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I have read over a dozen questions about the speed of light -- "why it $c$ constant?", "why can't anything travel faster than light?", "how do we know this?"

The responses are quite clear:

  • The invariance of light speed is determined empirically (e.g. from the Michelson-Morley experiment).

  • The speed of light is simply an axiom for physics and was discovered experimentally.

  • The invariant value of $c$ is a fundamental property of the universe.

My question is why can't the invariance of $c$ not be deduced theoretically with the following logic.

  1. As an object's velocity increases, its kinetic energy also increases.

  2. The kinetic energy growth is asymptotic, meaning it approaches infinity as the velocity approaches some value.

  3. This makes it impossible for anything with a mass to reach this velocity because it would require infinite kinetic energy.

  4. Therefore velocity must be have a limit.

See, this makes much more sense to me than the claim that the invariance of $c$ is just a postulate from lab work and that there's no reason for it to be invariant other than "it's just the way things are".

I suspect I've made a mistake. Perhaps the idea that an object's mass increases rapidly as speed increases comes from special relativity itself, which is derived from the assumption that $c$ is invariant. This doesn't, however, seem obvious to me since we should be able to observe this effect in experiments.

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    $\begingroup$ Just to clarify (since this misunderstanding seems to be common in these types of questions) I am not talking about the actual value of c, but its constancy. c is just used to represent some fixed value. $\endgroup$ – Frostbitten Mar 10 at 11:10
  • $\begingroup$ I've removed a number of comments that were attempting to answer the question. Please keep in mind that comments should be used for suggesting improvements and requesting clarification on the question, not for answering. $\endgroup$ – David Z Mar 10 at 12:14
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    $\begingroup$ as a sidenote - you can indeed arrive at invariance of c purely theoretically from taking seriously electromagnetism together with relativity principle as - I think - it was indeed done historically by Einstein. $\endgroup$ – Umaxo Mar 10 at 13:57
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    $\begingroup$ What do you mean? The constancy of the speed of light was deduced from classical physics. en.wikipedia.org/wiki/… $\endgroup$ – Solomon Slow Mar 10 at 14:19
  • $\begingroup$ I always thought c was variable but only constant in some frame $\endgroup$ – Frank Mar 10 at 22:44
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The kinetic energy growth is asymptotic, meaning it approaches infinity as the velocity approaches some value.

Unfortunately, this result already assumes that you know that there is an invariant speed. Without the invariant speed the formula for KE is $KE=\frac{1}{2}mv^2$ which has no limiting speed. It is only after you already know about the invariant speed that you get the expression $KE=((1-v^2/c^2)^{-1/2}-1)mc^2$ which goes to infinity as $v$ approaches $c$.

photons don't have mass so they can

This also requires already knowing about the invariant speed. Without the invariant speed there is no known relationship between all three of mass, energy, and momentum. With the invariant speed we learn $m^2 c^2=E^2/c^2-p^2$ which given the energy and momentum of light implies that light is massless.

So yes, those things, if known independently somehow, would have led to the conclusion of an invariant speed. But how could they have been known? Perhaps they could have simply been measured and known experimentally first, but historically it didn’t happen that way. Historically the invariance of c was postulated prior to measurements experimentally showing those points. Furthermore, such measurements would have been considered violations of classical physics.

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  • $\begingroup$ Interesting... do you know if the discrepancy between the two models' predictions can be measured experimentally, or is it negligible for speeds nowhere near c? I guess this is the source of my confusion, because I assumed the effect would be easily observed. $\endgroup$ – Frostbitten Mar 10 at 11:43
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    $\begingroup$ The discrepancy between the classical formula for KE and the relativistic formula for KE is $\frac{3}{8}m v^4/c^2 +O(v^6)$ so it is a fourth order effect, meaning it is usually small and only gets big at quite large $v$ $\endgroup$ – Dale Mar 10 at 11:55
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In classical physics the kinetic energy of an object is $$E_{\text{kin}}=\frac{1}{2}mv^2,$$ which clearly has no asymptotes. That is, the second step in you argument is wrong.

In special relativity the kinetic energy is $$E_{\text{kin}}= \left(\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}-1\right)mc^2,$$
which does have an asymptote. However, this formula is derived with the constancy of $c$ as a starting assumption.

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  • $\begingroup$ Thanks for your answer. I am confused as to why the second equation cannot be derived without the constancy of c as a postulate. Surely a classical experiment can evidence that the first is incorrect? $\endgroup$ – Frostbitten Mar 10 at 11:29
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    $\begingroup$ @Frostbitten See Dale’s comment on his own answer. You can use the formula he gives to estimate the speeds at which the relativistic correction becomes noticable $\endgroup$ – e4f5 Mar 10 at 12:10
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    $\begingroup$ @Frostbitten If you plug in some numbers you'll see that the difference is very small for any reasonable experiment performed on earth: E.g. at a velocity of 1000m/s the difference is 0.001%. So historically the invariance of the speed of light was measured first. $\endgroup$ – Almoturg Mar 10 at 12:12
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    $\begingroup$ @Almoturg and e4f5, your clarifications answered my question. Thank you. $\endgroup$ – Frostbitten Mar 10 at 13:57
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In classical mechanics, particles can have arbitrarily high speeds and still have finite energies, so your statement 2 fails. Given any finite speed $v$, the energy is $1/2 mv^2$. We can keep on increasing $v$ and the energy will increase and still remain finite.

Suppose you think the maximum speed is $C$. Then a particle with such a speed has energy $1/2mC^2$ (which is finite). Why can't a particle have speed $C + \delta C$, which has energy $1/2 m(C+ \delta C)^2$, which is still finite?

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As you have phrased it, your argument appears to be circular, because the asymptotic approach of energy to infinity at some finite maximum velocity is is deduced from the existence of a finite maximum velocity. However, we can tighten up the reasoning.

Either there is, or there is not, a finite maximum speed in nature. If there is not, then we would have Newtonian relativity which is empirically false. Consequently there is a finite maximum velocity and special relativity follows. (Newton himself argued that instantaneous action at a distance was an absurdity. I am inclined to agree, but I don't use that as a basis for a logical argument).

Note that relativity does not depend specifically on the speed of light, but only on the existence of a maximum speed. It just happens that, to the accuracy of measurement, light travels at the maximum speed.

The constancy of the maximum speed follows because, in natural units, all speeds are fractions of the maximum, and because, to carry out science at all, we must assume that the fundamental behaviour of matter is always and everywhere the same (Hume's principle of uniformity in nature).

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  • $\begingroup$ The second paragraph: is it really clear that there could not in theory be high velocity corrections to Newtonian mechanics that still allow arbitrarily large speeds? If not, this argument does not hold $\endgroup$ – e4f5 Mar 10 at 11:42
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    $\begingroup$ Yes, it is clear. Relativity was discovered from Maxwell's equations, not Newton's laws. The Michelson Morley experiment was a test of one prediction of Maxwell's equations (and apparently not even known to Einstein). Maxwell's equations were empirically established from many experiments. The fundamental problem studied by Lorentz, Poincare, Fitzgerald et al was that they are not consistent with Newtonian mechanics. The inconsistency is that arbitrarily high speeds would allow the definition of absolute time, necessitating a complete change to observed laws. $\endgroup$ – Charles Francis Mar 10 at 11:57
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    $\begingroup$ @CharlesFrancis I am unable to upvote your question so least I can do is comment to let you know it has helped me. $\endgroup$ – Frostbitten Mar 10 at 13:53
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As other posters have noted, your assumptions about classical mechanics are not correct. Newtonian mechanics cannot predict a theoretical maximum speed.

What you are looking for in your argument is to solve Maxwell's equations for the speed of an electromagnetic wave, and find that the result is completely invariant at $$c=\frac{1}{\sqrt {\mu_0 \epsilon_0} }$$ This leads to either some sort of universal medium (aether?) with which c is constant relative to, or the start of the ideas in special relativity that c is constant in every frame, which then leads to the Lorentz equations.

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  • $\begingroup$ That last paragraph was very interesting. Thanks for the explanation. $\endgroup$ – Frostbitten Mar 11 at 10:18

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