Why is the speed limit of special relativity $C$ equal to the speed of light $c$ and why must there be only one?

Question

DISCLAIMER: this question is different from other questions posted here previously despite accidentally similar titles. This question is not a duplicate of the one linked bc the linked question is distinct and none of the answers there provide what I've asked for.

It is known that the Maxwell equations lead to a speed $c=(\mu_0\epsilon_0)^{-1/2}$, and the numerical agreement of this speed with that of light led to Maxwell's prediction that light is an electromagnetic wave. However, it was unclear at the time which reference frame $c$ is to be measured in. It was also recognized by various mathematical physicists of the time that Maxwell's equations are invariant under Lorentz transformations. This invariance led to Minkowski's vision of spacetime, which (it appears) before Einstein was thought only to apply to electromagnetic phenomena.

It was Einstein's great insight in his paper "On the Electrodynamics of Moving Bodies" to elevate $c$ to the status of a truly universal speed limit, applying to all known physical phenomena (not just those of an electromagnetic nature). Many experiments have since confirmed Einstein's postulate (or failed to invalidate it, depending on your view of the scientific method), and his more general theory of relativity (which additionally incorporates the equivalence principle) is among the most successful physical theories ever conceived.

My question, which will be made much more precise at the end, is what is the theoretical, physical reason why the characteristic speed of special relativity $C$ is exactly equal to the speed of light $c$ and is it unique? A slightly similar question was asked here, but it does not ask (nor do any of the answer attempts provide) the more fundamental physical reason why $c$ should be the local speed limit of all physical phenomena, not just of those that are electromagnetic in nature as was understood before Einstein.

Despite much searching, I have been unable to find a satisfactory theoretical explanation. Knowing from my experience talking with colleagues that some will see this question and not understand the subtlety involved, I wish to point out that many of the usual answer attempts are circular arguments. The following are a few I've come across:

1. The invariant interval. Answers that involve defining a special-relativistic invariant from the outset assume the framework of special relativity, which is precisely what they are supposed to demonstrate.
2. Galileo + Maxwell. The usual argument here is the following: "As Galileo argued, (i) the laws of physics should not vary between inertial reference frames. (ii) Maxwell's equations give a constant speed $c$. Therefore, (iii) $c$ is constant in all inertial reference frames." This argument is inconclusive because (ii) does not necessarily imply that $c$ is the same in all inertial frames (and not just one special frame) unless one has already assumed special relativity, which is precisely what is supposed to be proven. Einstein himself clearly saw the circular reasoning involved, which is why he saw the need to postulate (iii) from the outset. Now, if someone could demonstrate that (ii) is true for all inertial reference frames, then this argument would be sound, but no one has done this (to my knowledge). To summarize, Galileo + Maxwell ≠ Relativity.
3. Finite energy. There are different flavors of this argument, which is at best an attempt at proof by contradiction. One is that it is impossible to accelerate an object without bound given finite (mass-)energy. While true, this argument presents merely a practical limitation, which is not a sound theoretical argument. Therefore, it does not prove. Another flavor of argument is that it would take infinite energy to accelerate an object to an infinite speed and at some point the object would become so massive as to form a black hole. Again, the only reason we know objects become more or less massive is special relativity. Therefore, this argument too is circular and does not prove. We could go on like this, but I'm sure you get the idea.
4. Speed of causality. This is just the assertion that $C = c =$ speed of causality. This is not a legitimate answer.

Having given a general sketch of the problem and addressed what appears to be an all too common logical pitfall, these are different aspects of my question that I believe an answer attempt should address:

1. The existence of a relativistic speed limit for all physical phenomena. It seems well-established experimentally that there is a characteristic speed governing the physics of the visible universe and that it is exactly equal to the speed of light. How can this be explained from first principles? The closest thing to a real answer I've seen is given in this paper. Other papers, for example this one, suggest the discreteness of spacetime, i.e., there being a non-zero minimum length or time, as an explanation. Naturally, I would expect a complete answer to this question to involve quantum mechanical considerations. Here is a quote from Einstein posted on Sabine Hossenfelder's blog that highlights, I think, some of the difficulty involved:

But you have correctly grasped the drawback that the continuum brings. If the molecular view of matter is the correct (appropriate) one, i.e., if a part of the universe is to be represented by a finite number of moving points, then the continuum of the present theory contains too great a manifold of possibilities. I also believe that this too great is responsible for the fact that our present means of description miscarry with the quantum theory. The problem seems to me how one can formulate statements about a discontinuum without calling upon a continuum (space-time) as an aid; the latter should be banned from the theory as a supplementary construction not justified by the essence of the problem, which corresponds to nothing “real”. But we still lack the mathematical structure unfortunately. How much have I already plagued myself in this way!

2. The uniqueness of a characteristic speed. Given that there is a characteristic speed, can it be proven that this is the only characteristic speed in our universe and that it must apply to every particle, known and unknown? This question is different from the one presented here because the author there is too vague, and here or here because the authors there limit themselves to forces which we already know obey relativity or they present a question that begs a circular argument. Suppose for concreteness that there are two types of matter particles A and Ω. Let A have the speed of light $c$ as its limit and let Ω have a speed limit $e$. Suppose further that these particles do not interact directly although they are both "massive" insofar as they both apparently exhibit gravitational attraction. (Obviously I have in mind dark matter). Does it necessarily follow that $e=c$? Please be rigorous.

ADDENDUM: Some quickly pointed out that I didn't give a starting point. I had intended this as a strength for the answerer by giving them their choice of where to start, but this was instead perceived as a weakness of the author. The isotropy and homogeneity of spacetime as well as Galileo's principle (which now forms the first postulate of special relativity) would have been excellent starting points to me. For example, I find the paper linked by Maximal Ideal in their solution is helpful. Obviously "all explanations in physics start with observations which inspire physicists to create a model which describes them." But every model has a motivation, and once we learn what we can from our model, we seek better models. Why? Because models fail! Because we don't want just a model; we want to know how things truly are! This is the dream, even if all we have at the moment is a model. All I wanted was a reasonable starting point that any reasonable person would accept (e.g., isotropy, homogeneity, etc.) and an argument that is not circular. But apparently my question hit a nerve. Why?

Answer 1

There is no a priori reason that the invariant speed (your $C$) must equal the speed of light (your $c$). In fact, it is still possible that these two speeds will eventually be found to be unequal.

It seems well-established experimentally that there is a characteristic speed governing the physics of the visible universe

As you say, the evidence strongly indicates that there is an invariant speed $C$. This and the isotropy and homogeneity of spacetime leads to special relativity, the Lorentz transform, and the spacetime interval, regardless of whether or not light happens to move at $c=C$ or at some $c\ne C$.

In other words it is not true that

Answers that involve defining a relativistic invariant from the outset assume the framework of relativity, which is precisely what they are supposed to demonstrate

Relativity follows from the invariance of $C$. That light travels at the invariant speed, $c=C$, is historically the first derivation. But that assumption is not necessary and can be replaced with the aforementioned experimental data in the one-postulate derivations of relativity.

With special relativity in place we find the following relationship between energy, mass, and momentum: $$m^2 C^2=E^2/C^2-p^2$$ and between energy, velocity, and momentum: $$\vec v=\frac{\vec p\ C^2}{E}$$ Again, these relationships are based entirely on the invariant speed $C$ and hold regardless of whether or not light happens to move at $c=C$ or at some $c\ne C$.

Combining those two equations we find that anything that is massless, $m=0$, travels with velocity $v=C$ regardless of its energy $E$.

So the statement that light travels with $c=C$ is simply the statement that light is massless in a relativistic universe. This is not a matter of logical necessity or assumption, but an experimentally falsifiable fact. So far, experiments place a very low upper bound on the photon mass of $10^{-18}\mathrm{\ eV}/c^2$.

However, if the photon were eventually found to have some small mass, the only necessary change to relativity would be to call $c$ the “invariant speed” instead of the “speed of light”. The standard model would be in trouble, but relativity would be fine.

Answer 2

(I wanted to write this as a comment but it became too long and I would consider my post as an attempt at an answer.)

First, it's not clear what you're willing to accept as an acceptable starting point. Every theoretical argument must start with some postulates/axioms or empirical observations that are generalized to laws.

Anyways, one possible approach to your question is to consider this paper. It implies the only "type of relativity" that satisfies "isotropy, homogeneity and a principle of relativity" is either Galilean relativity or Einstein relativity. Some invariant speed $C$ needs to be invoked as a tie breaker between the two (this must be empirical).

Once you deduce that the universe obeys Einstein relativity, the velocity addition formula of Lorentz transformations show there must be only one speed that is invariant. Further, my post here argues the universe cannot obey both Einstein and Galilean relativity at the same time. So we know the universe obeys Einstein relativity in which exactly one speed $C$ is invariant.

Now independently of all the above reasoning, one can show that Maxwell's equations are invariant under Lorentz transformations with invariant speed $c$. So either Maxwell's equations only hold in one reference frame and the principle of relativity is not preserved or it is preserved but only under Lorentz transformations with invariant speed $c$.

If $c\ne C$, then we have different parts of the universe obeying Einstein relativity with different invariant speeds. This means there is a part of the universe not obeying Einstein relativity with invariant speed $c$, and there is a part of the universe not obeying Einstein relativity with invariant speed $C$. Therefore, the total universe as a whole obeys no principle of relativity whatsoever. The only alternative left is that $c = C$.

In all cases, the only way the principle of relativity is preserved is if $c = C$. Indeed, there can only be at most one invariant speed, as you inquire in #2.

(There can be relativity violating particles and it can be the case that Maxwell's equations are not completely accurate or complete, but this is up to empirical science to answer.)

Answer 3

You are asking for a demonstration that the speed limit of special relativity is the same as the speed of light. That is a fine question; the short answer is that light happens to travel at the speed limit of the universe because light is massless. Other massless particles also travel at the "speed of light."

The longer answer to your question is that modern QFT starts by assuming that spacetime is Lorentzian, asks what type of particle-interaction-lagrangians are possible within this framework, and then you can show more rigorously that only massless particles will travel at the speed of light.

But then it seems like you are asking for too much, because you say:

The invariant interval. Answers that involve defining a relativistic invariant from the outset assume the framework of relativity, which is precisely what they are supposed to demonstrate.

What framework are we allowed to assume then in order to answer your "why" question?

This argument is inconclusive because (ii) does not necessarily imply that 𝑐 is the same in all inertial frames (and not just one special frame) unless one has already assumed special relativity, which is precisely what is supposed to be proven.

What exactly do you want here, a proof that the speed of light is equal to the speed limit of relativity...without assuming relativity? Again, what framework and premises are allowed as starting assumptions?

Regardless, if you assume that the speed of light is the same in all reference frames, then you can go through the usual derivation to show that the Lorentz transformation is the only (physically simple) transformation which accomodates this. From this, you get relativistic mechanics, which shows energy and momentum become infinite for a massive particle as you approach $c.$

Does it necessarily follow that 𝑒=𝑐? Please be rigorous.

Well you would get two different Lorentz transformations with different values, which is incoherent.

To summarize:

1. You can assume Lorentzian spacetime and show that massless particles travel at the universal speed limit.
2. Or you can assume that the speed of light is the same in all reference frames, and derive the usual relativistic mechanics.

Neither of these are circular.