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

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Solution 1

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.

Solution 2

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 is approximated 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.

Solution 3

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.

Solution 4

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.

Solution 5

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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Updated on August 29, 2021

Comments

  • SMUsamaShah
    SMUsamaShah about 2 years

    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?

    • Carlos
      Carlos over 9 years
      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.
    • dmckee --- ex-moderator kitten
      dmckee --- ex-moderator kitten about 7 years
      @Carlos From an experimental point of view interferometric experiments don't really try to "measure the speed of light"; instead they are sensitive explicitly to variations in the speed of light without being sensitive to what the speed is. And if well designed and constructed they can be exceedingly precise in excluding such variations.
    • David Spector
      David Spector about 3 years
      See my answer at physics.stackexchange.com/questions/361488/… . The true speed of light is infinite, when measured properly.
  • SMUsamaShah
    SMUsamaShah almost 12 years
    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?
  • Mark Eichenlaub
    Mark Eichenlaub almost 12 years
    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.
  • Ender
    Ender over 10 years
    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?
  • Jerry Schirmer
    Jerry Schirmer almost 10 years
    @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.
  • Mark Eichenlaub
    Mark Eichenlaub almost 10 years
    @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.
  • DJphy
    DJphy over 8 years
    A photon may travel slightly slower or faster than light, and only travels at light speed on average. The above line by Mark , when photons are the constituent of light and nothing else(light == photons, no doubt) why does he say the above statement[In fact the question can be interpreted as why photons travel at a constant speed 3,00,000 km/s in vacuum]
  • 1110101001
    1110101001 about 8 years
    Doesn't the MM experiment only prove the first point, that speed of light is independent of the speed of its source?
  • Žarko Tomičić
    Žarko Tomičić about 8 years
    Actually, it was designed to detect aether, a special kind of medium some proposed to exist..
  • Žarko Tomičić
    Žarko Tomičić about 8 years
    In order to explain MM exp results you have to assume length contracion etc...all the things connected to the lorentz transf and constancy of the speed of light...
  • user12262
    user12262 almost 7 years
    JakobH: "Unfortuantely, "Why" questions almost always get here the answer "Because that's how nature is!"." -- Indeed unfortunate, as far as they might be answered instead by Due to the experimental method by which we attempt to find out "Whether"!. "In order to move faster than $L_m / T_m$ an object would need to travel the minimal distance $L_m$ in a time interval shorter than minimal time interval $T_m$." -- But it might make discrete jumps $L > L_m$ during $T_m$; especially discrete jumps $L < 2~L_m$.
  • jak
    jak almost 7 years
    @user12262 Nice thought! However I find it quite plausible that no one is allowed to skip points on a fundamental spacetime lattice. Nevertheless, ideas beyond the standard ones are always a matter of taste. What usually bothers me is that people act as if there were no ideas at all.
  • user12262
    user12262 almost 7 years
    JakobH: Well, I also meant to express that I find your present choice of notation disagreeable, among other things. (However, to be fair, I have yet to submit my own explicit answer/rant about "How the experimental method by which we define speed and determine speed values implies finite and invariant signal front speed" on this particular page ...)
  • anna v
    anna v almost 7 years
    What you do not realize is that the current physics model, particularly the particle standard model, incorporates both the postulate for the speed of light and special relativity, but also an enormous number of data derived from experiments that cannot be described by handwaving an a la cart solution for the speed of light. To be taken seriously one has to use mathematical model that will also embed the standard model and the data it represents as accurately as the standard model does. The time of debating the heliocentric versus the geocentric system has passed since the introduction
  • anna v
    anna v almost 7 years
    calculus and mathematics in the modeling of the behavior of nature, and since physics separated from philosophy and mathematics , and is a discipline which models mathematically existing data and predicts new behaviors. When a question "why" in physics hits a postulate, the answer is "because". Challengers have the weight of proving that their new postulate explains all the data that hang from the displaced postulate.
  • anna v
    anna v almost 7 years
    to continue, for example string theories are also beautiful and people study them because they offer quantization of gravity and, in my experimentalist's opinion, they have the possibility of embedding the standard model with its symmetries elegantly, and in this sense , to be validated. The model needs new predictions, as for example supersymmetry , which is also speculative, if it is discovered at the LHC, that would be a validation also.
  • jak
    jak almost 7 years
    @annav I'm really not sure what you're trying to imply here. The idea that spacetime is quantized is one of the main results of Loop Quantum Gravity and not something "handwavy". In addition there is a lot of serious research along similar lines of thoughts, e.g. doubly special relativity. Expressed mathematically a quantized spacetimes requires that we understand the Poincare group as a Wigner contraction of the De Sitter group. This has far reaching experimental consequences that can be tested in experiments.
  • jak
    jak almost 7 years
    @annav I don't understand why you think that a quantized spacetime as a explanation for the constant speed of light somehow is in contradiction to the standard model. The De Sitter group becomes in the Limit $R \rightarrow 0$ the Poincare group, where $R$ is the invariant length scale. This is ananlogous to how the Poincare group becomes in the limit $c \rightarrow \infty$ the Galilei group. Effects of the the De Sitter spacetime only become relevant at very high energies (= small length scales), comparabe to how effects of the Poincare group only become relevant at high velocities.
  • anna v
    anna v almost 7 years
    @JakobH the standard model is based on c being a constant, as all the experimental numbers are extracted from measurements using special relativity four vectors and energy momentum conservation. It is not enough to have the transformations you need the constant numbers.
  • jak
    jak almost 7 years
    @annav uhm what? I hope you do know that there is a close connection between constants and transformations. The poincare group is DEFINED as the set of transformations that preserve c... The de sitter group preserves c, too... Where did you get the idea that I think c isn't constant?
  • anna v
    anna v almost 7 years
    Sorry, I have a head cold and was obviously confused last night. You are making space discrete. Look at this question and the answer by Motl physics.stackexchange.com/questions/67211/… about loop quantum gravity and the standard model . I think I will stop here because this is beyond my level. When I see a "granularity of space "theory come out naturally with the symmetries of the standard model I will get interested.
  • jak
    jak almost 7 years
    @annav no problem. Ah now I see... Motl is known for hating everything which isn't string theory and everything he says needs to be taken with a grain of salt (c.f. rationalwiki.org/wiki/Lubo%C5%A1_Motl#cite_note-1 ). I'm not an advocate of LQG, but Motl obviously claims nonsense in order to promote his favored string theory, c.f. physics.stackexchange.com/questions/4027/…
  • skan
    skan over 6 years
    yes, but in order to derive that equation you first must accept some hypothesis and accept that the speed of light is constant.
  • skan
    skan over 6 years
    the speed of the sound waves front it's not equivalent to the speed of light becuase it does depend on the speed of the reference frame used to measure it.
  • SMUsamaShah
    SMUsamaShah almost 6 years
    This sound analogy helped me visualize speed of light now, thanks for that. This means that for a constant distance, speed of light is also constant and speed of emitter simply does not matter. If it is this simple, why it is not explained like this (as you just did)?
  • SMUsamaShah
    SMUsamaShah almost 6 years
    But as @Zarko said, speed of light will be measured the same constant even in relative motion. Now that doesn't fit sound wave analogy. e.g. If two objects are moving away from each other and one measures the speed of sound from second object, it will be lesser than actual speed.
  • SMUsamaShah
    SMUsamaShah almost 6 years
    Is there any visual explanation of these 4 dimensions?
  • PM 2Ring
    PM 2Ring over 5 years
    This answer would be better if it didn't use the deprecated concept of relativistic mass.
  • Saravanabalagi Ramachandran
    Saravanabalagi Ramachandran almost 5 years
    In actual fact, the relative speed rule does not apply, ever, if speed of light is constant in all frames of reference.
  • Saravanabalagi Ramachandran
    Saravanabalagi Ramachandran almost 5 years
    "How can it be that things don't always go in the direction of the force on them?" Because when you push an object it moves in that direction. "How can it be that the speed of light is constant?" It has always been like that :'(
  • Sklivvz
    Sklivvz almost 5 years
    @SaravanabalagiRamachandran yes, in another universe it might not apply
  • Software Engineer
    Software Engineer almost 4 years
    As appealing and obvious as quantized space-time is, it is ultimately a preferred frame of reference and incompatible with relativity.
  • jak
    jak almost 4 years
    @SoftwareEngineer can you elaborate?