On the invariant speed of light being the upper speed limit

[Mister T]

I would here interject that "relativistic mass" is a largely obsolete concept that is prone to misunderstanding.

And a large part of that misunderstanding results from errors made in the way the concept is presented.

It is Newton's third law: Force = Mass times acceleration. Since mass approaches infinity, an infinite force is needed to accelerate a massive particle to the speed of light, which in turn will require an infinite amount of energy.

This explanation tends to plant within the recipient's mind the erroneous notion that the relativistic mass $\gamma m$ is a genuine relativistic generalization of the Newtonian mass $m$. Moreover, using that particular explanation in that particular context, one is referring to $\gamma^3 m$ as the mass, not the relativistic mass $\gamma m$.

It is this particular point that resulted in authors of introductory physics textbooks revisiting this particular erroneous explanation during the 1990's. By the end of that decade the concept of relativistic mass had virtually disappeared from those textbooks, never to return.

Oh, and by the way, it's Newton's 2^nd Law, not Newton's 3^rd Law, that relates force to mass and acceleration.

 

[Orodruin]

And a large part of that misunderstanding results from errors made in the way the concept is presented.

Of course it does, but I believe it is also redundant because we already have a different name for it that is less prone to confusion: "total energy". Of course, in a unit system that does not use c = 1 there is a conversion factor between mass and energy, but it is just that - a conversion factor. I see no need to introduce a new name just because I divide a quantity by an arbitrary constant that may or may not be convenient (compare with Planck's constant $h$ vs $\hbar$). If insisting on this, I would instead call it "energy equivalent mass" or something to that effect.

 

[Mister T]

Of course it does, but I believe it is also redundant because we already have a different name for it that is less prone to confusion: "total energy".

To me that's a less satisfying reason. Rest energy and mass are equally redundant, yet there's something valuable to be taught by retaining both terms, namely their equivalence. Placing relativistic mass in the lexicon obscures the meaning of that equivalence, in my opinion.

 

[Orodruin]

Rest energy and mass are equally redundant, yet there's something valuable to be taught by retaining both terms, namely their equivalence.

I disagree. I do not think you need to call it two different things in order to get the main insight, that the energy content at rest is the same as the inertia at rest.

 

[vanhees71]

First of all one should stress that energy and mass are conceptual different. Take a classical particle: Its energy together with its momentum combines to a Minkowski four-vector,
$$(p^{\mu})=\begin{pmatrix} E/c \\ \vec{p} \end{pmatrix}.$$
The mass of the particle in modern times is understood always and only as its invariant mass, i.e., a Minkowski scalar. It is related to the four-momentum vector by the energy-mass relation
$$p_{\mu} p^{\mu} =\eta{\mu \nu} p^{\mu} p^{\nu} = \left (\frac{E}{c} \right)^2-\vec{p}^2=m^2 c^2,$$
or solved for the energy
$$E=c \sqrt{m^2 c^2+\vec{p}^2}.$$
It's really much easier than to use the very confusing notions of "relativistic mass" in the very early beginning of Special relativity. Before Minkowski everything was pretty much a mess, and we should be very thankful that Minkowski worked out the mathematics of the manifestly covariant formalism.

As to the question of the title, I'd say the most satisfactory line of arguments towards Special relativity is to investigate the question how the mathematics of spacetime must look if you assume the special principle of relativity, i.e., the existence of an inertial frame, where test particles move uniformly if no forces are acting on them and any inertial observer observes a homogeneous time (no point in time is distinguishable from any other) and observes space as a Euclidean 3D affine manifold (implying homogeneity and isotropy of space).

After some math you come to the conclusion that there are two spacetime geometries compatible with these symmetry assumptions: Galilei-Newton and Einstein-Minkowski spacetime. The latter has a universal speed, $c$, which you can call the "univeral speed limit", i.e., no particle can move with a speed faster than $c$ relative to any inertial observer.

Now it turns out that Maxwell's equations of electrodynamics in a vacuum (in this context most naturally formulated in terms of Gaussian or Heaviside-Lorentz units) which also contain a fundamental speed, namely the phase velocity of electromagnetic waves in a vacuum, are in fact a relativistic field theory, and the speed of light in the vacuum coincides with the universal speed limit.

As any fundamental principle in physics, it's an empirical question (a) whether Galilei-Newton or Einstein-Minkowski space-time is a more accurate description of nature and (b) whether really the speed of light in Maxwell's vacuum equations coincides with the universal speed limit of Einstein-Minkowski space-time. Of course, (a) is answered in favor of Einstein-Minkowski spacetime which describes all phenomena so far observed in nature with high accuracy (except gravity, whose incorporation into relativity theory leads to the extension to the General Theory of Relativity (Einstein 1915)). Concerning (b) the modern formulation is the measurement of the photon mass. The best upper limit today is $m_{\gamma} < 10^{-18} \, \text{eV}$ which is damn small, and there's no hint of deviations from $m_{\gamma}=0$ at all. So it's pretty save to say that Maxwell's equations (and its quantum-theoretical extension, QED) with a massless photon, i.e., the speed of light coinciding with the universal speed limit of relativity.