When beginning a study of the special theory of relativity, one discovers that the theory of special relativity has as an axiom that the laws of physics are invariant with respect to transformations between inertial frames. The theory then states that Maxwell's equations are laws of physics and thus invariant between transformations between inertial frames. Furthermore, from Maxwell's equations we find that the speed of light is a constant and therefore must be invariant between inertial frames. Thus there exists a finite, constant speed limit to any physical process.

This is the only physical argument on purely theoretical grounds that I've ever heard which argues that there must exist a finite speed for any physical process.

My question is this: Are there any other physical arguments for the existence of a finite speed limit on which relativity can use as an axiom without appealing to the existence of the constant of the speed of the wave in Maxwells equations?


I would view the situation slightly differently.

The constant speed of light is a result of Lorentz symmetry, and the key assumption is that the universe is Lorentz symmetric. Any mathematical model postulated to describe the universe must include Lorentz symmetry, and therefore any such model will predict a constant speed of light. The point is that your argument seems to be:

  1. Maxwell's equations are laws of physics
  2. Maxwell's equations are Lorentz invariant
  3. therefore physics must be Lorentz invariant

But I would say you have this the wrong way round. We start with the assumption that Lorentz symmetry is fundamental, and if we assume this then Maxwell's equations are one possible mathematical model to describe the universe (and of course they've been proven a correct description by experiment).

So the physical arguments for the finite speed of light are every experiment that observes Lorentz invariance e.g. measurements of time dilation.

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  • $\begingroup$ I think I see what you are saying here. It's not simply that the speed of light is a constant in Maxwell's eqs, but that there is a Galilean asymmetry with respect to electromagnetic phenomena (for example the existence of magnetic and electric forces depends on your inertial frame) and the so we assume that Lorentz symmetry must be the real deal since the eqs are Lorentz invariant. Hence the speed of light must be constant. Is this consistent with your solution? $\endgroup$ – a_a Aug 25 '14 at 18:31
  • $\begingroup$ Yes, sort of. For example quantum electrodynamics also obeys Lorentz invariance, and I think QED would be regarded as more fundamental then Maxwell's equations. So Lorentz invariance is more fundamental than both of them. In fact, as far as we know everything respects Lorentz invariance making it about as fundamental as it's possible to get. $\endgroup$ – John Rennie Aug 26 '14 at 5:14
  • $\begingroup$ Hmm I thought that Lorentz invariance is violated when gravitational fields are involved? $\endgroup$ – a_a Aug 26 '14 at 6:13
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    $\begingroup$ @user117421: In GR Lorentz invariance is a local symmetry, i.e. the speed of light is always $c$ when measured locally. $\endgroup$ – John Rennie Aug 26 '14 at 6:15

From the today's view all interactions between physical bodies take place as between photons. Or as between electrons in their EM field. Einstein was genius as he declared, that c is the maximum speed. After clarification what the much of atoms volume is empty and yet we are not shrink to neutron mass because of the electrons repulsion based on EM fields we could realize that bodies could not be accelerated to a higher speed then c.

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that the theory of special relativity has as an axiom that the laws of physics are invariant with respect to transformations between inertial frames.

Technically, that's known as one of the two postulates.

Axioms would be rather the self-evident (inevitable) operational foundations upon which notions such as "(pairwise membership in an) inertial frame" could be defined and evaluated in the first place.

Such an axiom of the theory of relativity is in particular expressed by Einstein as the requirement that

All our well-substantiated space-time propositions amount to the determination of space-time coincidences [such as] encounters between two or more [...] material points.

Arguably such coincidences are in turn observable; and arguably coincidences of some (individual) "material point" with a corresponding signal front may be considered (axiomatically) as well. This also involves the (self-evident) notion of "recognizing a signal" as the first indication which a signal receiver has due to the signal having been stated by a source.

Are there any other physical arguments for the existence of a finite speed limit [...] without appealing to the existence of the constant of the speed of the wave in Maxwells equations?

The argument necessarily depend on how the axioms indicated above would have been used in order to define what's meant (by Maxwell, or anyone) by "speed" in the first place, as some quantity to be measured. I don't know if Einstein has been more explicit on this than to recognize that

In accordance with experience we shall assume that the magnitude $\frac{2~AB}{t'_A - t_A} = c$ is a universal constant [the speed of light in empty space].

(Note that the English translation appears a bit shorter than the German original.)

J.L. Synge ("Relativity. The general theory", Ch. III §2) put it (later, i.e. 1960) more thoroughly:

For us time [duration] is the only basic measure. Length (or distance), in so far as it is necessary or desirable to introduce it, is strictly a derived concept [and consequently, so is speed].

Accordingly, the "distance" of two suitable "ends", $A$ and $B$, to each other is defined as


if it had been established that meanwhile $A$ and $B$ had been "at rest to each other" (i.e. both having been members of the same "inertial frame") whereby it is established that their mutual ping durations are equal:

$$\tau^{\text{ping}}_{ABA} = \tau^{\text{ping}}_{BAB},$$

such that both $A$ and $B$ obtain the same value of their "distance" between each other, in mutual agreement that they, as a system, are thereby characterized.

Sensibly, $c$ is chosen there as some non-zero (and not infinite) parameter, since ping durations between different pairs are in general different from each other to begin with.

Together with the subsequent definition of "(average) speed" as ratio of

  • the distance "from start to finish", i.e. specificly as the distance of start and finish to each other, and

  • a suitable duration "from starting until finishing", which can involve determinations of simultaneity,

the constant $c$ is established as the "speed of the signal front", also known as "the speed of light in empty space"; which consequently means the same for all experimenters who follow this definition, based on having adopted the same axioms.

[...] speed limit

The signal speed obviously constitutes a limit on the speed of "material points" (participants) who took part together in one coincidence event (as signal event). Any other participant who did not take part in that signal event could have first found out about that at the latest at some meeting with anyone who had already taken part in the signal event, or generally even earlier. Consequently the speed of anyone having travelled from one meeting to the other is necessarily smaller (or at most equal to) the signal speed.

(This limit doesn't appliy to certain other notions of speed, such as phase speed.)

The main point is that from observations of coincidence events (as signal events) follow determinations of who had been at rest with respect to whom, and determinations of (ratios of) ping durations, and finally determinations of speed values in the first place.

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