I On the invariant speed of light being the upper speed limit

[e2m2a]

Guys, upon reading your replies, I must state that, although the information about how one could discover the metric from experiments are more than helpful, I was really interested in the mathematics behind it. Let me be more clear. I am now learning about GR and SR for the first time. Our professor just gave us the spacetime interval saying that it's a quantity which was found to be invariant under Lorentz transformations and then we calculated some interesting things. But, without knowing anything else except from the metric or the invariant interval could one deduce mathematically that there is a speed limit in the universe and that it is that of the speed of light? Or, put differently, how is the upper limit speed information encoded into the metric and the spacetime interval?

You have to start with grasping the physical concept first. Don't get sidetracked in the math. EInstein "postulated" that the speed of light was constant for all inertial frames, and showed from the Lorentz transformation equations that you can deduce that "c" is a limiting speed for material objects. But it goes deeper than this. Einstein postulated the constancy of c because of his conviction that the laws of nature are the same for all inertial frames. Einstein knew you could derive the speed of light from Maxwell's equations of electro-magnetism. Specifically, a solution to a second order, partial differential equation-- the wave equation The answer you obtain is that the speed of the electro-magnetic wave equals one over the square root of the permitivity of vacuum space times the permeabiliy of vacuum space. If you want to look at this mathematically, the permitivity and permeability of free or vacuum space are constants. So, intuitively, if you multiply a constant times a constant, you get a constant. If you take the square root of this constant, you get another constant. This constancy is derived from Maxwell's equatios, Now this, as far as Einstein was concerned, determined the speed of light's limiting speed. Why? Because a physical constant is a law of nature. And therefore, because the laws of physics are the same for all inertial reference frames according to the special theory , the speed of light must be the same. Focus on this, the physical concept. The speed of light is somehow determined by the permitivity and permeability of free space. If these constants ever changed, then the speed of light would change. Minkowski's spacetime ideas came after Einstein's special theory. Once you accept the physics, then the mathematics of Minkowski will make sense to you.

 

[Dale]

But, without knowing anything else except from the metric or the invariant interval could one deduce mathematically that there is a speed limit in the universe and that it is that of the speed of light?

How would you mathematically define the speed limit in the universe? If you can do that then you should be able to derive it mathematically.

I think that you need some physics in addition to the math to define what you even mean by the "speed limit".

 

[Herman Trivilino]

Have all of these experiments (or similar experiments) been conducted before? For example the lightning train one.

A version of the lightning train thought experiment is performed every day, all day long. Engineers who synchronize the GPS clocks must take this effect into account lest the system become so inaccurate that you might only know what city you're in instead of which street intersection you're near.

Thousands of other engineers and scientists must take this and the other relativistic effects into account every hour of every day to do things like steer subatomic particles or calibrate proton therapy devices.

 

[Jeronimus]

The way i understand it is as follows.

I imagine two laser beams emitted at the location i am standing, in opposite directions.

In a minkowski diagram this would constitute two worldlines at a 45° angle.

Now imagine yourself accelerating near instantaneously to 0.99999999 c (seen from the perspective of an observer who remains at rest in your former rest frame).

In your new rest-frame, the worldlines of the laser beams are at a 45° angle still since light always travels at c no matter which rest frame you are measuring it from.

You can accelerate again to 0.9999999 c from your new rest frame, but again, the laser beams will be at a 45° angle still again in your new rest frame(second postulate).

In your rest frame, you are always traveling towards the t axis. Hence, it is impossible for you to travel to any point at the area below the laser beams which you would have to if you wanted to travel at a speed higher than the laser beams travel, which is exactly c.

Therefore, the second postulate stating that light always travels at c in a vacuum absent of gravity also seemingly prevents us from traveling to any point included in the area below the two laser beams described above (minkowski diagrams) using acceleration.From the perspective of other observers traveling at arbitrary velocities relative to you, your worldline can approach a 45° angle arbitrary close, but never reach exactly 45° or below.

 

[Herman Trivilino]

In your rest frame, you are always traveling towards the t axis. Hence, it is impossible for you to travel to any point at the area below the laser beams which you would have to if you wanted to travel at a speed higher than the laser beams travel, which is exactly c.

I know what you mean, but the way you're stating it can be misleading. Your worldline can't trace a path there, which is not the same thing as you not traveling somewhere. In your example you and the observer and the light beams are all traveling through a one-dimensional space. The worldlines you describe trace a path through a 2-dimensional spacetime.

You don't travel towards a t-axis. You travel either away from or towards an origin along a straight line. Your worldline traces a path that you've described as towards the t-axis.

The fact that your worldline never has a slope less than one (can't cross a light line) is equivalent to saying that as you travel along that straight line your speed can never reach c.

 

[Herman Trivilino]

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.

 

[Herman Trivilino]

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.