B A couple of questions on mass increase

[Dale]

I'd appreciate, as long as you are at it, if you cared to explain (with your usula clarity) what is the big deal in abjuring mass increase?

In relativity usually we use units where c=1, so things that differ only by factors of c are considered to be the same thing. This means that relativistic mass is just another name for energy.

On the other hand, the invariant mass is a fundamentally different quantity ($m^2=E^2-p^2$ in units where c=1). This quantity turns out to be very useful.

Thus we have three useful quantities (m, E, p) and three relevant words (mass, energy, momentum). So it makes more sense to assign one word to each quantity than it would make to assign two words to E and none to m.

 

[Nugatory]

what is the big deal in abjuring mass increase?

Over the past century, the way we think about relativistic mechanics has evolved in ways that make the concept of relativistic mass positively harmful less useful. This is somewhat to be expected, as Einstein was working his way through unknown territory - the mathematical formalism of modern relativity was developed after the fact. It's not surprising that with 20/20 hindsight and the benefit of knowing the answer ahead of time, Einstein's 1905-vintage formulation can be improved upon.

Einstein proposed the notion of relativistic mass increase in 1905, perhaps because he was reluctant to sacrifice the three-dimensional $F=\frac{dp}{dt}$, an understandable position in historical context (even if the different behavior of transverse and parallel forces is seriously disconcerting). Only several years after that did Minkowski publish the four-vector mathematical formulation of Einstein's discovery that we use today - and in that formalism relativistic mass is somewhere between unnecessary and downright confusing. It was a full decade after that that Einstein discovered general relativity - and one of the prerequisites for learning GR is unlearning relativistic mass and learning Minkowski's formalism.

Another historical factor is that during the first few decades of the 20th century, some of the most practical laboratory tests of relativistic kinematics involved measuring the acceleration of a moving body when subjected to a transverse force - and this is one of the very few problems that is simplified by the concept of relativistic mass. Nowadays, modern particle accelerators provide far more compelling support for special relativity, so experiments of this sort are no longer an area of active investigation. You'll have noticed that the paper Dale pointed to is about how to do an undergraduate-level demonstration, not about developing any new understanding (and it's been that way for decades - when I did a similar experiment as an undergraduate in the 1970s the proposition being tested was not "Are the predictions of SR supported by experiment?", it was "Is nugatory competent to set up and run an experiment?").

So the answer to your question "what is the big deal in abjuring mass increase?" is, in no particular order:
- It gets in the way of understanding the modern mathematical formalism of SR.
- If you learn it you have to unlearn it before you can progress beyond SR.
- The only problems that it makes easier are no longer especially interesting.
as well as the issues that others have already touched on in this thread.

 

[vanhees71]

Energy is energy and mass is invariant mass and invariant mass only. It is easy to calculate an electrons velocity with a given kinetic energy. First you calculate the total energy, including its rest energy $E_0=mc^2$ ($m$ invariant (!) mass)
$$E=E_0+E_{\text{kin}}.$$
Then the momentum is given by
$$\vec{p}^2 =\frac{E^2}{c^2}-m^2 c^2,$$
and finally it's three-velocity is
$$v=\frac{|\vec{p}| c^2}{E}.$$

 

[vanhees71]

I am not asking for an exact value, but for an appproximation of 5 digits and surely an experiment at LHC or the like produces such approximation.

- The predicted value for an increase of 9 rest masses is v = 0.994 987 437 (2) with the accuracy of 9 digits or, if we take all 10 digits, we get the value od 10.000 000 09 masses)
- At LHC or at any other sinchrotron, you know the exact energy provided, the magnetic field and the radius r of the collider

I suppose the can get a result with a five-digit accuracy, am I wrong?

Now I use natural units $c=\hbar=1$. Then for $E_{\text{kin}}=9m$ you get $E=10m$ and thus $\vec{p}^2=E^2-m^2=99 m^2$ or $|\vec{p}|=\sqrt{99}m$ and $v=|\vec{p}|/E=\sqrt{99}/10 \simeq 0.994987$.