Re: New version of a relativity FAQ

Pmb wrote:
A new version of the FAQ "Does mass increase with speed?" was written and is now online at
http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html

This is a well written article on this whole relativistic mass thing. The author makes a great point
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A common argument against the use of relativistic mass is the fact that the equation E=mc^2 says that a body's relativistic mass equals its total energy, so why should we use two terms for what is essentially the same quantity? We
should just stay with energy, and use the word "mass" to refer only to rest mass. But this argument neglects the definitions of the words mass and energy. Mass is a property of a body that we have an intuitive feel for;
its definition as a resistance to acceleration is very fundamental. Energy, on the other hand, is defined in physics in rather ad hoc ways. Neither concept is even remotely understood by modern physics.
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I think we're going to have to review changes to the FAQ more carefully. This page now just mentions a few straw-man objections to "relativistic mass", and does not discuss at all the actual reason why "mass" is best considered to be an invariant. And worse: it is internally inconsistent in just this area.

For instance, I would not say "[mass's] definition as a resistance to acceleration is very fundamental", I would say its definition as "how much 'stuff' is present" is more fundamental. "relativistic mass" does not obey this, but the usual terminology does.

In the physics 101 I took many years ago, the example given
was to differentiate mass from weight, as the former should
not vary with position, but for a given object the latter
diminishes at the top of a mountain or in orbit. Having
"mass" vary with velocity is equally inappropriate, as is
having it vary with direction. "How much stuff is present"
is CLEARLY independent of such variations.

A more serious objection is that the author says "Mass is a property of a body" (which I agree with), but "relativistic mass" IS NOT A PROPERTY OF A BODY (it varies with the body's motion and its physical situation). This article is internally self-inconsistent.

Another inconsistency: "resistance to acceleration" is dependent on the direction of the applied force, but "relativistic mass" is not direction dependent. The author has abandoned his own "very fundamental" definition.

Like PMB, tears come to my eyes from this page :-). But not
because of its relationship to my personal beliefs (which PMB
calls "quality of writing"), but rather because it is internally
inconsistent and therefore wrong. And it is wrong in subtle
ways that can easily confuse its target audience. That is
diametrically opposed to the purpose of a FAQ page.

The first paragraph ends "which concept is more useful?", but the article only mentions elementary uses and never touches on the real issue: the relationship between kinematics and dynamics.



The underlying question is: What is the best way to generalize terminology and concepts from Newtonian physics to relativity? Note I said "relativity", not "SR", and that is an important aspect of this.

There are two very different generalizations:

The one which has become standard:

Term Newton SR GR
------- ------------ ---------- ----------
velocity 3-vector 4-vector 4-vector
momentum m*v m*V m*V
mass invariant invariant invariant
force 3-vector 4-vector 4-vector
acceleration 3-vector 4-vector 4-vector

And the one that article advocate:

Term Newton SR GR [*]
------- ------------ ---------- ----------
velocity 3-vector 3-"vector" (none)
momentum m*v gamma*m*v (none)
mass invariant gamma*m (none)
force 3-vector 3-"vector" (none)
acceleration 3-vector "matrix mess" (none)

[*] These are not discussed at all, and I base them on the
fact that in GR any coordinates are equally valid.

Note, please, that these terms are those that appear in the DYNAMICS of a physical theory, but their definitions are KINEMATICAL. The whole point of kinematics is to make the dynamics simpler by making fundamental symmetries be incorporated into the terminology and notation. This second table completely abandons that. I challenge advocates of this second table to write down the Lagrangian of classical electrodynamics using those quantities, and compare to the standard formulas. Then for extra credit try the Lagrangian of QED (this will be exceedingly perverse, and I'll be surprised if it fits on one page).

Comparing these tables shows why the standard terminology has become standard. "Relativistic mass" and the other entries in the second table are of use ONLY pedagogically, and ONLY in teaching elementary SR. They are essentially useless in advanced SR and in real physical theories like QED and GR, beause those definitions do NOT reflect the underlying kinematical symmetry (local Lorentz invariance); the standard meanings do reflect it, which is at base why they are so much better. And I challenge the pedagogical use of "relativistic mass" -- IMHO stressing symmetries is an EXTREMELY important part of teaching physics.

Remember in Newtonian mechanics different 3-vectors
transform differently under boosts, but in SR all 4-vectors
transform identically. This gets amplified in the differences
between the two tables, and the "matrix mess" for acceleration
is truly perverse compared to the transformation of
4-acceleration. But that is minor compared to the complexity
of writing dynamical equations using the second table, and
the difficulty of generalizing them to GR....

At the turn of the last century J.W. Gibbs went on a crusade to establish the 3-vector notation we all use today. My arguments are quite similar to his: the notation should reflect the underlying symmetry. Gibbs obviously won that dispute; I have no doubt about a similar ultimate result of this terminological and notational dispute. For the same reasons.


I think this page needs to be re-written again. It should discuss the origins of the terms "mass" and "relativistic mass", and why the latter has gone out of favor in research and advanced classrooms, and why the current terminology is better -- it simplifies the DYNAMICS of any relativistic theory by making the KINEMATICAL relationships obey the underlying symmetry. As is, it takes a rather superficial and elementary viewpoint.


Tom Roberts
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