# Problem about postulate of the invariance of the speed of light

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In summary, the conversation discusses the issue of redshift and its relation to the postulate of the invariance of the speed of light. One person argues that redshift is a result of the light traveling slower in relation to us, while the other explains that it is due to the distance between the light source and the observer increasing. The conversation also touches on the concepts of separation rate and relative velocity, as well as the incorrect assumption that a stationary observer would measure the light moving slower. Overall, the conversation highlights the need for a better understanding of relativity in order to reconcile these concepts.f

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TL;DR Summary
A simple thought experiment seems to contradict this postulate.
Hello,

I have a problem with the postulate of the invariance of the speed of light.
When we move away from a light source it is redshift, it is the sign that the relative velocity between us and the light source has changed. If a stationary observer observes the phenomenon, he will measure that the light does not move at c relative to us but less quickly. The redshift is thence due to the fact that the light goes slower in relation to us, each peak of light taking longer to reach us. The speed of light is therefore not constant with respect to us. This seems to contradict Einstein's postulate. If the speed of light was constant, I think we should not see any redshift when we move away from it.

PeroK
If a stationary observer observes the phenomenon, he will measure that the light does not move at c relative to us but less quickly.
You are mixing up separation rate and relative velocity. Separation rate is the rate of change of distance between two objects as measured by a third party in that party's rest frame, and it can be up to 2c for someone observing light pulses traveling in opposite directions. Relative velocity is the velocity one object measures for another, and this is the quantity that never exceeds c and is always c when you are observing light.

The two concepts are the same in Newtonian physics but not in relativity.

vanhees71, malawi_glenn, topsquark and 2 others
When we move away from a light source it is redshift, it is the sign that the relative velocity between us and the light source has changed.
No, it isn't. Even a constant relative velocity will result in a redshift.

If a stationary observer observes the phenomenon, he will measure that the light does not move at c relative to us but less quickly.
No, he won't. Check the relativistic velocity addition law.

The redshift is thence due to the fact that the light goes slower in relation to us
Wrong. See above.

each peak of light taking longer to reach us.
This is true, but it has nothing to do with the light going slower. It has to do with the light source being further away as each successive peak is emitted, so each peak takes longer to reach us than the one before.

If the speed of light was constant, I think we should not see any redshift when we move away from it.
You are incorrect. See above.

vanhees71, malawi_glenn and topsquark
No, it isn't. Even a constant relative velocity will result in a redshift.
Yes, but there is a relative velocity different from c.

No, he won't. Check the relativistic velocity addition law.
I have said if a "stationary observer" measures the relative velocity between the light and me who is receding from the light. Moreover I place myself at low speed (<c/10) in the Newtonian approximation.

Wrong. See above.
This is how the observer will interpret it.
This is true, but it has nothing to do with the light going slower. It has to do with the light source being further away as each successive peak is emitted, so each peak takes longer to reach us than the one before.
If the source is moving away from us, it is going slower in relation to us.

PeroK
there is a relative velocity different from c.
Yes, the relative velocity between the source and observer will always be different from (and less than) c.

I have said if a "stationary observer" measures the relative velocity between the light and me who is receding from the light.
A stationary observer can't measure that. As @Ibix pointed out, he can only measure the "recession speed" between the light source and you, which is not the same thing as the velocity of the light source relative to you.

If the source is moving away from us, it is going slower in relation to us.
No, it isn't. See above.

topsquark
I leave you to ponder on that. For my part, I cannot reconcile it with the postulate of the invariance of the speed of light, especially since it is impossible to directly measure the relative velocity of the source which is moving away from me, the only way to do so is by interpreting the redshift.

PeroK
For my part, I cannot reconcile it with the postulate of the invariance of the speed of light
Then that is a problem with your understanding of relativity, not with relativity itself. We can only point out that your understanding is erroneous or incomplete at best and point you to how you may amend that. If you refuse to do so, it does not change the fact that relativity itself is perfectly compatible (and in fact based on) the invariance of the speed of light.

malawi_glenn and topsquark
This is how the observer will interpret it.
Not if the observer correctly understands relativity, and the difference between recession speed and relative velocity.

Also, you appear to be confusing the speed of the light with the speed of the light source.

Perhaps it might help to actually look at the math for this. As I understand your scenario, you have a stationary observer, O, a light source, S, and a light receiver, R. You have R moving at some speed ##v_R## relative to you (let's say in the positive ##x## direction), and S moving at some speed ##v_S## relative to you, where ##v_S > v_R##. Then S emits successive light pulses towards R at a constant rate according to S's clock. The question is at what rate R will receive the pulses according to R's clock.

First, let's dispose of the relative velocity issue. According to the relativistic velocity addition law, the speed of S relative to R will be:

$$v = \frac{v_S - v_R}{1 - v_S v_R}$$

Note the extra factor in the denominator: it is not the case that ##v = v_S - v_R##. The quantity ##v_S - v_R## is what @Ibix called the "recession speed", and is not the same as relative velocity, as the above shows. In fact, ##v## will be larger than ##v_S - v_R##; the receiver R will see the source moving away faster than the difference in their speeds as you see them.

You appear to be assuming that the light pulses move at speed ##-1## (I am using units in which ##c = 1##) relative to S, and then asserting that the pulses move slower than ##- 1## according to you. But according to the relativistic velocity addition law, if ##v_L## is the velocity of the light pulses relative to you, we have

$$v_L = \frac{v_S - 1}{1 - v_S} = -1$$

So relativity says the light pulses move at speed ##-1## relative to you. (A similar calculation using ##v## as obtained above shows that the light pulses also move at speed ##-1## relative to R.)

Now, let's look at the redshift. Light pulses are emitted by S at a constant rate relative to S's clock. But relative to you, S's clock is time dilated by a factor ##\sqrt{1 - v_S^2}##. So if S emits light pulses at a rate ##\nu##, the rate at which you see S's light pulses being emitted is ##\nu \sqrt{1 - v_S^2}##.

The receiver R, however, is also time dilated relative to you, by a factor ##\sqrt{1 - v_R^2}##, so if S emits light pulses at a rate ##\nu##, the rate at which you calculate that R would receive S's light pulses, not allowing for the change in light travel time, is ##\nu \sqrt{1 - v_S^2} / \sqrt{1 - v_R^2}##.

[Note: I have edited the previous paragraph and added the next one to correct the analysis; the final comparison has also been edited.]

However, since S is moving away from R, there is an additional effect due to the change in light travel time between pulses. This adds an extra factor which, from your point of view, can be expressed as the ratio of the "recession speed" of the light relative to S and R, i.e., a factor ##\left( 1 + v_R \right) / \left( 1 + v_S \right)##. So your final prediction for the rate at which R receives S's light pulses is ##\nu \sqrt{1 - v_S^2}\left( 1 + v_R \right) / \sqrt{1 - v_R^2} \left( 1 + v_S \right)##

What does the relativistic Doppler shift formula tell us? It tells us that, if S is moving at speed ##v## away from R (where ##v## is given by the formula we derived above), and S emits light pulses at the rate ##\nu##, then R receives light pulses at the rate

$$\nu \sqrt{\frac{1 - v}{1 + v}}$$

Now we just substitute and do the algebra:

$$\nu \sqrt{\frac{1 - v}{1 + v}} = \nu \sqrt{\frac{1 - v_S v_R - v_S + v_R}{1 - v_S v_R + v_S - v_R}} = \nu \sqrt{\frac{\left( 1 - v_S \right) \left( 1 + v_R \right)}{\left( 1 + v_S \right) \left( 1 - v_R \right)}} = \nu \sqrt{\frac{\left( 1 - v_S^2 \right) \left( 1 + v_R \right)^2}{\left( 1 - v_R^2 \right) \left( 1 + v_S \right)^2}}$$

which is the same as the formula that we derived above. So you calculate that R will receive S's light pulses at exactly the same rate that the relativistic Doppler shift formula predicts.

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malawi_glenn, Dale and topsquark
For my part, I cannot reconcile it with the postulate of the invariance of the speed of light
You reconcile it by doing the math. See my post #8.

malawi_glenn and topsquark
since it is impossible to directly measure the relative velocity of the source which is moving away from me, the only way to do so is by interpreting the redshift.
What's wrong with a ruler and a clock?

Dale and topsquark
Yes, but there is a relative velocity different from c.
Sure. The second postulate says that light propagates at c in all inertial reference frames. In never claims that what you call the “relative velocity” is always c. Indeed, I don’t think that would be a self consistent claim

For my part, I cannot reconcile it with the postulate of the invariance of the speed of light
That is largely because your statement of what the postulate says is very wrong.

Ibix, PeterDonis and topsquark
You reconcile it by doing the math. See my post #8.
Ok, thank you, the relativity calculations are working fine.
But suppose I'm at low speed and the relativistic effects are negligible. I see the source from which I move away redshifted. What does this redshift mean (the classical redshift) if not that the light moves slower with respect to me? I see no other explanation.

What's wrong with a ruler and a clock?
It is impossible to measure the speed of light in a single direction, we only measure the average speed of a round trip.

PeroK
suppose I'm at low speed and the relativistic effects are negligible. I see the source from which I move away redshifted. What does this redshift mean (the classical redshift) if not that the light moves slower with respect to me?
It means that each successive light pulse has to travel farther than the previous one did. If time dilation effects are negligible, that is the only effect that is left.

topsquark
It is impossible to measure the speed of light in a single direction, we only measure the average speed of a round trip.
Yeah, but (a) you were talking about the source not the light, and (b) interpreting the redshift doesn't get around the problems with one-way speeds. So what's wrong with a clock and a ruler that you think isn't wrong with interpreting the redshift?

topsquark
I see the source from which I move away redshifted. What does this redshift mean (the classical redshift) if not that the light moves slower with respect to me? I see no other explanation.
In still air you can easily measure the speed of sound and find that it actually is isotropic. And you can also directly measure that emitters moving towards you are blueshifted and emitters moving away are redshifted. So your “explanation” of the Doppler shift doesn’t even work for sound. The anisotropic frequency shift occurs even without an anisotropic speed of sound. This fact is easy to derive and is completely consistent with both logic and observation.

Your misunderstanding here is not just a misunderstanding of relativity or the second postulate, but a misunderstanding of the classical Doppler effect. Doppler shift even classically does not imply a change in wave propagation velocity. Your assertion that it is implied is completely unsupported logically and is in contradiction with ordinary waves in classical physics and everyday experience.

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Sagittarius A-Star and topsquark
I see the source from which I move away redshifted.
In your rest-frame, the source is moving away from you.

What does this redshift mean (the classical redshift) if not that the light moves slower with respect to me? I see no other explanation.
Does the equation ##c = f \lambda## help?

Source:

topsquark and Ibix
It means that each successive light pulse has to travel farther than the previous one did. If time dilation effects are negligible, that is the only effect that is left.
In still air you can easily measure the speed of sound and find that it actually is isotropic. And you can also directly measure that emitters moving towards you are blueshifted and emitters moving away are redshifted. So your “explanation” of the Doppler shift doesn’t even work for sound. The anisotropic frequency shift occurs even without an anisotropic speed of sound. This fact is easy to derive and is completely consistent with both logic and observation.
In your rest-frame, the source is moving away from you.

If we take the example of sound, we see that doppler shift comes from two effects that are indistinguishable from each other.
1. I am stationary in the air and the sound source is moving away. In this case, as you say, the shift
comes from the fact that the wave is lengthened by the motion of the source in the air.
2. I am moving relative to the air and the source is stationary. In this case the shift comes from the fact that the speed of sound relative to me is reduced.
These two effects are indistinguishable but nonetheless real.
I think that by assuming an ether we can reason the same with light and by removing the ether only case 1 remains possible.

These two effects are indistinguishable
You say that, but if you measure them accurately enough they are distinguishable.

In case (1)$$f = \left ( \frac {c_\text{sound}}{c_\text{sound} + v_\text{s}} \right ) f_0 \, ,$$and in case (2)$$f = \left ( \frac {c_\text{sound} - v_\text{r}}{c_\text{sound}} \right ) f_0 \, .$$And the relativistic case for light is neither of the above$$f =\sqrt{\frac{c - v}{c + v}} f_0 \, ,$$ as in post #8.

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PeterDonis, topsquark, Ibix and 1 other person
I think that by assuming an ether we can reason the same with light and by removing the ether only case 1 remains possible.
And the relativistic case for light is neither of the above$$f =\sqrt{\frac{c - v}{c + v}} f_0 \, ,$$
The analogy between the relativistic formula for light and the classical case 1 for sound (moving source) can be seen, when re-writing the relativistic formula for light as follows (time-dilation of the moving source, with reference to the receiver's rest-frame, times classical Doppler formula):$$f =\frac{1}{\gamma(1 + v/c)} f_0 \$$

topsquark
These two effects are indistinguishable but nonetheless real.
The issue is that the claim in your OP is illogical. Your claim is that merely observing a Doppler shift implies a change in the speed of the wave. Since your case 1 exists, that implication is clearly false. You must do something more than merely observe a Doppler shift to conclude case 2.

Your repeated statements of “no other explanation” are also patently false given that you have succinctly stated two explanations.

Also, it is not correct that the effects are indistinguishable, but let’s focus on this logical mistake before diving into that experimental mistake.

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PeterDonis, topsquark and vanhees71
The redshift is thence due to the fact that the light goes slower in relation to us, each peak of light taking longer to reach us.
Each successive peak takes longer to reach you because the peaks are further apart (and also because of time dilation), not because they are moving slower

topsquark and vanhees71
Here's a couple of animations I did a while back that might help.
First light waves emitted from a source at rest with respect to two observers.

No blue or red shift.

Now with the source moving towards the blue observer.

Each part of the emitted waves still travel outward, at c, from the emission point. But, the emitter is moving to the right, so each successive part is emitted a bit closer to the Blue observer and further from the red one. Since the emitter is "chasing after" the light it emits to the right, and "running away" from the light it emits to the left, the troughs and peaks are compressed to the right and spread out to the left.

Now, if you were to view this as someone at rest with respect to the emitter, you would see the concentric circles like in the first animation, but the blue observe would be approaching from the right, and the red receding to the left. The blue observer would meet wave peaks at a higher rate than the red observer, so you would still expect the blue observer to measure a higher frequency than the red one.

vanhees71, Sagittarius A-Star, phinds and 2 others
The issue is that the claim in your OP is illogical. Your claim is that Doppler shift implies a change in the speed of the wave. Since your case 1 exists, that implication is clearly false. You must do something more than merely observe a Doppler shift to conclude case 2.

Your repeated statements of “no other explanation” are patently false given that you have succinctly stated two explanations.
It's because at the beginning I thought only of an object which moves away from a light source which would be stationary. This is case 2. In fact if we assume that the speed of light is isotropic in both directions, like Einstein, case 2 is impossible, but if we do not make this assumption case 2 becomes possible. Einstein's assumption amounts to saying that everyone is stationary with respect to the ether, which is strange.
Each successive peak takes longer to reach you because the peaks are further apart (and also because of time dilation), not because they are moving slower
This is case 1.

Einstein's assumption amounts to saying that everyone is stationary with respect to the ether, which is strange.
No. There is no evidence for the existence of an ether with a certain state of motion. An invariant speed must exist, because Newton's assumption of an absolute time (##t' = t##) was proven false by time dilation experiments.

topsquark
No. There is no evidence for the existence of an ether with a certain state of motion. An invariant speed must exist, because Newton's assumption of an absolute time (t′=t) was proven false by time dilation experiments.
It seems to me that no ether or always being stationary with respect to it amounts to the same thing.
The average velocity of the round trip must be invariant, but not the velocity of the outward or return trip taken separately.

PeroK
It seems to me that no ether or always being stationary with respect to it amounts to the same thing.
That is wrong. Two frames moving relative to each other cannot be at rest relative to the same ether with a certain state of motion.

The average velocity of the round trip must be invariant, but not the velocity of the outward or return trip taken separately.
That effects only the definition of coordinate systems, not physics itself. A non-standard synchronization makes only the mathematical description more complicated.

topsquark and PeroK
That is wrong. Two frames moving relative to each other cannot be at rest relative to the same ether with a certain state of motion.
In fact, if we assume that everyone is motionless with respect to the ether, the ether cannot exist as an entity with a certain state of motion, so it certainly does not exist at all. So that's what I'm saying, the two hypotheses are equivalent.

That effects only the definition of coordinate systems, not physics itself.
I confess that I do not understand this. It seems to me that the physics is not the same.

A non-standard synchronization makes only the mathematical description more complicated.
The hypothesis of the anisotropy of the speed of light is simply Lorentz's ether theory and physics is not the same between the two theories.

It's because at the beginning I thought only of an object which moves away from a light source which would be stationary.
So it appears that you recognize that the state of motion must be determined separately from the Doppler shift.

Given that the Doppler shift is insufficient to determine the speed of a wave, the speed must be determined with other information. In fact, the two way speed can be measured directly. When such measurements are performed we find that the second postulate is consistent with the results.

In fact if we assume that the speed of light is isotropic in both directions, like Einstein, case 2 is impossible, but if we do not make this assumption case 2 becomes possible.
It actually is not impossible. That is related to what Einstein showed in 1905. In fact, he has a section on the Doppler effect in his seminal paper.

Einstein's assumption amounts to saying that everyone is stationary with respect to the ether, which is strange.
Yes, “strange” is an apt description.

topsquark
I confess that I do not understand this. It seems to me that the physics is not the same.
Here is a good link to learn it:
https://www.mathpages.com/home/kmath229/kmath229.htm

The hypothesis of the anisotropy of the speed of light is simply Lorentz's ether theory and physics is not the same between the two theories.
LET and SR make the same predictions for experimental results. An argument against LET is Occam's razor.

topsquark
LET and SR make the same predictions for experimental results. An argument against LET is Occam's razor.
In the case of Lorentz, the contraction is real and mechanical and it must be explained by new physics (wave matter theories), and time is not dilated, it is the physical processes that slow down. There is no Minkowski spacetime either and therefore the cosmology cannot be the same. So the physics is not the same.

Occam's razor is not against LET, I read it the other day in a scientific document I don't know where. The LET assumptions are all necessary in a non-Minkowskian spacetime. The question is: is space-time Minkowskian or euclidean ?

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The question is: is space-time Minkowskian or euclidean ?