# Why does light have invarient speed?

1. Dec 30, 2007

### mdeng

What is the physics answer to the question of why light has an invariant speed
to anyone and everyone, other than this is what light is? There must be a
reason why light behaves this way (or perhaps not necessarily this way
always). I'd think something must have happened external to the light to give
it this peculiar property. Put it in another way, what's wrong with the
classical physics where velocity would follow the law of vector arithmetics,
when applied to light?

Thanks,
- Ming

2. Dec 31, 2007

### haushofer

I think the appropriate answer to this should be looked in the Maxwell equations. These equations say that electromagnetic fields obey wave equations, and that the velocity of these waves only depends on the properties of the material. Because this velocity is also the speed of light we measure, it can be postulated that light is an electromagnetic wave.

Now, this has to be compared with measurements, and indeed we measure that the speed of light doesn't depend on the observer, only on the material in which the observer observes.

3. Dec 31, 2007

### Staff: Mentor

The only other "why" I can think of for the invariance of the speed of light is that photons are massless.

4. Dec 31, 2007

### lightarrow

There is not an explanation in the present physics.
However I could give you a sort of idea to which think about.

Imagine to be born in another planet and to have studied physics in a different way: there velocity is not defined as s/t, s = space, t = time, but in a different way, exactly in the way mass was defined taken a sample of Platinum-Iridium; you take an object moving at a constant speed (and this one you could measure as you like, e.g. in our way) and you put on it another object with exactly the same speed. In this way you have defined what is a "double" speed, and so on. What comes out is a quantity called "rapidity".
The interesting fact is that rapidity has no limit, it can go to infinite. The more interesting thing is that light's rapidity is infinite. So, if you lived in that planet, it wouldn't be so strange for you that light's speed is independent on the relative velocity v between source and observer: infinite + v = infinite!

5. Dec 31, 2007

### mdeng

I am aware of the consequence of the Maxwell equation. Intuitively though, why would the set of equations give rise to c? And what mechanism (i.e., cause-and-effect) is behind light that gives it this peculiar property?

6. Dec 31, 2007

### mdeng

I think masslessness must have something to do with invariance of c. I am not clear though, whether masslessness of photon is a consequence of assuming that c is constant. In other words, does Einstein’s relativity theory require photon to be massless (at rest)? If yes, then we would be going in circles. Another question I had, as posted in the Quantum group, what happens to non-photon particles that are moving at a speed close to c and are moving against each other? Is the relative speed of the two particle beams (whose sum is > c) capped by c? Relativity theory says yes. But what would be the mechanics behind this phenomenon? And would this be called "invarance of upbound of relative speed"?

Last edited: Dec 31, 2007
7. Dec 31, 2007

### belliott4488

I would say that historically we first discovered that the speed of light is invariant and then from that learned the properties of space and time (as described by Special Relativity). Now that we know those properties, however, I would venture to say that it is a property of space and time that massless particles always move at the maximum speed that any object can obtain, which is also invariant for different observers. Light happens to be an example but is otherwise not special.

In other words, I'd say the invariance of the speed of light is a by-product of the underlying properties of space-time, so the question becomes, why are space and time the way they are? I doubt there's a definitive answer for that yet.

8. Dec 31, 2007

### mdeng

Are you saying with your example that for any non-photon particle, if we keep cranking up their speed, they will go faster and faster but never attain the speed of light, as the relativity theory dictates? In this regard, light speed appears to be infinite as it’s not attainable unless it’s a photon.

9. Dec 31, 2007

### belliott4488

The lower a particle's mass, the faster it accelerates in response to any force acting on it. In the zero-mass limit, all particles must move at speed c. Nothing in Relativity requires that light be massless, but the observation that it moves at speed c implies that it is indeed massless.

Yes, you're right, two particles moving at .999c and -999c relative to one observer still see a relative speed less than c in their own frames. This is a result of the velocity addition theorem from Special Relativity; you can't add two velocities and exceed speed c.

As I said above, I would attribute all of this to the properties of space and time - they are what dictate the relative velocities of inertially moving objects.

10. Dec 31, 2007

### JesseM

It follows from the formula for velocity addition in relativity...if you're moving at velocity v to the right in the first particle's frame (meaning it's moving at v to the left in your frame), and the second particle is moving at velocity u to the right in your frame, then the first particle will see the second particle moving at this velocity:

$$\frac{u + v}{1 + uv/c^2}$$

As long as both u and v are below c, this formula gives a velocity below c as well. If you're confused about why the velocity is not just u + v as it would be in classical mechanics, it has to do with the fact that each observer uses rulers and clocks at rest relative to themselves to define speed in terms of distance/time, but each observer sees the other observer's rulers shrunk relative to his own, and the other observer's clocks slowed down relative to their own.

Last edited: Dec 31, 2007
11. Dec 31, 2007

### mdeng

Interesting answer, though it raises more questions than answered.

Could it be that space really has some other dimension where light takes “shortcuts” and its trajectory projection onto our 3-D space is invariant. And photon’s m = e/c^2 may just happen to be a mathematical convenience in 3D world we currently understand.

The following from wikipedia offerst an interesting insight.

If it is true that there must “exists a theoretical maximal speed of information transmission which must be invariant”, it may explain the existence of light which happen to satisfy this postulate. What would be the mathematical proof of this postulate? And does it require whatever these particles may be to have zero mass at rest?

Last edited by a moderator: May 3, 2017
12. Dec 31, 2007

### belliott4488

I can't imagine why we'd want to say that. The current theory predicts observed phenomena just fine - why mess with it? If it ain't broke ...

I don't know what you're asking - you don't prove postulates, that's why they're postulates. Besides, this principle wasn't stated as a postulate, but more as an observation, I think.

In any case, it doesn't explain the existence of light, but it puts the same "speed limit" on light as on anything else. Light - as well as other massless particles - moves at that speed for the reasons given in the theory of Special Relativity.

13. Dec 31, 2007

### mdeng

Hi JesseM and Belliott,

Thanks for pointing to the velocity addition formula. It’s pretty amazing, clean, concise, and even intuitive.

14. Dec 31, 2007

### mdeng

As for “If it ain't broke …”, yes and no. It’s working as it seems to agree with all the observations we have so far, but it also made some assumptions which we all would like to have an answer to.

The invariant max speed principle (pardon my use of terms above) seems a reasonable one. Intuitively, I can see that it has to be constant, or else some “instantaneity” will occur which violates locality law. I am not quite sure why it has to be invariant except that it might otherwise enable arbitrary travel into to the past.

You are probably aware of the “instantaneity” observation in some quantum physics experiments. That seems to suggest there must exist something that travels faster than light (in fact, infinitely faster). What’s the thought on this for those in the relativity area?

15. Dec 31, 2007

### rbj

photons have inertial mass of

$$m = \frac{E}{c^2} = \frac{h \nu}{c^2}$$

but their rest mass (or "invariant mass")

$$m_0 = m \sqrt{1 - \frac{v^2}{c^2}}$$

is zero because v=c.

i think it's the other way around. photons are "massless" (have no rest mass) because they are believed to move at the same speed c as the wavespeed $1/\sqrt{\epsilon_0 \mu_0}$ for light waves. the first principle is that photons move at speed c for any observer and the consequence is that their rest mass is zero.

this is maybe more than you are asking for, but it is among similar questions asked in the past ("Why Light"), so i am collecting stuff that i said then, that i gleaned from a few email conversations that i have had with physicists like Michael Duff and John Baez in the past. i think the physics is kosher (Integral or Pervect will come down on it if it isn't), but i am not a physicist, but an electrical engineer.

it's not just light. it's the speed of propagation of any "instantaneous" interaction, whether it's E&M, gravity, or nuclear interactions (or something that hasn't been discovered yet).

thought experiment #1
in the case of EM, imagine that you and i are standing some distance apart and facing each other. you're holding a positive charge and i am holding a negative charge and that we both are restricting our charges so they cannot move toward each other but they can move up and down and left and right (just not forward or backward). so i move my charge up a meter. since your charge is attracted to mine, your charge also wants to move up a meter and you allow that. then i move it down and your charge follows it down. now i move it to my right (your left) and your charge moves toward your left. then to my left (your right) and your charge follows it.

now i move my charge up and down repeatedly and your charge follows it up and down. that is an electromagnetic wave that originated with me moving my charge around and that wave moved toward you (at the speed of propagation of E&M waves which is "c") and causes your charge to move correspondingly. in a very real sense, my moving charge is a "transmitting antenna" and your moving charge is a "receiving antenna". if, somehow, i could move my charge up and down a million times per second, you could tune your AM radio to 1000 kHz and hear a signal (a silent carrier). if i could move it up and down 100 million times per second, you could tune it in with your FM radio just between the 99.9 and 100.1 settings (provided no other stations were close by). if i could move it up and down 500 trillion times per second, you would see it as a blur of orange colored light. now i can't move it up and down an entire meter 500 trillion times per second because the speed of that movement would exceed c. but i can have a whole pile of like charges and move them up and down maybe 10 microns at a frequency of 500 trillion Hz. that is what happens in a transmitting antenna or something that emits visible light. charges are moving and that causes other charges to move. but they don't react instantaneously (as observed by a third party that is equi-distant to you and i).

that is what light is (from a wave-property perspective, no mention of photons here) and it required no medium for these waves to travel. they just are there because unlike charges attract and like charges repel (that's the fundamental physics) - there need be no medium in between for that to happen.

why is the speed of light constant for different observers moving relative to each other? it's because there is no way to prefer one inertial, but moving (at least from the POV of someone else) observer to any other inertial (but also moving) observer. if you can't prefer one over the other, the laws of physics must be the same.

the postulate (of SR) is that no inertial frame is qualitative different (or "better") than any other inertial frame of reference and that if we can't tell the difference between a "stationary" vacuum and a vacuum "moving" past our faces at a high velocity, that there is no meaningful difference between a stationary vacuum and a moving vacuum and that Maxwell's Equations should work the same for any and all inertial frames so then the speed of E&M must be measured to be the same in all inertial frames, even if it is the same beam of light viewed by two observers moving relative to each other. from that, we got time dilation, then length contraction, then Lorentz transformation, and so on.

besides the fact that there was a very important experiment, the Michealson-Morley experiment, where they were specifically looking for evidence of a change in the speed of light, given the realistic assumption that if the aether existed, our planet oughta be moving through it at least some season of the year and at sufficient speed that they could measure the difference in c parallel to this movement and perpendicular to this movement and the experiment came out negative . no such change in c was detected. besides that experimental fact, Einstein had a thought experiment about it that i paraphrase below:

you understand that "light" is the propagation of electromagnetic (E&M) fields or "waves" and the physics that describes that propagation are "Maxwell's Equations").

i would not call the constancy of c (for all frames of reference) an axiom or postulate for which there is no idea why such principle exists (and we just notice it experimentally). it's because we can detect no intrinsic difference between different inertial frames of reference (two observers moving at constant velocities relative to each other both have equal claim to being "stationary", there is no good reason to say that one is absolutely stationary and the other is the one that is moving) and that the laws of physics, namely Maxwell's equations, apply to both frames of reference equally validly. if two different observers, neither accelerated but both moving relatively to each other, are examining the very same beam of light (an electromagnetic wave), for both observers, when they apply and solve Maxwell's equations for the propagation of the EM wave, they both get the same speed of c out of solving Maxwell's eqs.

so we do have a good idea for why the speed of propagation of E&M is the same for all inertial observers that may or may not be moving relative to each other. it's because, we cannot tell the difference between a "moving" vacuum and "stationary" vacuum, that there is no difference between a moving and stationary vacuum and then there is not apparent reason for the observed speed of light to be different.

this is different than for sound. the physics of Maxwell's Equations make no reference to a medium that conducts the electromagnetic field (and, indeed, the Michaelson-Morley experiement failed to show that such a hypothetical medium, called "aether" exists - if it does exist, it seems to be moving around in the same frame of reference as the Earth going around the sun because no matter what time of day or season of the year, no one could detect with the M-M apparatus any motion through this aether). but for sound, the physics describe it as compressions and rarefractions of the air (or whatever other matter medium). there is no such thing as sound in a vacuum (but there is light). so if you feel the wind moving past your face from left to right (say at a speed of 20 m/s), you will also measure the speed of sound from a source on your left to be 20 m/s faster than sound from a source in front of you and 40 m/s faster than a sound that is at your right. now you can repeat that setup and get an identical result if there is no wind but you are moving (toward your left) through the air at a speed of 20 m/s. so the observer that is stationary (relative to the air) will look at a sound wave and measure it at something like 334 m/s, but you, moving through the air toward the source at 20 m/s will measure the speed of that very same sound wave to be 354 m/s.

thought experiment #2
now think of the same thing, but instead you two observers are out in some vacuum of space somewhere and are looking at the same beam of light. the other observer is holding the flashlight and measuring the speed of light to be 299792458 m/s. you are moving toward that observer at a speed of, say, 1000 m/s and looking at the very same beam of light that the other observer is. you are thinking that you would measure it at a speed of 299793458 m/s, right? but why should it be any different for you? you have equal claim to being stationary (and it's the guy with the flashlight is moving toward you at 1000 m/s). you cannot feel the vacuum moving past you at a speed of 1000 m/s, in fact there is no physical meaning to the vacuum moving past your face at 1000 m/s like it's a wind. you cannot tell the difference between you moving and the other guy as stationary or if the roles were reversed and there is no meaning to any notion of who is stationary absolutely and who is moving.

so then, if there is no meaningful difference, if both of you have equal claim to being stationary (and it's the other guy that is moving), then the laws of physics (particularly Maxwell's Equations) have to be exactly the same for both of you, both in a qualitative sense, and in a quantitative sense. both of you have the same permittivity of free space ($\epsilon_0$) and permealbility of free space ($\mu_0$). so when you apply Maxwell's equations to this E&M wave (of this flashlight beam), you will see that this changing E field is causing a changing B field which, in turn, is causing a changing E field which is causing a changing B field, etc. now for both of you, the laws (Maxwell's Eqs. and the parameters $\epsilon_0$ and $\mu_0$) are the same. then it turns out, when we solve Maxwell's Equations for this case, we get a propagating wave and the wave speed is

$$c = \sqrt{\frac{1}{\epsilon_0 \mu_0}} [/itex] but that's the same for both you and the other observer!! (even though you are both moving relative to the other.) there is no reason that the other guy should solve the Maxwell's equations and get a different $c$ than you get (because you have the same $\epsilon_0$ and $\mu_0$)! even if you two are looking at the very same beam of light. now, to repeat and sum up (my, this is long): it's not just light. it's the speed of propagation of any fundamental interaction. if, say, gravity (or some other action) could propagate faster (like instantaneously), we could conceivable devise a device that could use the interaction of gravity to communicate information at a speed that is faster than c. but nothing moves faster than that. it is not just a speed limit for moving objects, it is actual and finite speed that the fundamental interactions of nature (all of them) move. why at this speed (299792458 m/s)? because we cannot measure any physical quantity except by measuring it against a like dimensioned physical quantity (that we might call a "standard" or a "unit") and not only is that the way we measure things, it's overall how we experience or perceive things (relative to something else, often us, our bodies or our thinking). it's not like Nature is decreeing that "light, E&M, gravity, nuclear and all other fundamental physical interactions shall propagate at a speed of 299792458 m/s", it's only that Nature decrees that this speed be finite and the same finite speed for all of these interactions. whatever finite speed that is doesn't matter because it (along with G and $\hbar$) defines the scaling in reality of length, time, and mass. all the physics says is that this speed of interaction is finite, not infinite. this is what Planck Units are fundamentally about. we, by a historic accident, have chosen a unit of length to be the meter (about as big as we are) and the unit of time to be the second (about as long as a fleeting thought that our biological brains can do), so because of that, we observer that c is 299793458 m/s, but the speed of light is always, fundamentally 1 Planck Length per Planck Time. Now, I don't know why an atom's size is approximately 1025 LP, but it is (and that seems to me to be a legitimate question for physicists), or why biological cells are about 105times bigger than atoms, but they are (and that seems to me to be a legitimate question for micro-biologists), or why we sentient human beings are about 105 times bigger (in one dimension) than the cells that make up our bodies, but we are (a good question for biologists) and if any of those dimensionless ratios changed, life would be different. We would know the difference. But if none of those ratios changed, nor any other ratio of like dimensioned physical quantity, we would still be about as big as 1035 LP , our clocks would tick about once every 1044 TP, and, by definition, we always perceive the speed of light (not just light or E&M but the speed of propagation of all instantaneous interactions, such as gravity) to be c = LP/ TP which is the same as how we do now, no matter how some "god-like" manipulator changes it. Now if some dimensionless value like the Fine-structure constant $\alpha$ changed, that's different. We would perceive the difference. But to attribute that change to a change in c, that case is not defensible. You could argue that the change in alpha is due to a change in the speed of light, and I could argue it's a change in Planck's constant or the elementary charge and there is no way to support one view over the other. So, rather than asking "Why is the speed of light equal to 299792458 m/s?", which really is a meaningless question, we would ask why is the meter (which is about as big as we are) about 1025 LP? And why is the second (which is about a fleeting moment of thought for our species) about 1044 TP? (Both are asking about dimensionless quantities, which are meaningful questions.) When those questions get answered, then we have an answer for why the speed of light is equal to 299792458 m/s. 16. Jan 2, 2008 ### lightarrow Yes. 17. Jan 2, 2008 ### lightarrow Maybe there is also another reason: if a massless particle moved at v < c, it would have zero energy and zero momentum. How could we detect it? 18. Jan 4, 2008 ### ||spoon|| When not using relativity, or when ignoring its effects, total velocity is simply: [tex]V_t=v_1+v_2$$

When we take into account the consequences of relativity the equation becomes:

$$V_t=\frac{v_1+v_2}{1+\frac{v_1v_2}{c^2}}$$

where c is the einstein constant (or speed of light in vacuum).

now let v_1 be equal to c, and our own velocity be equal to v_2. (of course our velocity is of no consequence as you will see :) )

then what you arrive with is the following:

$$V_t=\frac{c+v_2}{1+\frac{c v_2}{c^2}}$$

by cancelling c within the denominator and multiplying by c we arrive at:

$$V_t=\frac{(c+v_2)c}{c+v_2}$$ (because multiplying the c cancels c in denominator)

Then by caneclling (c+v_2) we are left with:

$$V_t=c$$

Therefore it does not matter what our speed is at all... the speed of light when will always remain at this constant (depending on the medium) so lights speed will not vary. Light does adhere to arithmetic... but only when you use the correct equations :)

note: first time i used latex thing so sorry if i screwed up anywhere

Last edited: Jan 4, 2008
19. Jan 5, 2008

### mdeng

r b-j, thanks for gathering all the info.

That’s what Einstein (any many other experiments) observed. But why should vacuum, and consequently, light have this property? Note that when light travels in other mediums such as water, light lost its peculiar property of speed constancy.

I am starting to like this idea. More generally, it’s what some calls the locality principle which postulates “there exists a theoretical maximal speed of information transmission which must be invariant”, and light in vacuum just happens to possess this property. It would be convincing to me if we can prove the locality principle and derive from it the c constant (in some unit). It would then entail the whole special relativity theory and likely provide better insight to relativity. And perhaps, if there is another world where locality principle does not always apply (as some claims in quantum world), we can derive something else.

20. Jan 5, 2008

### mdeng

Thanks! I also found this at http://en.wikipedia.org/wiki/Velocity-addition_formula.

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