The postulates of the special theory of relativity say that there is a limiting speed - the speed of light. But this is a postulate. There are experiments confirming that the speed of light is approximately $3 * 10 ^ 8 m / s$. From what fact does the finiteness of signal propagation come from? All theories, whether it is special relativity or general relativity, confirm the postulate that the speed of light is finite, but in Maxwell's electrodynamics there is no limit on exceeding the speed of light. There, the magnitude of the speed of light is the constant of propagation of an electromagnetic wave in a vacuum as $\frac{1}{\sqrt{\epsilon_0\mu_0}}$. Lorentz transformations and, as a consequence, the incorrectness of the formula for adding the speeds of Hallei $v=v_1+v_2$ are also a consequence of the fact that the postulate of the special relativity declares that the maximum speed is the speed of a photon. If in the special theory of relativity it was said about the limiting speed as $2c$, then the Lorentz transformations would still be correct, but the maximum speed would be different. Where does the finiteness of the signal propagation come from?
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1$\begingroup$ That's like asking why the mass of the electron is the value it is, isn't it? $\endgroup$– AllureCommented Oct 24, 2020 at 19:54
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8$\begingroup$ This post seems confused - you start by saying that this is a postulate, then say its empirical, then ask for a mathematical proof. What do you actually want to know? $\endgroup$– jacob1729Commented Oct 24, 2020 at 19:57
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2$\begingroup$ Maxwell's equations for electromagnetism are covariant (i.e. invariant in form) under Lorentz transformations which are valid with limiting speed $c$. If the limiting speed would be $2c$ neither Lorentz tranformations would be correct nor Maxwell's equations would be covariant under them. $\endgroup$– VoulkosCommented Oct 25, 2020 at 11:38
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$\begingroup$ TLDR: The postulate is that light travels at the same speed in any coordinate frame. That came from Maxwell. The conclusion is that if anything travels at a constant speed regardless of coordinate frame, then nothing else in the universe can travel any faster. See answers, below, for more detailed explanation. $\endgroup$– Solomon SlowCommented May 5, 2022 at 16:57
5 Answers
The postulates of the special theory of relativity say that there is a limiting speed - the speed of light.
This statement of yours is incorrect. The second postulate of special relativity is-
The speed of light in free space has the same value c in all inertial frames of reference.
This postulate doesn't say that the speed of light is the highest allowed speed. But using this postulate we can get the Lorentz transformations. Then we define the momentum of a particle with mass m as- $$\bar{p}=\gamma m\bar{u}=\frac{m\bar{u}}{\sqrt{1-(\frac{u}{c})^2}}$$ The reason we define like this is if we define $\bar{p}=m\bar{u}$ momentum won't conserve. But the above definition conserves momentum. From using this definition of momentum we get the energy as-$$E=\gamma mc^2=\frac{mc^2}{\sqrt{1-(\frac{u}{c})^2}}$$
You can easily see that any finite energy cannot take a non zero mass m to the speed c. So no signals can propagate faster than c. If a particle has zero mass it should always travel at c and if its mass is non zero then its speed is always less than c.
Additional Note
If you are wondering why kinematical laws are connected to light, they really aren't connected much to light. If we replace the second postulate with-
There exists a speed b which is the same in all inertial frames of reference.
Then we can get the same laws of special relativity and we can show that anything which has 0 mass has to travel at the speed b (which coincides with c as photons have zero mass/rest mass). In fact, there CAN be other massless particles also, for example, the hypothetical Graviton is massless (Its existence is needed for String theories but not for Loop Quantum Gravity).
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$\begingroup$ And if its mass is imaginary then it should always travel faster than c. $\endgroup$ Commented Nov 3, 2020 at 13:48
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$\begingroup$ @user253751 well tachyons are hypothetical as of now, so neglected them. $\endgroup$ Commented Nov 3, 2020 at 14:12
The speed of light in vacuum is measured or like you said determined by maxwells law. The R.Th says it is independent of the relative velocity of the light source, so one cannot increase it by starting from a moving source . This also can and is be proved by experiments. So how will you get to 2c?
The speed of light was known to be finite long before Einstein. As you point out, Maxwell figured it out from $\frac{1}{\sqrt{\epsilon_0\mu_0}}$. This was experimentally confirmed as invariant by the Michelson-Morley experiment, which led on to the Lorentz-Fitzgerald contraction and thence to Einstein's inspiration.
So really, you are asking why the permittivity and permeability of free space turn out to be the minimal values permissible.
Well, firstly, in every environment other than high vacuum they are higher. This is down to the nature of matter as charged particles and the way they interact with passing photons. Light goes fastest when there is nothing slowing it down.
But then, where do those limiting numbers come from? We might hope to find an answer in the zero-point fields of quantum field theory, the density of virtual particles and suchlike. But here we hit a famous discrepancy of some 120 orders of magnitude - we have not the faintest idea what is going on. We just pull the measured numbers out of the sky and plug them in, as Maxwell did.
Since we do pretty much that with well over a hundred similar fundamental constants and parameters, we don't worry about it too much.
Yes, relativity says there is a limiting speed, call it $c$ but don't give it a name at this point, and that gives the Lorentz transformations.
If you apply the Lorentz transformations to electrostatics, which contains $\epsilon_0$ to describe the force between charges, then you get magnetism, which contains $\mu_0$ to describe the force between currents, and this shows that $\mu_0=1/(c^2 \epsilon_0)$.
That gives you Maxwell's equations, which turn out to have wave solutions, and the wave velocity of these EM waves is $1/\sqrt{\mu_0 \epsilon_0}$ which is $c$. At this point we can say "$c$ is the speed of light".
The speed of C (which is also the speed of anything that is massless (including light)). Is just a 45° angle in a spacetime diagram. It reveals and illustrates the relation between space and time.
Now that space and time are "connected" that "connection" is the speed of C.
And as the story goes, nothing can travel either faster neither slower than C, in spacetime.
Initially we thought that we were living inside a flat Euclidian spacetime. But experiments showed that maybe we are not. And the 1st confirmation of this was when we tried to measure the speed of something massless (like light) and we found that its speed was invariant.
So now we have a theory that changed our relativity, from the Galilean relativity that we had before, to the Einstein relativity that we use now, where spacetime is not Euclidian but hyperbolic.
In such a spacetime velocities in space and time are both relative and the way they are related is called the Lorentz factor which includes the speed of C in it.
In such a hyperbolic spacetime, the speed of light is effectively infinite.
By effectively, I mean the following:
If you are traveling at the speed of light, you would be able to travel ANY distance at ANY time. You could travel at some distant star that is 500 million light years away, in ZERO time - not because you were indeed traveling that fast, but because you simply didn't experienced any time.
Ofcourse it did took you 500 million years to reach that star - but you just didn't experience those 500 million years. To you, that trip looked instantenous.
Those 500 million years did pass, you just didn't experience them. That is the idea behind [proper velocity][1]
$${\displaystyle {\textbf {w}}={\frac {d{\textbf {x}}}{d\tau }}}$$
It is an alternative to ordinary velocity, the distance per unit time where both distance and time are measured by the observer.
Now in Galilean relativity, indeed the speed of light was thought as infinite, but also, Galilean relativity preserves causality by having a fixed and invariant speed for time.
Everything including light, (in Galilean Relativity) is traveling in time at 1s every second... So If we call that speed in time λ then $$ λ= \frac{1s}{s}$$
and thus λ is a constant and invariant speed.
But ofcourse back then, nobody questioned why λ was constant or invariant, because it was obvious (although not anymore) that if we were to synchronize our clocks, and you get in a Ferrari and drive away - when you come back our speeds would still match... So everything is moving towards the future at the speed of λ which was our philosophical idea about the reality that we were living in, the idea that by having a constant and invariant speed for time, would preserve causality.
Einstein's Relativity has a diffrent constant and invariant speed that also does the same job, preserves causality. That speed is not a speed in time, (because now speeds in time are relative) but a speed in spacetime. And that speed is C.
And C is a speed that indicates that there is a relation between space and time. That relation is give by the Lorentz factor γ
$${\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {dt}{d\tau }}}$$
In relativity, you can go as fast in space or in time as you want, as long as, when your speed in time and space are combined they yield C. So our speed in spacetime is always C.
The finiteness of the propagation of light, comes from our "struggle" to preserve causality inside a hyperbolic spacetime like the Minkowski one that Einstein's relativity is using.
If anything was to travel in spacetime faster than C, (even light) then causality wouldn't be preserved. The same ofcourse happens if anything was to travel in spacetime slower than C.
C is not just the speed limit in spacetime, its the ONLY SPEED allowed in spacetime and nobody can go slower or faster than it.
But ofcourse as I showed above, since time and space are relative, we can effectively travel "faster than C" because there will be a huge amount of time that we wouldn't experience. But that will only apply to those who measure speeds incorrectly. Once we measure one's speed correctly (taking into account both his speed in space and time) then everything in the Universe travels at C in spacetime.
Edit: I don't understand why people downvoted this post. Nothing of what I said is wrong and they illustrate perfectly the core ideas of Einstein's relativity. I wish people would at least tell me why they are downvoting this :/ I'm ok if you downvote me, but please at least tell me why.
