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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

7 Answers 7

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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The fact that different observers in relative motion can measure the same light ray to move at a speed of c has to do with the fact that each observer defines the "speed" in terms of distance/time on rulers and clocks at rest relative to themselves. It's crucial to understand that different observers use different rulers and clocks to measure speed, because in relativity each inertial observer will see the rulers of other inertial observers to be shrunk (length contraction), and the clocks of other inertial observers to be running slow (time dilation) and to be out-of-sync with one another (relativity of simultaneity). Each observer can be imagined to measure the speed using a pair of clocks at different positions along a ruler (the clocks synchronized in their own frame using the Einstein synchronization convention), measuring the time T1 on the first clock as the light wave passes it, and the time T2 on the second clock as the light passes it, and then if their ruler shows the clock to be a distance D apart, this observer concludes the speed of the light ray was D/(T2 - T1).

But now consider how the rulers and clocks of this observer will look in my frame, if I see the observer to be moving at some velocity v along my x-axis (with the ruler parallel to the x-axis). From my perspective, the ruler which the moving observer used to measure the distance is shrunk by a factor of $ \sqrt{1 - v^2/c^2} $ due to length contraction, the time between ticks on the clocks of the moving observer expands by $ 1 / \sqrt{1 - v^2/c^2} $ due to time dilation (or equivalently, in $T$ seconds of time in my frame I only see the moving observer's clock tick forward by $ T \sqrt{1 - v^2/c^2} $), and the rear clock's time-reading is ahead of the front clock's reading by $ vL/c^2 $ due to the relativity of simultaneity, where $L$ is the distance between the clocks in the observer's own frame, as measured by their own ruler.

Let's look at a numerical example. Say that the ruler is 50 light-seconds long in its own rest frame, moving at 0.6c in my frame. In this case the relativistic gamma-factor $ 1 / \sqrt{1 - v^2/c^2} $ (which determines the amount of length contraction and time dilation) is 1.25, so in my frame the ruler's length is 50/1.25 = 40 light seconds long. At the front and back of the ruler are clocks which are synchronized in the ruler's rest frame; because of the relativity of simultaneity, this means that in my frame they are out-of-sync, with the front clock's time being behind the back clock's time by $vL/c^2$ = (0.6c)(50 light-seconds)/$c^2$ = 30 seconds.

Now, when the back end of the moving ruler is lined up with the 0-light-seconds mark of my own ruler (with my own ruler at rest relative to me), I set up a light flash at that position. Let's say at this moment the clock at the back of the moving ruler reads a time of 0 seconds, and since the clock at the front is always behind it by 30 seconds in my frame, then in my frame the clock at the front must read -30 seconds at that moment. 100 seconds later in my frame, the back end will have moved (100 seconds)*(0.6c) = 60 light-seconds along my ruler, and since the ruler is 40 light-seconds long in my frame, this means the front end will be lined up with the 100-light-seconds mark on my ruler. Since 100 seconds have passed, if the light beam is moving at c in my frame it must have moved 100 light-seconds in that time, so it will also be at the 100-light-seconds mark on my ruler, just having caught up with the front end of the moving ruler.

Since 100 seconds passed in my frame, this means 100/1.25 = 80 seconds have passed on the clocks at the front and back of the moving ruler. Since the clock at the back read 0 seconds when the flash was set off, it now reads 80 seconds; and since the clock at the front read -30 seconds, it now reads 50 seconds. And remember, the ruler was 50 light-seconds long in its own rest frame! So in its frame, where the clock at the front is synchronized with the clock at the back, the light flash was set off at the back when the clock there read 0 seconds, and the light beam passed the clock at the front when its time read 50 seconds, so since the ruler is 50-light-seconds long, the beam must have been moving at 50 light-seconds/50 seconds = c as well! So you can see that everything works out--if I measure distances and times with rulers and clocks at rest in my frame, I conclude the light beam moved at 1 c, and if a moving observer measures distance and times with rulers and clocks at rest in his frame, he also concludes the same light beam moved at 1 c.

If you want to also consider what happens if, after reaching the front end of the moving ruler at 100 seconds in my frame, the light then bounces back towards the back in the opposite direction towards the back end, then at 125 seconds in my frame the light will be at a position of 75 light-seconds on my ruler, and the back end of the moving ruler will be at that position as well. Since 125 seconds have passed in my frame, 125/1.25 = 100 seconds will have passed on the clock at the back of the moving ruler. Now remember that on the clock at the front read 50 seconds when the light reached it, and the ruler is 50 light-seconds long in its own rest frame, so an observer on the moving ruler will have measured the light to take an additional 50 seconds to travel the 50 light-seconds from front end to back end.

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Points usually end up low if you explain, but high if you say that there is no reason nor explanation. Go figure. You might like to watch my Special Relativity videos at goo.gl/fz4R0I. They explain, and thus have been ignored in general. –  Sean Oct 23 at 12:14

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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