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I will try to be as explanatory as possible with my question. Please also note that I have done my share of googling and I am looking for simple language preferable with some example so that I can get some insight in this subject.

My question is what is so special about $c$? Why only $c$. Its like chicken and egg puzzle for me. Does Einstein reached to $c$ observing light or does he got to light using some number which turned out equal to $c$.

Why is $c$ not relative. If something has zero rest mass like a photon why they only travel at $c$ in vacuum and not with $c+1$ or $c-1$?

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See my answer here, where one can show by symmetry arguments that a fundamental speed limit $c$ exists (possibly infinite). From there, it's a matter of experimental testing to find something with the behavior foretold by these symmetry arguments. The Michelson Morley experiment found that the speed of light had this special behavior. See John Rennie's answer too. There are many ways to look at this. –  WetSavannaAnimal aka Rod Vance Oct 11 '13 at 9:02
    
    
why using the "infinite" word ? we know that c is about 3*10^8 km/s, so its not infinite. and it is verifiable everyday in telecommunications (lag). –  v.oddou Jan 10 at 3:44
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3 Answers 3

up vote 18 down vote accepted

Special Relativity is based on the invariance of a quantity called the proper time, $\tau$, which is the time measured by a freely moving (i.e. not accelerated) observer. The proper time is defined by:

$$ c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 - dz^2 $$

This is similar to Pythagoras' theorem as learned by generations of schoolchildren, except that it includes time (converted to a distance by multiplying by $c$) and it has a mixture of plus and minus signs. The mixture of signs is responsible for all the weird effects like time dilation and length contraction, and because there is a mixture of signs the value of $d\tau^2$ can be positive, negative or zero.

If $d\tau^2$ is less than zero then $d\tau$ must be imaginary, and therefore unphysical. A quick bit of maths will show you that $d\tau^2$ can only be negative if you travel faster than light, and therefore that $c$ is the fastest speed anything in the universe can travel.

So $c$ is special because it determines a fundamental symmetry of the universe.

Footnote:

I've said $c$ is special while Kostya has said the opposite, but actually we are both right.

Kostya is right that there is nothing special about the speed 299,792,458 m/s (though if you change it by much you'll change physics enough that we may not be here :-). However the speed at which light travels is very special because anything travelling at this speed follows a null geodesic, i.e. $d\tau^2 = 0$. This is the sense in I mean that $c$ is special.

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+1 Let me offset that drive-by, though I actually wonder why you don't mention the Michaelson-Morley experiment which (unless I'm wrong) made people try to get their heads around the speed of light being observer-invariant, thus leading to that concept of proper time. (not to mention some Einstein thought-experiments) –  Mike Dunlavey Oct 11 '13 at 20:18
    
@MikeDunlavey That's Michelson-Morley, not Michaelson. –  called2voyage Jan 9 at 15:20
    
@called2voyage: Right. Thanks for the correction. –  Mike Dunlavey Jan 9 at 15:35
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Nothing. From Nature's perspective speed of light is entirely artificial number.

Imagine that you've discovered an alien culture that measured horizontal length $\ell$ and height $h$ in different units. They live on a planet with very strong gravitational force, and for them it is very difficult to rotate stuff in vertical plane. Such kind of rotations are really unnatural and counter-intuitive for alien-layman. While alien-physicists discovered them and have introduced a special transition coefficient $\alpha$ that transformed one dimension into another, allowing aliens to understand that both of these quantities are just projections of a more general thing called "distance": $$ d^2 = \alpha^2 h^2 + \ell^2 $$

Then imagine that there is an alien-physics.stackexchange site and someone asked there "What is so special about that $\alpha$?" And the answer is, again, "nothing". Nothing is special about $\alpha$ -- these aliens are just used to special conditions.

Same thing applies to homo sapience -- we are just used to very low speeds, which makes us think that time and space are completely unrelated and cannot be "rotated" into each other. While non-alien-physicists discovered that this is not the case, introducing a transition coefficient $c$...

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I really like your alien analogy, but I think one could take the conclusion to be that there is actually something very special about $\alpha$. It's the most (in fact the only) "natural" way of converting height into an equivalent horizontal length. It also corresponds to the angle $45^\circ$, which is arguably a special angle in the sense that it contains equal components vertical and horizontal. So I'd say there is some flexibility of interpretation here. –  David Z Oct 11 '13 at 18:50
    
Hmm... Were you mentioning "Homo sapiens" (at the end), or intentionally mentioned "sapience"? ;-) –  Waffle's Crazy Peanut Oct 13 '13 at 3:49
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@CrazyBuddy pun intended... –  Kostya Oct 13 '13 at 10:35
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Einstein in 1905 derived $c$ from the Maxwell's equations. Exact title of the paper in which special relativity was published was On the Electrodynamics of Moving Bodies. He essentially just resolved the problem of two electrons moving relative to each other. Moving electrons create a changing magnetic field and at the same time they are accelerated by Lorentz force and electrical force originating from the field.

In an inertial frame of reference associated with any electron there is no magnetic force ($q\mathbf{v}\times\mathbf{B}$, $\mathbf{v}$ being $\mathbf{0}$), only electric field acts on the electron. The solution is: if we change our frame of reference to a frame of reference which is moving at constant speed there is a transfer between magnetic and electric fields. There inherently is a constant in the transformation which units are metres per second. It was called "an Einstein's constant".

Einstein calculated speed of electromagnetic waves using well known at the time magnetic and electric constants (it is equal to Einstein's constant, by the way), and noted that resemblance with observed speed of light is... intriguing. It was just a suggestion of light being an electromagnetic wave.

So:

  • Einstein's constant
  • speed of light

two separate things. But since light is an electromagnetic wave, it's speed in vacuum is equal to Einstein's constant $c$.

Today commonly we call it just "speed of light" for convenience, but in many contexts it could be called "Einstein's constant".

$$c = \sqrt{\frac{1}{\epsilon_0\mu_0}}$$

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