Theoretical proof of the constant of speed of light $c$ in vacuum in all frames of references

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Solution 1

As WillO says, one has to state one's theory precisely through a definition of one's axioms (and allowable rules of inference).

But something you may find intellectually fulfilling is the following. Beginning from very basic symmetry principles - homogeneity and isotropy of spacetime as well as continuity of transformations between frames and continuous dependence on relative velocity and, finally, causality, see the Pal and Levy-Leblond papers I cite in my special relativity resource recommendation answer here. One can prove from these assumptions that there must exist a frame-invariant speed $c$ (it could be infinite i.e. Galilean relativity is included in the possible outcomes) and also one derives the form of the Lorentz transformations.

One then experimentally finds that $c$ is finite because the speed of light is experimentally found to behave in this frame invariant way.

Solution 2

There is a theoretical "prove" that $c$ is constant, though it could be considered "weak". Let me explain.

It is well known from mathematical physics that any quantity $f=f(x,t)$ obeying the equation $\frac{∂^2f}{∂t^2}-v^2\nabla^2 f=0$ represents a wave moving with velocity $v$. Now, if one uses Maxwells equations for vacuum, it can be shown that the eletric and magnetic fields obey exactly an equation of that type, that is (lets use just the electric field for simplicity)

$$\frac{∂^2E}{∂t^2}-\left(\frac{1}{\mu_0 \epsilon_0}\right) \nabla^2 E=0$$

This imply an electric field wave moving with speed $c=\frac{1}{\sqrt{\mu_0\epsilon_0}}$. That is the electric component of light.

Now, as you see, both $\mu_0$ and $\epsilon_0$ are constants coming from completely unrelated experiments like Coulomb's and Biot-Savart's experiments. They were added as "force transforming" constants. $c$ here is constant and there is no way to remedy it. It doesn't admitted any other value and it came from the theory of electromagnetics (electrodynamics).

It would be over simplifying to say electrodynamics summarizes just to the wave equation. However, this wave is at the core of the entire theory. The slightest change here affects all the theory. Nonetheless, any one accepting Galileo's transformation could in principle say that Maxwell's electrodynamics is wrong and needs change. That why I say it is "weak".

At the end of 19th century it became clear that the following very well established theories: a)Newton's mechanics; b)The (Galileo's) relativity principle; and c)Maxwell's electrodynamics; could not be simultaneously true. One could choose 2 of then but the remaining would request for reforms.

One could make $c$ constancy as a postulate, saving electrodynamics and strengthening it. If you admit the relativity principle also, then voila! you'd be a genius (Einstein to be precise).

From this 2 postulates one can get Lorentz transformation. You can see from this post that it keeps the speed of the light wave constant and equal to $c$, as expected.

In summary: electrodynamics is the "prove" theory for $c$ to be constant.

PS: Einstein actually made this twice in his relativity theory. He also transformed the weak correspondence principle, considering the evidence from Eötvös experiments that gravitational mass must be the same as inertial mass into the correspondence principle (making it strong) by postulating it.

Solution 3

The proof might have two steps totally based on geometry.

Given an arbitrary metric for curved spacetime in general relativity you can always find a local inertial frame. It is an approximation to first degree that helps you switch from general relativity to special relativity around a point for which the first partial derivatives of the metric all vanish. The metric is now equal to the Minkowski metric at this point and around.

This was the rigorous wording of the basic insight by Einstein that "the physics of curved spacetime must reduce over small regions to the physics of simple inertial mechanics."

Now consider the Minkowski metric for the inertial frame that remains invariant under Lorentz transformations.

This metric demands that there should be a constant multiplier of timelike dimension with the opposite sign of spacelike dimensions, i.e. signature, in the geometry of the four dimensions, if you want to have an invariant definition for the interval between points or events in spacetime after rotations, boosts, and translations.

This constant of the geometry of spacetime, with the dimension of space over time, in the inertial frame of reference is identified with the speed of light.

This constant should be there theoretically if you want to have an invariant notion of distance or interval in spacetime undergoing the Lorentz transformations.

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AravindM

Software Engineer by profession but physics enthusiast by heart.

Updated on December 28, 2020

Comments

  • AravindM
    AravindM almost 3 years

    Is there any theoretical proof for the constant speed of light in all frames of references? I know it is experimentally proven but just curious.

    • WillO
      WillO almost 7 years
      Of course there is a theoretical proof. There is in fact a one-line theoretical proof in any theory that takes the constancy of the speed of light as an axiom. There are other theories in which there are proofs, but none quite so short, and others in which there is no proof at all. To make this question meaningful, you need to specify what theory you're talking about.
    • Alfred Centauri
      Alfred Centauri almost 7 years
      @WillO, perhaps I misunderstand your comment, but in a theory that takes the constancy of the speed of light as an axiom, there is no theoretical proof of said constancy, one-line or other, to speak of because, if there were, the constancy of the speed of light would not be an axiom in the theory.
    • Kalpak Gupta
      Kalpak Gupta almost 7 years
      To be finicky, shouldn't it say experimentally verified instead of proven?
    • WillO
      WillO almost 7 years
      @AlfredCentauri: A proof, according to the standard definition, is a finite list of statements, each of which is either an axiom or follows via specified rules of inference from previous statements on the list. A list of length one is a proof if and only if the one item on that list is an axiom.
    • Alfred Centauri
      Alfred Centauri almost 7 years
      @WillO, I see your point in the context of a formal system but I must confess that I find the notion that 'stating' an axiom proves the axiom preposterous if not vacuous.
    • WillO
      WillO almost 7 years
      @AlfredCentauri: Do you find it equally preposterous (in the same formal context) that every proof begins with an axiom?
    • Alfred Centauri
      Alfred Centauri almost 7 years
      @WillO, not at all; why?
    • WillO
      WillO almost 7 years
      @AlfredCentauri: Recognizing that we're straying off-topic here, it's probably best not to belabor this, but: If $A$ is an axiom, and if $B$ follows from $A$, then you are (if I understand you) willing to consider $(A,B)$ a non-preposterous proof of $B$, but not willing to consider $(A)$ a non-preposterous proof of $A$. But both proofs rest on exactly the same foundation, so I'm having trouble seeing why one would count as more preposterous than the other.
    • Alfred Centauri
      Alfred Centauri almost 7 years
      @WillO, the key difference for me is "B follows from A", i.e., the axiom A and some rule of inference leads to B and thus, B is proved (B is a theorem) in the system.
    • WillO
      WillO almost 7 years
      @AlfredCentauri: But that is not a difference, because it is equally true that "A follows from A" --- the axiom A, together with the rule of inference that says any statement can be derived from itself, leads to the theorem A. So I do not think that your comment can be an accurate description of whatever difference you are perceiving.
    • Alfred Centauri
      Alfred Centauri almost 7 years
      @WillO, "any statement can be derived from itself" - I'll just leave it at that.
  • jim
    jim almost 7 years
    "One can prove from these assumptions that there must exist a frame-invariant speed c" where c is the speed of light?
  • J. Manuel
    J. Manuel almost 7 years
    What about the fact that $c$ is constant in electrodynamics ($c=\frac{1}{\sqrt{\mu_0 \epsilon_0}}$) with the particular twist that you cannot change that, even when the reference frame changes without "destroying" the theory. Maxwells Electrodynamics demands for constancy of the speed of light.
  • Selene Routley
    Selene Routley almost 7 years
    @jim Yes, see the papers by Pal and Levy-Leblond I cite in the answer I link here.
  • Selene Routley
    Selene Routley almost 7 years
    @J.Manuel Well, that's yet another motivation and the one that Einstein used.