? Another erxample: There are only two known fields in
whjich a particle's trajectory is not self-intersecting closed curve, due to specific fine-tuning between vectors of force and acceleration: k/r and k*r^2. Based only on this information, and using ONLY the concept of the invariant mass, can one predict, wthout any equations, relativistic precession of the orbit, say, in the Coulomb's field? If yes, I am curious, how?With relativistic mass, this effect can be predicted before writing any equations. But I cannot (and do not) use these examples as an argument against the invariant mass. Both concepts are useful and meaningful aspects of reality. "Let us know how that works out for you".
That was addressed (I hope, rather extensively) in my book "Special Relativity and How It Works", e.g., Ch. 5, 9, 12.