B What does the constancy of speed of light mean?

[thaiqi]

The second postulate says the speed of light is constant c independent of all inertial observers.
Does it mean the speed of the wave front relative to the observer , that is, the relative speed between the wave front and the observer?

 

[sweet springs]

Not always. Yes, for wave front goes away form or come toward the observer. No, for other wave goes transverse to it.

 

[Orodruin]

The second postulate says the speed of light is constant c independent of all inertial observers.

It says that something traveling at the speed of light in one inertial frame does so in all inertial frames.

Not always. Yes, for wave front goes away form or come toward the observer. No, for other wave goes transverse to it.

Wrong. The wave front always moves at c.

 

[sweet springs]

Wrong. The wave front always moves at c.

Yea, I was wrong. Thanks.

 

[Ibix]

The second postulate says the speed of light is constant c independent of all inertial observers.
Does it mean the speed of the wave front relative to the observer , that is, the relative speed between the wave front and the observer?

Yes. If I see you doing half the speed of light, I will say that the distance between you and a pulse of light is changing at $c/2$ (assuming that you're traveling in the same direction). However, length contraction, time dilation and the relativity of simultaneity conspire so that you measure the speed of the pulse as $c$.

You will say the same about me, except that from your perspective I am traveling in the opposite direction as the light pulse, so you'll say the gap opens at $(3/2)c$. But you will give the same reason why I measure $c$.

 

[thaiqi]

When we say speed of light in vacuum is c, the (propagation) "speed" here, versus the wave front speed above, are the two the same thing? Aren't they a little different?

 

[Ibix]

In vacuum, no, they're the same thing. There are various different ways to define "speed" in dispersive media, but vacuum isn't dispersive. All electromagnetic waves travel at $c$ in vacuum.

There are some subtleties around non-plane waves, but one can always decompose a wave into small areas that are locally flat, and these propagate at $c$ in vacuum.

 

[thaiqi]

Thanks.
How about that case when we say (roughly) "the speed of sound in air is 340m/s" while its wave front speed is not invariant to different moving observers?

 

[Ibix]

What about it? Relativity requires that there be an invariant speed. Light travels at that invariant speed. You can't have more than one invariant speed, so all speeds other than $c$ are frame-dependent.

 

[thaiqi]

So the "speed" in the statement "the speed of sound in air is 340m/s" has different meaning to the "speed" in "speed of light"?

 

[PeroK]

So the "speed" in the statement "the speed of sound in air is 340m/s" has different meaning to the "speed" in "speed of light"?

No. Speed is distance/time in both cases.

The difference with sound in air is that the air is a physical medium and an observer can be traveling relative to the air. The speed a sound wave travels relative to you depends on your speed relative to the air.

But you cannot measure your speed relative to empty space. There have been lots of experiments (like the Michelson-Morely) to try to detect motion though space. The failure of these experiments is one fact that led Einstein to the postulate.

 

[thaiqi]

I mean, though we refer to "wave front propagation speed" in "speed of light", we generally don't say so for sound , that is, I don't talk about "wave front speed of sound is 340m/s in air".

 

[PeroK]

I mean, though we refer to "wave front propagation speed" in "speed of light", we generally don't say so for sound , that is, I don't talk about "wave front speed of sound is 340m/s in air".

What else would you mean by the speed of sound?

 

[thaiqi]

Say, phase velocity, (wavelength/period).

 

[Dragon27]

I mean, though we refer to "wave front propagation speed" in "speed of light"

When we say "the speed of light", we mean 299,792,458 m/s. Whatever moves with that speed in one inertial frame of reference, moves with the same speed in all other inertial frames of reference. It's a fundamental physical constant.

 

[FactChecker]

The definition of "speed" for sound and light are the same although they are traveling by a different mechanism. The difference is that space-time is shaped in such a way that "speed" or "velocity" behaves very differently when it approaches c then it does at the low speed of sound.

 

[thaiqi]

What else would you mean by the speed of sound?

Say, phase velocity, (wave-length / period)

 

[PeroK]

Say, phase velocity, (wave-length / period)

In any case, speed of propagation of energy is the critical factor.

 

[thaiqi]

The definition of "speed" for sound and light are the same although they are traveling by a different mechanism. The difference is that space-time is shaped in such a way that "speed" or "velocity" behaves very differently when it approaches c then it does at the low speed of sound.

For light, "the wave front speed" is constant c. While for sound, its wave front speed relative to the observer is decided by: its phase velocity + the velocity of medium + the velocity of the observer. If I made it right as aforementioned, isn't this a difference between light and sound?

 

[bhobba]

I think the OP needs to see a modern presentation of relativity:
http://physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

I used to suggest Rindler, but recently became aware of Morin which I prefer:
https://www.amazon.com/dp/1542323517/?tag=pfamazon01-20

Bottom line - its the other way around - because of relativity the speed of light is constant even though initially Einstein did it the other way around. It's all part of Einstein's genius - as Feynman said - knowing what Einstein did he could not have invented Relativity. Einstein was like a sleepwalker - while groping in the dark he made all sorts of 'mistakes':
https://www.amazon.com/dp/0393337685/?tag=pfamazon01-20

Thanks
Bill

 

[thaiqi]

I mean, though we refer to "wave front propagation speed" in "speed of light", we generally don't say so for sound , that is, I don't talk about "wave front speed of sound is 340m/s in air".

For light, "the wave front speed" is constant c. While for sound, its wave front speed relative to the observer is decided by: its phase velocity + the velocity of medium + the velocity of the observer. If I made it right as aforementioned, isn't this a difference between light and sound?

 

[PeroK]

For light, "the wave front speed" is constant c. While for sound, its wave front speed relative to the observer is decided by: its phase velocity + the velocity of medium + the velocity of the observer. If I made it right as aforementioned, isn't this a difference between light and sound?

The fundamental difference is that light does not need a medium.

And the question then is: in which frame of reference is the speed of light $c$?

The answer is: in all inertial reference frames.

For sound the answer is: the speed of sound is constant in the rest frame of the medium in which it is travelling.

 

[FactChecker]

For light, "the wave front speed" is constant c. While for sound, its wave front speed relative to the observer is decided by: its phase velocity + the velocity of medium + the velocity of the observer. If I made it right as aforementioned, isn't this a difference between light and sound?

It is one difference, but not the critical one where velocities near c are concerned. No matter what medium something is in or what inertial reference frame is used, the relativistic units of distance and time guarantee that there is no such thing as a speed greater than c. That means that anything moving at the speed of c would be at that speed in any inertial reference frame. It applies to any electromagnetic wave and also to gravity waves.
It also means that there are small effects at slower speeds. Suppose an airplane is traveling at the speed of sound and a person in the back of the airplane yells at a person in the front. In the airplane's relativistic units of time and distance, the speed of his yell would travel forward at the speed of sound in the cabin air. But the speed of the airplane cannot be simply added to the speed of the yell to get the speed of the yell sound in the Earth inertial frame. The relativistic units of the airplane's time and distance change to make that speed slightly less than the arithmetic sum.

 

[PeroK]

Actually, to give the maths behind what @FactChecker has said. If something has a constant speed $v$ in some inertial frame and an object has a speed $u$ in the "moving" reference frame, then the speed of that object in the original inertial frame is given by:
$$u' = \frac{v + u}{1 + uv/c^2}$$
And that equation is, in fact, equivalent to the speed of light postulate. It's called relativistic velocity addition. In this case I've assumed the speeds are in the same direction.

Now, if we try $u = c$. We get:

$u' = \frac{v+c}{1 + v/c} = c$

In other words, the speed of light is invariant across all inertial reference frames - if we assume this velocity addition formula.

On the other hand, if $u = s$, the speed of sound, and $v, s << c$, then we get:

$u' \approx v + s$

For normal speeds like those of sound in air and aircraft etc., we get approximately the simple classical velocity addition formula.

 

[thaiqi]

For sound the answer is: the speed of sound is constant in the rest frame of the medium in which it is travelling.

and relative to a "rest observer"?

 

[Ibix]

and relative to a "rest observer"?

It depends what you mean by a "rest observer".

Speed is always specified relative to something we define as "at rest". When you talk about the speed of a car, typically you regard the local surface of the Earth as "at rest". If we talk about the speed of the Earth in its orbit around the Sun we are typically regarding the Sun as "at rest". Often, though, when speaking casually you don't say this - you just say your car is going at 30mph. You actually mean "30mph relative to the surface of the Earth (unless otherwise specified)".

The speed of sound is defined relative to the medium in which it travels, because that's the only sense in which there's a unique speed for sound. If I sprint along at 10m/s then the speed of sound (340m/s relative to the air) relative to me varies between 330m/s and 350m/s. When someone talks about the speed of sound, they mean the speed relative to the medium.

Where there is not an obvious choice for speed relative to what, people specify. Pilots, for instance, make a clear distinction between ground speed and air speed, because both are important to aeroplanes for different reasons.

So when we talk about "the speed of light", the question is: speed relative to what? It doesn't have a medium, so we can't mean relative to the medium. So what do we mean? It turns out that this one case does not matter. Everyone who measures the speed of light relative to themselves comes up with the same answer, 3×10⁸m/s, no matter if they are moving compared to each other. So the speed of light is always being specified relative to you, or to whatever you are defining as "at rest".

 

[thaiqi]

and relative to a "rest observer"?

I mean: As to that PeroK said: "For sound the answer is: the speed of sound is constant in the rest frame of the medium in which it is travelling", it must be that the speed of sound is constant only when it is in the rest medium frame and it is relative to a rest observer stationary to the medium.

 

[thaiqi]

It depends what you mean by a "rest observer".

……
whatever you are defining as "at rest".

I mean: As to that PeroK said: "For sound the answer is: the speed of sound is constant in the rest frame of the medium in which it is travelling", it must be that the speed of sound is constant only when it is in the rest medium frame and it is relative to a rest observer stationary to the medium.

 

[Ibix]

There's a distinction between "constant" and "invariant". Constant means that it doesn't change with time. Invariant means that it does not change for different reference frames.

If I don't accelerate, my speed is constant, but not invariant. Different frames will disagree about the value, but not that it doesn't change. On the other hand, the number of atoms making up my body is invariant, but not constant.

The speed of light is both invariant and constant.

The speed of sound is constant (more or less), but whether or not it is invariant depends on definition. Defining "the speed of sound" as its speed as measured in the rest frame of the medium is invariant, but if you define "the speed of sound" as the speed measured by you then it isn't invariant.

 

[thaiqi]

The speed of sound is constant (more or less), but whether or not it is invariant depends on definition. Defining "the speed of sound" as its speed as measured in the rest frame of the medium is invariant, but if you define "the speed of sound" as the speed measured by you then it isn't invariant.

Thus generally the "speed of light" is that relative to and measured by the observer, but the "speed of sound" is that relative to the medium.

 

[Ibix]

Thus generally the "speed of light" is that relative to and measured by the observer, but the "speed of sound" is that relative to the medium.

If you want a unique "speed of sound", yes, you have to define it that way.

There's nothing wrong with defining the speed of a sound wave as its speed relative to some observer, but then there isn't a particular value. It's more useful for everyone to know the speed of sound relative to the medium; then if you know your speed relative to the medium then you can figure out the speed relative to you.

This is a different behaviour to light, where the speed relative to you is always $c$. It's worth noting that the constant $c$ emerges from the maths of relativity whether light travels at that speed or not.

 

[thaiqi]

Thanks.

 

[bhobba]

The speed of light is both invariant and constant.

Of course true. But again I, for the OP's benefit, want to emphasize that purely from the symmetries of an inertial frame, having nothing to do with light, you get the Lorentz Transformations where the C in those equations is a speed that is the same for all observers. It could be infinity, in which case you get the Galilean Transformations - the usual transformations found in Newtonian Physics. The question is, is the C in the equations finite or infinite. If its finite then it really can be any value because you can change it by simply changing the units used.

Its relation to light is, interestingly, you can derive Maxwell's equations from Coulomb's Law and Relativity::
http://www.cse.secs.oakland.edu/haskell/Special Relativity and Maxwells Equations.pdf

Now we see that C must be finite otherwise many phenomena we know to be true would not happen eg you would not have magnetism or even the existence of light. So we see it is not the constancy of the speed of light that 'really' determines relativity. The very existence of light itself, magnetism and other EM phenomena, all described by Maxwell's equations, implies the C in relativity is finite, the speed of light, and is the same speed in all inertial frames.

So while it is not wrong to derive relativity as Einstein did from the axiom of the constancy of the speed of light, a deeper analysis shows that the very existence of EM phenomena, including light, tells us the C in the Lorentz Transformations is finite which implies its speed is the same in all inertial frames - and all the other EM phenomena.

Again, interestingly, if you read the first chapter of Schwinger's book on EM, you will see a derivation of Maxwell's equation from Coulomb's law and the existence of EM radiation. So the very existence of EM radiation, along with relativity, implies its speed is constant, and the C in the Lorentz Transformations is the speed of light.

Just as an aside most seem to have Jackson as their reference for Electromagnetism, but IMHO Schwinger is better and is my EM reference
https://www.amazon.com/dp/0738200565/?tag=pfamazon01-20

Because its so widely used Jackson it still worthwhile to also get. But he has some views I am not particularly happy with, such as his view derivations of Maxwell's equations are silly - at least in discussing it with someone they mentioned it to me. I hope the above shows that's not quite the case. I personally don't have a copy but am thinking of getting one.

Thanks
Bill