# Why and how is the speed of light in vacuum constant, i.e., independent of reference frame?

I was told that the Galilean relative velocity rule does not apply to the speed of light. No matter how fast two objects are moving, the speed of light will remain same for both of them.

How and why is this possible?

Also, why can't anything travel faster than light?

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The speed of light being constant is a starting point for theorizing, rather than a conclusion. By that I mean we've mean we've tried to measure it, and it seems to be constant. Relativity theory is saying "if this is true, then what are the consequences"? Well, one of the consequences is that nothing can travel faster. –  Carlos Jul 23 at 16:08

The view of most physicists is that asking "How can it be that the speed of light is constant?" is similar to asking "How can it be that things don't always go in the direction of the force on them?" or "How can it be that quantum-mechanical predictions involve probability?"

The usual answer is that these things simply are. There is no deeper, more fundamental explanation. There is some similarity here with the viewpoint you may have learned in studying Euclidean geometry; we need to start with some axioms that we assume to be true, and cannot justify. Philosophically, these ideas are not precisely the same (mathematical axioms are not subject to experimental test), but the constant speed of light is frequently described as a "postulate" of relativity. Once we assume it is true, we can work out its logical consequences.

This is not to say that, in physics, postulates stay postulates. For example, many people are especially concerned about probability in quantum mechanics, and are trying to understand it based on more fundamental ideas (see decoherence as one example). As another example, Newton's laws of motion were originally taken as unprovable postulates, but are now explained via quantum mechanics (see Ehrenfest's theorem).

At this time, the constancy of the speed of light, or more generally the principle of Lorentz symmetry, is not justified by anything considered to be more fundamental. In fact, the assumption that it is true has been a guiding light to theoretical physicists; quantum field theory was invented by thinking about how quantum mechanics could be made to respect the ideas of relativity.

Although we do not have a theoretical justification for the constancy of the speed of light, we do have very accurate experimental tests of the idea. The most famous is the Michelson-Morley experiment, which measured the relative speed of light in different directions to see if it was affected by the motion of the Earth. This experiment rejected the hypothesis that the motion of the Earth affects the speed of light. According to the Wikipedia article I linked, a modern version of this experiment by Hils and Hall concluded that the difference in the speed of light along directions parallel and perpendicular to Earth's motion is less than one part in $5*10^{12}$. In addition to direct tests of the speed of light, there have also been many other experimental tests of special relativity. (I haven't read this last page carefully, but, on flipping through, it looks good.)

There are a few caveats worth mentioning. In general relativity, the speed of light is only constant locally. This means that the distance between two objects can increase faster than the speed of light, but it is still impossible for light to zip past you at a speed faster than the normal one. Also, in quantum theory, the speed of light is a statistical property. A photon may travel slightly slower or faster than light, and only travels at light speed on average. However, deviations from the speed of light would be probably be too small to observe directly.

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I have exactly the same thought, if distance b/w two objects can obviously increase with more speed, what will happen? The relative speed of one object will be more than the speed of light? –  LifeH2O Dec 17 '11 at 10:29
Yes, one might say the relative speed exceeded $c$ if the distance increased faster than $c$. The speed of light being a maximum is only a local constraint on the speeds. –  Mark Eichenlaub Dec 17 '11 at 15:52
I always assumed it was because (in an nutshell) light travels so fast, that we have nothing to compare it to, so therefore, nothing can be faster. Isnt that simpler? –  Ender Jun 2 '13 at 20:55
@MarkEichenlaub: isn't the amplitude for any off-shell process zero? I'm pretty sure the S-matrix is predicted in such a way that any superluminal degrees of freedom have zero amplitude. –  Jerry Schirmer Jan 27 at 22:26
@JerrySchirmer To be honest I was describing physics that was beyond me. I simply remembered reading this in Feynman's QED. Looking it up, on pp 89 it says "The major contribution occurs at the conventional speed of light... but there is also an amplitude for light to go faster (or slower) than the conventional speed of light. You found out that in the last lecture that light doesn't go only in straight lines; now, you find out that it doesn't go only at the speed of light!" Maybe I misunderstand just what this means, though. I don't know quantum field theory. –  Mark Eichenlaub Jan 27 at 22:58

In actual fact, the relative speed rule does not apply, ever.

The relativistically correct speed addition rule is the following:

$$s=\frac{v+u}{1+\frac{vu}{c^2}}$$

When $\frac{vu}{c^2}$ is close to zero (in other words when the velocities invloved are much less than the speed of light, then the correct formula reduces to the Galilean version $s=u+v$.

Nothing can be faster than light, fundamentally, because as you accelerate you not only gain speed, but also mass. As you approach the speed of light, the energy given to you by the force causing the acceleration basically contributes more and more to the increase of your mass and less and less to the increase of your speed. It does this precisely so you never reach the speed of light. Instead, massless particles like photons always travel at the speed of light.

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As to part 2 of your question "Also why nothing can be more speedy than light?", the answer is that it's not just light. The point is that c is the maximal velocity of any causal, information transmitting interaction in the universe, mediated by anything travelling forwards in time (see footnote). Its just that photons, having 0 rest mass, travelling in a vacuum approach that fundamental limit, c.

Footnote: Except maybe 'tachyons' - never seen and traveling backwards in time because they go faster than c. (Note that Norbert Wiener once pointed out that for a causal influence travelling backwards in time, we would experience it as "random", since it would apparently be an event without an antecedent cause to us).

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John Moffat and Moffat and Albrecht and Magueijo have variable speed of light theories where the speed varied in the early universe and is not a constant. Majueijo has a poplular book Faster Than The Speed of Light outlining his theories. IMO the book is quite outrageous and insults various people. I mention this answer for completeness only as I believe the speed of light in a vacuum is constant.

Space can expand faster than the speed of light, but no information can be transmitted. See the Alcubierre warp drive for some fun.

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The speed of light it's the speed limit in the universe because in an informal sense it's infinite. If a spacecraft was built to travel at a constant 1 g acceleration it would very fast reach 99.9% the speed of light, enabling traveling through the whole observable universe in a lifetime due to the effect of time dilation. There is no rest frame for the photon in relativity but approaching it's speed makes you experience more an more a subjective close to infinite speed. So from the hypothetical 'point of view' of the photon it travels an arbitrary distance in zero time. Emission is the same point as absorbtion for a photon. Now answering why it's constant for all observers and not infinite, I have to say it comes down to the laws of causality and locality.

Also in relativity physics, rapidity (φ) is used as an alternative to speed as a measure of motion. The equation is φ = artanh(v/c). Substituting v = c you get artanh(1) = infinity. So the rapidity of light is infinite.

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That the speed of light is invariant is a property of Minkowski spacetime, and there should be plenty on that in Wiki - or search for 'geometric algebra' or Clifford Algebra.

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