Understanding Einstein's second postulate of special relativity

[Erland]

TL;DR Summary

In Einstein's 2nd postulate, it is the motion of the light source that is important, not the motion of the observer. The observer is assumed to be "stationary".

Einstein's two postulates of special relativity reads, according to Wikipedia:

1. First postulate (principle of relativity)

The laws of physics take the same form in all inertial frames of reference.

2. Second postulate (invariance of c)

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. Or: the speed of light in free space has the same value c in all inertial frames of reference.

The postulates are most often formulated similarly to this. But in my opnion, the second postulate shouldn't be formulated as above, because then one misses the point. This is particular true for the second sentence: "the speed of light in free space has the same value c in all inertial frames of reference". This is certainly true, but it is not what the postulate is about. The postulate is not about "all inertial frames of reference" it is about one particular intertial frame, which is considered stationary.

Indeed, Einstein himself writes, i his original 1905 paper:

Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good. In order to render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the “stationary system.”

In the next section in this paper, he states the postulates:

1. The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion.
2. Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. Hence
   
   
   where time interval is to be taken in the sense of the definition in § 1.

So, in Einstein's wording, the 2nd postulate does not talk about the value of the speed of light in all intertal frames, but only in one particular frame, which Einstein singled out and called "stationary". Einstein also writes what this really is about: "whether the ray be emitted by a stationary or a moving body".
That is the point. For observer at rest in the "stationary" frame, it doesn't matter if the light source is at rest, or moves towards or away from him/her, or sideways, (s)he will always measure the speed of this light to c.
This is different from how it is for slowly moving objects. For example, if a car is moving with velocity V and a ball is thrown from the car with velocity v relative to the car, then the ball will get the velocity v+V relative to an observer on the ground, if the car moves towards the observer, and v-V if the car moves away from the observer.
It would be natural to think that the same holds for light rays: i.e. if the light source is at rest in the "stationary" frame, then the speed light is measured to c by the "stationary" of observer, but if the light source moves towards the observer with velocity v, then the observer measures the light speed to c+v, and to c-v if the light source moves away from the observer with velocity v.
But the 2nd postulate says that it is not so: the stationary observer measures the light speed to c, independently of the motion of the light source.
Again: This is the point of the second postulate.

Of course, what Wikpedia says is also true: the light speed has the same value c in all inertial frames. But that is not what the 2nd postulate says, and it does not follow from that postulate alone, we also need the 1st postulate to infer this: The constancy of the light speed is considered as a law of physics, and those are the same in all intertial frames, by the 1st postulate. Therefore, the speed of light is measured to c in all intertial frames.
When, as in the Wikipedia article and many other texts, this conclusion is lumped togther with the 2nd postulate itself, it is easy to miss the point and think that the 2nd postulate is about different observers all measuring the light speed to c, when it in fact is about the light speed being independent of the motion of the light source.
This might be one reason why Einstein chose to single out one particular inertial frame and call it "stationary" and formulate the 2nd postulate about this frame only.

The weakness with talking about this "stationary" frame is, of course, that it is arbitrary. There is no way to single ot such a "stationary" frame in reality. All inertial frames are equivalent physically, by the 1st postulate. It is therefore understandable that one in pedagocical representations of SR does not want to invoke such a "stationary" frame: the reader might believe that it is something real, and also: why introdue this concept when it is immediately getting rid of, by concluding that the light speed is the same in all intertial systems (as we did above)?

Still, I think the pedagogicial gain is greater that its difficulties: if one just does not take the "stationary" frame too seriously, and thinks of it as the "ground" or the "laboratory frame" or something like that, it is useful, and it leads to a better understanding of what the 2nd postulate really is about.

 

[PeroK]

Einstein's two postulates of special relativity reads, according to Wikipedia:

The postulates are most often formulated similarly to this. But in my opnion, the second postulate shouldn't be formulated as above, because then one misses the point. This is particular true for the second sentence: "the speed of light in free space has the same value c in all inertial frames of reference". This is certainly true, but it is not what the postulate is about. The postulate is not about "all inertial frames of reference" it is about one particular intertial frame, which is considered stationary.

Wikipedia is correct. It's you who is entirely missing the point.

Indeed, Einstein himself writes, i his original 1905 paper:In the next section in this paper, he states the postulates:So, in Einstein's wording, the 2nd postulate does not talk about the value of the speed of light in all intertal frames, but only in one particular frame, which Einstein singled out and called "stationary".

I'm not sure why Einstein decided to called this frame "stationary". "Stationary" here is only a label and implies nothing significant about that particular frame. Given the confusion it causes, I wish he had simply called it frame "A".

 

[Janus]

Note that He puts quotation marks around "stationary". This is because there is no frame that is stationary in an absolute sense. In fact, any inertial frame can treat itself as the "stationary" frame. Just substitute " the inertial frame from which the measurement is being made" for "stationary".
Besides that, it make no sense to belabor exactly how he phrased his 2nd postulate, as once you get in to the meat of the theory, it becomes clear as to what he means.

 

[Dale]

So, in Einstein's wording, the 2nd postulate does not talk about the value of the speed of light in all intertal frames, but only in one particular frame, which Einstein singled out and called "stationary".

Einstein had the first word on relativity, not the last. The second postulate is now understood and written as Wikipedia described, by generations of scientists since Einstein. We no longer use Einstein’s wording.

 

[Sagittarius A-Star]

TL;DR Summary: In Einstein's 2nd postulate, it is the motion of the light source that is important, not the motion of the observer. The observer is assumed to be "stationary".

So, in Einstein's wording, the 2nd postulate does not talk about the value of the speed of light in all intertal frames, but only in one particular frame, which Einstein singled out and called "stationary". Einstein also writes what this really is about: "whether the ray be emitted by a stationary or a moving body".
That is the point.

That is not the point. For example, the speed of sound from a moving police car is also independent of the velocity of the source in the reference frame, in which the air is at rest.

Also, Einstein 1905 does not single out a certain frame physically. As you quoted him, he used “stationary system” only to distinguish this frame verbally form the others. Today, we usually call this "reference frame".

This is different from how it is for slowly moving objects. For example, if a car is moving with velocity V and a ball is thrown from the car with velocity v relative to the car, then the ball will get the velocity v+V relative to an observer on the ground, ...

According to SR, if the ball is thrown with velocity $v$ with reference to the rest-frame of the car, then the velocity of the ball with reference to the ground frame is not $v+V$, but $\frac{v+V}{1+(vV/c^2)}$.

Still, I think the pedagogicial gain is greater that its difficulties: if one just does not take the "stationary" frame too seriously, and thinks of it as the "ground" or the "laboratory frame" or something like that, it is useful, and it leads to a better understanding of what the 2nd postulate really is about.

Yes.

 

[vela]

The way I read what Einstein wrote is that (1) any inertial frame can be considered the "stationary" frame and (2) in that frame, the speed of light is $c$ regardless of the motion of the source. It's not exactly a big leap to see that implies the speed of light is the same in all inertial reference frames.

 

[PeterDonis]

The postulate is not about "all inertial frames of reference" it is about one particular intertial frame, which is considered stationary.

But this frame could be any frame at all, since no assumptions are made that would distinguish it from any other frame. The term "stationary" is just for convenience; it does not pick out any property of that particular frame that would make it different from all other frames. So anything which is proven for that frame is valid for any frame.

 

[Erland]

The way I read what Einstein wrote is that (1) any inertial frame can be considered the "stationary" frame and (2) in that frame, the speed of light is $c$ regardless of the motion of the source. It's not exactly a big leap to see that implies the speed of light is the same in all inertial reference frames.

The way I understand what Einstein wrote is that one picks an inertial frame (in principle arbitrarily, since they cannot be distingushed from each other by any physical means, as many of you point out), but that it is only for this selected frame that the light speed is assumed to be measured to the same value regardless of the motion of the light source in this frame.
If my interpretation is correct, this would mean that the 2nd postulate could be reformalated like this:

There is (at least) one intertial frame such that the light speed is measured to the same value regardless of the motion of the light source in this frame.

Then, we need the 1st postulate to infer that the light speed is measured to the same value in all inertial frames.

We cannot know for certain what Einstein meant, but in my opinion, this interpretation has the advatage that it empasizes the motion of the light source, which is what the 2nd postulate really is about. That it is so can be seen from how the 2nd posultate is usually tested: Then, one measues the speed of light from moving sources with different velocities (such as double stars, or, as in the Alväger experiment, fast electrons), with the observer considerad as stationary. One doesn't measure the speed of light measured by different observers moving relative to ech other (or are there such experiments too?).

 

[PeroK]

The way I understand what Einstein wrote is that one picks an inertial frame (in principle arbitrarily, since they cannot be distingushed from each other by any physical means, as many of you point out), but that it is only for this selected frame that the light speed is assumed to be measured to the same value regardless of the motion of the light source in this frame.
If my interpretation is correct, this would mean that the 2nd postulate could be reformalated like this:

There is (at least) one intertial frame such that the light speed is measured to the same value regardless of the motion of the light source in this frame.

Then, we need the 1st postulate to infer that the light speed is measured to the same value in all inertial frames.

We cannot know for certain what Einstein meant, but in my opinion, this interpretation has the advatage that it empasizes the motion of the light source, which is what the 2nd postulate really is about. That it is so can be seen from how the 2nd posultate is usually tested: Then, one measues the speed of light from moving sources with different velocities (such as double stars, or, as in the Alväger experiment, fast electrons), with the observer considerad as stationary. One doesn't measure the speed of light measured by different observers moving relative to ech other (or are there such experiments too?).

The problem at the time was this: in which particular reference frame are Maxwell's equations of electrodynamics valid? And, Einstein's stunning answer was: all inertial reference frames. Not just in one special frame.

 

[Dale]

The way I understand what Einstein wrote is that one picks an inertial frame (in principle arbitrarily, since they cannot be distingushed from each other by any physical means, as many of you point out), but that it is only for this selected frame that the light speed is assumed to be measured to the same value regardless of the motion of the light source in this frame.
…

We cannot know for certain what Einstein meant, but in my opinion, this interpretation has the advatage that it empasizes the motion of the light source, which is what the 2nd postulate really is about.

Einstein was not a prophet and his words are not holy writ.

The current mainstream consensus is reflected in the Wikipedia article. What he may or may not have meant is not particularly important to decipher. Nor is it a particularly strong justification for your claim about what the 2nd postulate is "really about".

There is (at least) one intertial frame such that the light speed is measured to the same value regardless of the motion of the light source in this frame.

Then, we need the 1st postulate to infer that the light speed is measured to the same value in all inertial frames.

Another issue is that this doesn’t follow. Your 2nd postulate, combined with the first, does not produce the invariance of c. To see this, consider another postulate of the same form:

There is (at least) one intertial frame such that the speed of sound is measured to the same value regardless of the motion of the sound source in this frame.

This postulate (though not accurate) shows the logical hole in your formulation. In the inertial frame where the air is at rest the speed of sound indeed does not depend on the motion of the source. And yet this fact is compatible with a world in which the first postulate holds without requiring the invariance of the speed of sound. The presence of a medium allows this combination.

Similarly, if light were to propagate through a medium then the speed of light could be c in the rest frame of that medium, the principle of relativity could hold, and c could be frame variant. This is one reason that the usual description of the second postulate now explicitly mentions frame invariance.

 

[PeterDonis]

The way I understand what Einstein wrote is that one picks an inertial frame (in principle arbitrarily, since they cannot be distingushed from each other by any physical means, as many of you point out), but that it is only for this selected frame that the light speed is assumed to be measured to the same value regardless of the motion of the light source in this frame.

Since you can pick any frame, it makes no sense to say that the speed of light is only assumed to be $c$ for that frame. One can do the analysis in that chosen frame for convenience, but since the same analysis could be done in any frame, doing it in one particular frame is in no way the same as saying that the speed of light is only $c$ in that frame.

That it is so can be seen from how the 2nd posultate is usually tested: Then, one measues the speed of light from moving sources with different velocities (such as double stars, or, as in the Alväger experiment, fast electrons), with the observer considerad as stationary.

No, this does not justify your claim, because you are assuming that "moving" and "stationary" have some absolute meaning. But they don't. That was one of Einstein's key points.

The actual experimental tests are for sources that are moving relative to the observer. The relative motion is the only physically meaningful motion. Doing the analysis in the observer's rest frame and calling that frame "stationary" is, as has already been said several times, a matter of convenience. It does not have the implications you are claiming.

 

[Mister T]

TL;DR Summary: In Einstein's 2nd postulate, it is the motion of the light source that is important, not the motion of the observer. The observer is assumed to be "stationary".

The postulate is not about "all inertial frames of reference" it is about one particular intertial frame, which is considered stationary.

There's nothing to distinguish stationary from inertial reference frames. They are equivalent. You have at most a quibble over the language used.

 

[Erland]

Another issue is that this doesn’t follow. Your 2nd postulate, combined with the first, does not produce the invariance of c. To see this, consider another postulate of the same form:

There is (at least) one intertial frame such that the speed of sound is measured to the same value regardless of the motion of the sound source in this frame.

This postulate (though not accurate) shows the logical hole in your formulation. In the inertial frame where the air is at rest the speed of sound indeed does not depend on the motion of the source. And yet this fact is compatible with a world in which the first postulate holds without requiring the invariance of the speed of sound. The presence of a medium allows this combination.

Similarly, if light were to propagate through a medium then the speed of light could be c in the rest frame of that medium, the principle of relativity could hold, and c could be frame variant. This is one reason that the usual description of the second postulate now explicitly mentions frame invariance.

I meant of course the speed of light in vacuo. I took that for granted and didn't write out.

 

[Erland]

No, this does not justify your claim, because you are assuming that "moving" and "stationary" have some absolute meaning. But they don't. That was one of Einstein's key points.

The actual experimental tests are for sources that are moving relative to the observer. The relative motion is the only physically meaningful motion. Doing the analysis in the observer's rest frame and calling that frame "stationary" is, as has already been said several times, a matter of convenience. It does not have the implications you are claiming.

No, I don't assume that "moving" and "stationary" have some absolute meaning. I mean the same as you: "sources that are moving [or not moving] relative to the observer".
One could imagine making experiments where several observers moving with various velocities relative to each other all measure the speed of light, to see if they get the same or different values, in order to test the 2nd postulate (understanding it in the way Wikipedia and most of you do). But I don't know of any such experiments (but perhaps there has been such experiments unknown to me...).

 

[PeroK]

No, I don't assume that "moving" and "stationary" have some absolute meaning. I mean the same as you: "sources that are moving [or not moving] relative to the observer".
One could imagine making experiments where several observers moving with various velocities relative to each other all measure the speed of light, to see if they get the same or different values, in order to test the 2nd postulate (understanding it in the way Wikipedia and most of you do). But I don't know of any such experiments (but perhaps there has been such experiments unknown to me...).

I'm not sure what's the purpose of this thread. Many scientific theories are tested indirectly. Take Newton's law of gravity. You can't go out and physically measure the force between Jupiter and the Sun. Instead, you can measure the orbits of the various planets and steadily corroborate the predictions of the theory.

SR is no different. It's difficult, if not impossible, to set up a direct test of the second postulate. Instead, as indeed Einstein did in his 1905 paper (see section 8), he worked out some consequences of the postulate that distinguished it from the current theory.

SR is now so deeply embedded in modern physics that every time the LHC at CERN is operated it represents a rigorous test of SR! People may forget this and imagine that all we have as a basis for SR is the "holy writ" that is the 1905 paper with its second postulate.

Moreover, especially with modern insight, the second postulate is part of the historical road that led to SR, but by no means the only way to establish the theory. All of modern particle dynamics is based on relativistic energy-momentum, which is a direct consequence of the second postulate.

 

[Sagittarius A-Star]

I meant of course the speed of light in vacuo. I took that for granted and didn't write out.

But Einstein wrote the postulates also to exclude, that the light speed in "vacuo" depends on the state of motion of a hypothetical “luminiferous ether”:

Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the “light medium”
...
The introduction of a “luminiferous ether” will prove to be superfluous inasmuch as the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.

Source:
https://www.fourmilab.ch/etexts/einstein/specrel/www/

 

[Ibix]

But I don't know of any such experiments (but perhaps there has been such experiments unknown to me...).

Observations of binary stars do this, surely. At least if you accept that the only relevant motion is the relative motion of source and receiver.

 

[vanhees71]

That is not the point. For example, the speed of sound from a moving police car is also independent of the velocity of the source in the reference frame, in which the air is at rest.

But that's the crucial difference between electromagnetic waves in a vacuum and waves like sound waves of a medium, where the (local instantaneous) rest frame(s) of the medium is a preferred frame of reference, and it's used to define all intrinsic properties of this medium in terms of scalars (density, charge density, temperature, chemical potential, and all that).

If you treat the general case of the Doppler effect for such kinds of wave, you'll see that the observed frequency depends on the velocity of the source, the velocity of the medium, and the velocity of the detector separately. For em. waves, and only for em. waves, in vacuo does the observed frequency depend only on the relative velocity of source and detector, because em. wave (to the best of our knowledge) propagate with the "limiting speed" as phase velocity.

For details, see

https://itp.uni-frankfurt.de/~hees/pf-faq/rela-waves.pdf

 

[Sagittarius A-Star]

But that's the crucial difference between electromagnetic waves in a vacuum and waves like sound waves of a medium

The part "whether the ray be emitted by a stationary or by a moving body" is not crucial. The first part of the text in the 2^nd postulate in Einstein 1905 is sufficient.

Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body.

The first part "Any ray of light moves in the 'stationary' system of co-ordinates with the determined velocity c" is sufficient, because Einstein also wrote:

Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good.²
In order to render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the “stationary system.”

Source:
https://www.fourmilab.ch/etexts/einstein/specrel/www/

Easier understandable is of course the following modern formulation in Wikipedia:

the speed of light in free space has the same value c in all inertial frames of reference.

Source:
https://en.wikipedia.org/wiki/Postulates_of_special_relativity

 

[Dale]

I meant of course the speed of light in vacuo. I took that for granted and didn't write out.

That doesn't alter the point. At the beginning of the 20th century it was commonly believed that the vacuum contained the luminiferous aether, a medium for EM waves. It was expected that the speed of light would transform like any other speed. This would be consistent with your formulation of the postulates.

 

[vanhees71]

The part "whether the ray be emitted by a stationary or by a moving body" is not crucial. The first part of the text in the 2^nd postulate in Einstein 1905 is sufficient.The first part "Any ray of light moves in the 'stationary' system of co-ordinates with the determined velocity c" is sufficient, because Einstein also wrote:

Source:
https://www.fourmilab.ch/etexts/einstein/specrel/www/

Easier understandable is of course the following modern formulation in Wikipedia:

Source:
https://en.wikipedia.org/wiki/Postulates_of_special_relativity

I think the most simple and accurate formulation is that for any (inertial) observer the speed of light in a vacuum is independent of the motion of its source.

 

[Erland]

That doesn't alter the point. At the beginning of the 20th century it was commonly believed that the vacuum contained the luminiferous aether, a medium for EM waves. It was expected that the speed of light would transform like any other speed. This would be consistent with your formulation of the postulates.

I don't understand. Are you thinking of some kind of "aether drag" hypothesis?

 

[Dale]

I don't understand. Are you thinking of some kind of "aether drag" hypothesis?

Yes. Under an aether hypothesis your formulation of the second postulate cannot be combined with the first postulate to imply the invariance of c.

 

[Erland]

Yes. Under an aether hypothesis your formulation of the second postulate cannot be combined with the first postulate to imply the invariance of c.

Do I understand you correctly: You imagine that in one inertial frame, the one we call "stationary", there is no local aether drag and the light speed is measured to c everywhere, but in another inertial frame, there is local aether drag and the light speed is measured to different values at different places, depending upon the local aether drag?

 

[Dale]

Do I understand you correctly: You imagine that in one inertial frame, the one we call "stationary", there is no local aether drag and the light speed is measured to c everywhere, but in another inertial frame, there is local aether drag and the light speed is measured to different values at different places, depending upon the local aether drag?

Yes. And if the aether is a medium for light like air is a medium for sound, then this does not violate the first postulate.

 

[Ibix]

Do I understand you correctly: You imagine that in one inertial frame, the one we call "stationary", there is no local aether drag and the light speed is measured to c everywhere, but in another inertial frame, there is local aether drag and the light speed is measured to different values at different places, depending upon the local aether drag?

Note that @Dale is pointing out that this belief was the context in which Einstein was writing, not proposing it as correct.

 

[Nugatory]

One could imagine making experiments where several observers moving with various velocities relative to each other all measure the speed of light, to see if they get the same or different values, in order to test the 2nd postulate (understanding it in the way Wikipedia and most of you do). But I don't know of any such experiments (but perhaps there has been such experiments unknown to me...).

This might be one of the most performed experiments of all time.
The GPS satellites and receivers are all moving at different speeds relative to one another, and the accuracy of the positioning depends on the measured speed of light being the same for all of them. So every single GPS location fix....

 

[PeterDonis]

I don't assume that "moving" and "stationary" have some absolute meaning.

You have to if you want to make the claims you are making.

I mean the same as you: "sources that are moving [or not moving] relative to the observer".

Then the claim you are making in the OP is not justified. If all you mean is relative motion, you have no ground for claiming that Einstein's second postulate only applies to a single frame. Relative motion between the source and observer can be described in any frame; as has already been said several times, Einstein chose the observer's rest frame and called it "stationary" for convenience, that's all.

 

[Sagittarius A-Star]

I think the most simple and accurate formulation is that for any (inertial) observer the speed of light in a vacuum is independent of the motion of its source.

I would say, this formulation difficult to understand.

It clearly excludes the emission theory of Walter Ritz.

But for the exclusion of the ether hypothesis of the 19^th century, this formulation is difficult to understand, because according to this hypothesis, the speed of light is in general anisotropic. In a reference frame, it depends directly on the motion of the ether and on the direction of a light beam. So there is no unique light-speed in the reference frame, that could be independent of the motion of its source.

 

[PeterDonis]

I would say, this formulation is not even wrong.

You do realize that he is just stating one of the postulates of SR, correct? And that there is extensive experimental evidence that this postulate holds?

It clearly excludes the emission theory of Walter Ritz.

So what?

But for the exclusion of the ether hypothesis of the 19th century, this formulation is problematic

So what?

 

[Sagittarius A-Star]

You do realize that he is just stating one of the postulates of SR, correct?

No, he omits a part of it and wrote only the part, that is not crucial.

2. Second postulate (invariance of c)

As measured in any inertial frame of reference, light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body. Or: the speed of light in free space has the same value c in all inertial frames of reference.

Source:
https://en.wikipedia.org/wiki/Postulates_of_special_relativity

 

[PeterDonis]

omits a part of it and wrote only the part, that is not crucial.

What you quoted from Wikipedia says the same thing @vanhees71 said. The second sentence is not an additional part of the postulate; it's just a rewording of the first sentence. That's why the "Or" is there in between.

 

[Sagittarius A-Star]

What you quoted from Wikipedia says the same thing @vanhees71 said. The second sentence is not an additional part of the postulate; it's just a rewording of the first sentence. That's why the "Or" is there in between.

He omitted the first half of the first sentence.

 

[PeterDonis]

He omitted the first half of the first sentence.

No, he didn't. He said "for any inertial observer".

 

[Sagittarius A-Star]

No, he didn't. He said "for any inertial observer".

He omitted "is always propagated in empty space with a definite velocity c".