Exploring Relativistic Mass & Length to Help STEM Students

[hutchphd]

TL;DR Summary

QUERY for the RELATIVITY DOYENS:

The notion of "relativistic mass" was found to be cumbersome and pedagogically inutile and has fallen into disrepute. The term "mass" has supplanted "rest mass". In that spirit, why should we continue to call the measurement made by a moving observer the "length" of an object? Seems equally inappropriate to me.
Comments please
Is there good sobriquet to use for relativistic length ?
Just a thought to aid the struggling college frosh STEM students

Comments please
Is there good sobriquet to use for relativistic length ?
Just a thought to aid the struggling college frosh STEM students

 

[Ibix]

People do talk of the rest length and the contracted length when clarity is needed.

I think there is a difference between length and mass in that a rest frame isn't always particularly well defined for length. For the mass of a system you can always add the four-momenta of the parts and find the frame in which this is parallel to the time axis. With length (or distance between particles) it isn't obvious if you should average four velocities or four momenta to find a rest frame.

You could make a case for "length" to mean rest length when you are talking about a material object which has a well-defined rest frame, but then you have to draw a distinction between lengths of things and distances between things, so I don't think there's such a clear cut case here.

That's what I think off the top of my head, anyway.

 

[hutchphd]

I think defining the length of, say, an amoeba is fraught with difficulty independent of frame. I was tacitly considering a rigid object I guess. And the term "length" could not mean distance. A rotating object might be tricky...?

 

[Orodruin]

Length contraction and the definition of length itself is fraught with frame dependence to no end. However, it is an ingrained part of relativity history. When I teach special relativity, I do not put that much focus on length contraction but I do mention it and explain the origin after discussing other things that are relevant to the discussion such as relativity of simultaneity. The discussion also is always accompanied by a caveat regarding the use of the length contraction formula and an encouragement to use Lorentz transforms instead unless you are absolutely sure the application is correct.

 

[anuttarasammyak]

$$m^2=E^2-p^2$$
and
$$s^2=t^2 - l^2$$
So you may regard mass m and world interval s would correspond in mathematics.
m is energy measured at COM system or system with zero momentum.
s is time measured in IFR where the spatial distance of the two events are zero or i multiplied by
length measured in IFR where the two events are simultaneous or time difference of the two events are zero.

 

[Nugatory]

The notion of "relativistic mass" was found to be cumbersome and pedagogically inutile and has fallen into disrepute. The term "mass" has supplanted "rest mass". In that spirit, why should we continue to call the measurement made by a moving observer the "length" of an object? Seems equally inappropriate to me.

Length contraction is more useful because we need it to preserve the distance-speed-time relationship across frames.

”A spaceship leaves Earth traveling at .8c towards a destination one light-year away, arriving 15 months later. In the spaceship frame, the ship is at rest and the destination is rushing towards it at .8c - how long until it arrives?”​

cannot be sensibly explained at an introductory level without appealing to the length contraction of the distance. The alternatives, abandoning $v’=-v$ or “speed equals distance divided by time”, are worse (although that’s pretty much what we do when we get to GR, where coordinate velocities are treated with appropriate disdain).

Relativistic mass on the other hand does nothing except preserve $F=ma$ in the special case of transverse acceleration. If we never had to do anything except calculate the trajectories of charged particles in a uniform magnetic field (B only, no E) we would find relativistic mass to be a useful property but in most other problems we have better (more general and powerful) ways of describing the relationship between work, momentum, and velocity.

 

[vanhees71]

If you describe the properties of "massive point particles" it's useful to define all their intrinsic properties (mass, charge and, for quantum theory also spin) in their rest frame, which is clearly a preferred frame for a point particle. That's why in the modern description (where modern means from about 1908 on, when Minkowski clarified the mathematics behind special relativity once and for all) uses only such invariant definitions for intrinsic quantities, and "relativistic masses" of different kinds are abandoned once and for all from the scientific literature. Why there are still textbooks, even at the university level, where outdated stuff is printed again and again, is completely ununderstandable ;-)).

It becomes more complicated for extended objects like fluids (gases and liquids) and solid (necessarily elastic!) bodies. There you have local descriptions, i.e., you define intrinsic quantities in the local rest frame of macroscopically "infinitesimal" elements of the medium. For such many-body systems you get then scalar descriptions of temperature, chemical potential, particle-number density, etc. The confusion about this part of relativistic physics lasted pretty long with contradictory descriptions of thermodynamical and statistical quantities until about the end of the 1960ies, when the issue was clarified by van Kampen et al.