# Second postulate of SR quiz question

#### loislane

I recently quizzed physicists in my workplace with the following question: The speed c in the second postulate refers to:
a) the one-way speed of light
b) the round-trip speed of light
c) Both
d) Neither

I was surprised at the variety of answers. What do you say?

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#### PeterDonis

Mentor
I would say d). The "c" in the second postulate refers to an intrinsic property of spacetime: a conversion factor between units of time and units of distance.

#### DrGreg

Gold Member
loislane, as the second postulate can be worded in a number of different ways, it would do no harm to quote which version you are referring to. Or is that ambiguity part of the quiz?

#### stevendaryl

Staff Emeritus
I would say d). The "c" in the second postulate refers to an intrinsic property of spacetime: a conversion factor between units of time and units of distance.
Well, in Einstein's original paper seemed pretty definitely to be referring to the speed of light:

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 amoving body.​
http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf (top of page 4)

#### Orodruin

Staff Emeritus
Homework Helper
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2018 Award
Well, in Einstein's original paper seemed pretty definitely to be referring to the speed of light:

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 amoving body.​
http://hermes.ffn.ub.es/luisnavarro/nuevo_maletin/Einstein_1905_relativity.pdf (top of page 4)
While this is true, I would agree with Peter regarding the modern view of relativity, where $c$ is just a conversion factor between units of space and units of time. It then follows that massless fields propagate at $c$, which means that light propagates at $c$. I would say that the nomenclature "speed of light" is a historical remnant due to light being the first thing discovered to propagate with the invariant speed. At the time of Einstein, it was natural and had light not had the property of travelling at the invariant speed, relativity would have taken significantly longer to develop.

#### stevendaryl

Staff Emeritus
While this is true, I would agree with Peter regarding the modern view of relativity, where $c$ is just a conversion factor between units of space and units of time.
But the question was about Einstein's "second postulate", so it would seem to be more about Einstein's original presentation, rather than the modern understanding.

#### Orodruin

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But the question was about Einstein's "second postulate", so it would seem to be more about Einstein's original presentation, rather than the modern understanding.
Einstein is not mentioned in the OP. That there is a speed which is independent of the observer can still be taken as a postulate of SR.

#### PAllen

Going back to Einstein, c as two way speed was a measurement, one way speed was a postulate which produced the simplest models consistent with measurement. I believe Einstein was well aware that one way speed had to be a postulate of "immense convenience" but unknowable truth.

Mentor

#### MindWalk

Perhaps you could clarify what is meant by a two-way speed of light? If a beam of light strikes a mirror and gets bounced back, it travels at c (in vacuo) in each direction.

I do not see why we must choose between thinking of c as the speed of light in vacuo, on one hand, and thinking of c as a conversion factor between units of distance and units of time, on the other.

#### Nugatory

Mentor
Perhaps you could clarify what is meant by a two-way speed of light? If a beam of light strikes a mirror and gets bounced back, it travels at c (in vacuo) in each direction.
That is an assumption. It is a very good and very plausible assumption and most people are willing to accept it without argument, but it's still an assumption and not an experimentally proven fact. Many people have proposed experiments that appear to (at least in principle) measure the one-way speed of light, but if you dig deep into these proposals you'll find that there's a hidden assumption that the speed is the same in each direction. You'll find a number of threads about this if you search this forum.

I do not see why we must choose between thinking of c as the speed of light in vacuo, on one hand, and thinking of c as a conversion factor between units of distance and units of time, on the other.
That, I think, is a defensible position. The first is the historical path that brought us to special relativity. The second is the better (shorter, simpler, fewer hidden assumptions, fewer limitations, springboard for further progress) understanding that we found once we had arrived and realized what we had discovered.

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#### Nugatory

Mentor
I was surprised at the variety of answers. What do you say?
It's not that surprising when we consider that:
1) Neither Einstein's presentation of the "postulates" of SR nor his derivations based on them come anywhere near the level of precision that a mathematician would demand from a postulate and the proofs derived from it. That's an observation, not a criticism - but it does leave much room for difference of opinion about exactly what is being postulated.
2) The modern understanding of SR is quite different from the historical understanding, and that affects the interpretation of the second postulate.
3) The second postulate makes its point in a rather odd (in hindsight) way. If you accept the first postulate at face value, and accept Maxwell's electrodynamics, and apply Occam's razor, you'll find only two possibilities. Either the speed of light in vacuum is the same for all observers, or you have to make an additional assumption that there is a luminiferous aether or equivalent which allows us to distinguish the absolute state of motion of different observers. So why not state the second postulate as "And no additional assumptions needed" or "And I don't need no stinkin' aether!" or "And I really mean the first postulate, even when it comes to the electrodynamics of moving bodies"? The answer, of course, is that none of those formulations would have been convincing in 1905. Again, this creates much opportunity for the post-1905 crowd, blessed with hindsight, to disagree about exactly what truth lies behind the wording of the second postulate.

This might be a good time to quote F. Scott Fitzgerald: "The test of a first-rate intelligence is the ability to hold two opposed ideas in mind at the same time and still retain the ability to function". Considering the different ways that the second postulate can be interpreted is far more illuminating than arguing about which one is right.

#### stevendaryl

Staff Emeritus
Perhaps you could clarify what is meant by a two-way speed of light? If a beam of light strikes a mirror and gets bounced back, it travels at c (in vacuo) in each direction.

I do not see why we must choose between thinking of c as the speed of light in vacuo, on one hand, and thinking of c as a conversion factor between units of distance and units of time, on the other.
The difference between two-way speed and one-way speed is that two-way (or round-trip) speed doesn't require a convention for synchronizing distant clocks. You can measure (in principle) the round-trip speed of light using a standard meterstick and a single clock: Put a mirror at one end of the stick and measure the round-trip time for light to travel from the other end to the mirror, and back. That measurement gives you an average speed of light, but it doesn't give you the one-way speed unless you assume that light has the same speed in all directions.

#### stevendaryl

Staff Emeritus
Einstein is not mentioned in the OP. That there is a speed which is independent of the observer can still be taken as a postulate of SR.
Yes, but when someone says "THE second postulate of SR" without adding "according to the presentation in such-and-such book or paper" I would think that it would mean the original presentation by Einstein. No other presentation is famous enough to use the definite article.

#### stevendaryl

Staff Emeritus
Yes, but when someone says "THE second postulate of SR" without adding "according to the presentation in such-and-such book or paper" I would think that it would mean the original presentation by Einstein. No other presentation is famous enough to use the definite article.
But loislane has seemingly dropped out of the discussion; otherwise, she could confirm what she meant by "the second postulate".

#### harrylin

I recently quizzed physicists in my workplace with the following question: The speed c in the second postulate refers to:
a) the one-way speed of light
b) the round-trip speed of light
c) Both
d) Neither

I was surprised at the variety of answers. What do you say?
The way Einstein phrased it in 1905 was ambiguous.
He was clearer in 1907, as he phrased it in terms of a), but such that it effectively refers to b):

"We [...] assume that the clocks can be adjusted in such a way that
the propagation velocity of every light ray in vacuum - measured by
means of these clocks - becomes everywhere equal to a universal
constant c, provided that the coordinate system is not accelerated."

The essential point is that distant simultaneity is not postulated.

#### stevendaryl

Staff Emeritus
The way Einstein phrased it in 1905 was ambiguous.
He was clearer in 1907, as he phrased it in terms of a), but such that it effectively refers to b):

"We [...] assume that the clocks can be adjusted in such a way that
the propagation velocity of every light ray in vacuum - measured by
means of these clocks - becomes everywhere equal to a universal
constant c, provided that the coordinate system is not accelerated."

The essential point is that distant simultaneity is not postulated.
Yes. In a certain sense, one-way speed of light is purely a matter of convention, because it depends on how distant clocks are synchronized.

#### vanhees71

Gold Member
This thing with the mirror in the "two-way speed of light" definition brings me to another question. The question is, which "speed" is meant here. Is it (the magnitude of) phase velocity, group velocity, front velocity or whatever else you can think of?

Without thinking much about it, seen from the perspective of Einstein 1905 it's the constant occuring in Maxwell's equations in vacuo (nowadays it's hidden in the mess of the SI, but it's of course there in form of $c^2=1/(\epsilon_0 \mu_0)$, and that's the phase velocity, as seen when looking for the plane-wave modes. That's also what's measured in the Michelson-Morley experiment, where one looks on interference fringes of a stationary wave in the interferometer.

Now, what about the literal "two-way speed of light", where you send a signal (i.e., a wave packet) to a mirror and observe the reflected wave packet. Couldn't there be some delay since the signal has to reflect at the mirror? I'd have to do the calculation to check this. It's perhaps also not so easy to really do this as an experiment, I guess. So are there experiments measuring the "two-way speed of light" really in this way, i.e., sending a wave packet and measuring the arrival time of this back reflected wave packet and how accurate can this be made? Note that wave packets have a finite width and each photodetector has a threshold. Of course you can use the same detector for the outgoing and the reflected wave packet. You also need a large enough distance in order to measure well separated wave packets. If you are to close to the mirror you may measure some wave field which is a superposition of the incoming and the reflected partial waves. As I said, I have to do the calculation.

The same is of course also true for the one-way speed, but there's no possible delay due to a reflection at the mirror. There you'd of course need to photodetectors to measure the time the wave packet needs to travel the distance and consequently a convention to synchronize the clocks to measure the arrival time between the two detectors. A la Einstein that's done by assuming that the one-way speed is the same as the two-way speed (assuming that there's no time delay due to the reflection.

#### loislane

I would say d). The "c" in the second postulate refers to an intrinsic property of spacetime: a conversion factor between units of time and units of distance.
Actually that way out of the question was not considered valid by me because as commented by other posters considering c a conversion factor is independent of its being a speed and certainly in all the variants of the second postulate that is its meaning, even when defined as conversion factor either for the meter or the second it is referred to as distance travelled by light in a certain time or time it takes light to traverse a certain distance in vacuum.
loislane, as the second postulate can be worded in a number of different ways, it would do no harm to quote which version you are referring to. Or is that ambiguity part of the quiz?
Not intentionally, but then I realized that ambiguity is inevitable due to the ambiguous way the postulates are worded in different sources, and even within Einstein's first formulation in 1905. This has also been acknowledged by some posters.
Going back to Einstein, c as two way speed was a measurement, one way speed was a postulate which produced the simplest models consistent with measurement. I believe Einstein was well aware that one way speed had to be a postulate of "immense convenience" but unknowable truth.
I think he was quite aware of the ambiguity he was allowing into the theory. He was mainly after a way to rationalize the Lorentz transformations in a way completely different from Lorentz and Poincare and their absolute rest. A certain calculated ambiguity was essential for that basically interpretational goal.

The way Einstein phrased it in 1905 was ambiguous.
[...]
The essential point is that distant simultaneity is not postulated.
I find this an essential point too. But then the relativity of simultaneity was his very clever way to depart from the Lorentz ether. After all distant simultaneity is more philosophical than physical(in the sense of empirically showing whose clocks are really the correctly synchronized ones from their point of view, being a symmetrical situation there is no "correct" observer) . The real physics and math of the theory lies on the Lorentz transformations themselves.

What do you say?
I have to first say that even though the 4 options showed up most people answered b, maybe because the context of Einstein first paper was more implicitly assumed, and in that paper it is the case that c is defined in a formula as the average speed over twice the distance AB, but indeed there is room and arguments to choose any of the four options due to the commented ambiguity and lack of mathematical rigor of the postulates.
I went for d) basically because with any of the other three one can find ways to convince oneself that the postulates lead to contradictions that anyway cannot be proved precisely due to the ambivalence of the semantics of the postulates and the concept of distant simultaneity.

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#### bcrowell

Staff Emeritus
Gold Member
My answer would be that I don't know, because the question refers to a particular formulation of an obsolete axiomatization, and I don't think anyone in the year 2015 should be memorizing that kind of historical trivia (what's postulate #1, what's postulate #2, etc.). I think it's unfortunate if people are still teaching their students SR using Einstein's postulates, because they reinforce various misconceptions, such as the belief that c has something to do with the speed of light, or that light plays some fundamental role in relativity. Since Einstein himself had a view of SR that, looking back from 2015, seems to have been in many ways hazy and incorrect, why would it be of interest to anyone other than historians of science to try to figure out exactly what he had in mind?

#### PeterDonis

Mentor
considering c a conversion factor is independent of its being a speed and certainly in all the variants of the second postulate that is its meaning, even when defined as conversion factor either for the meter or the second it is referred to as distance travelled by light in a certain time or time it takes light to traverse a certain distance in vacuum.
The fact that we use light, as a practical matter, as our standard for the conversion factor does not mean the conversion factor, considered as a postulate, refers to the speed of light. We use light as the standard because it's the most easily accessible massless field we know of. If we discovered some other massless field that we could use to establish a more accurate standard than using light, we'd use that. Such a change would not change the second postulate of SR, in its modern form, at all.

(As others have commented, Einstein's original form did specifically mention the speed of light; but your OP didn't say "Einstein's second postulate", it only said "the second postulate", which, to me, means you're talking about the postulate in its best modern formulation, i.e., you're talking about physics, not history.)

#### harrylin

[..] 3) The second postulate makes its point in a rather odd (in hindsight) way. If you accept the first postulate at face value, and accept Maxwell's electrodynamics, and apply Occam's razor, you'll find only two possibilities. Either the speed of light in vacuum is the same for all observers, or you have to make an additional assumption that there is a luminiferous aether or equivalent which allows us to distinguish the absolute state of motion of different observers. So why not state the second postulate as "And no additional assumptions needed" or "And I don't need no stinkin' aether!" or "And I really mean the first postulate, even when it comes to the electrodynamics of moving bodies"? The answer, of course, is that none of those formulations would have been convincing in 1905. Again, this creates much opportunity for the post-1905 crowd, blessed with hindsight, to disagree about exactly what truth lies behind the wording of the second postulate.
The essential feature of Maxwell's electrodynamics for the derivation is contained in the second postulate, and Einstein clarified (1907):

"It is by no means self-evident that the assumption made here, which we will call "the principle of the constancy of the velocity of light," is actually realized in nature, but -at least for a coordinate system in a certain state of motion- it is made plausible by the confirmation of the Lorentz theory [Lorentz1895], which is based on the assumption of an ether that is absolutely at rest, through experiment [Fizeau].

Classical relativity remains a possible solution without the second postulate (and with Occam's razor, it can be argued to be the most plausible one).

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#### harrylin

[..] I have to first say that even though the 4 options showed up most people answered b, maybe because the context of Einstein first paper was more implicitly assumed, and in that paper it is the case that c is defined in a formula as the average speed over twice the distance AB, but indeed there is room and arguments to choose any of the four options due to the commented ambiguity and lack of mathematical rigor of the postulates.
I went for d) basically because with any of the other three one can find ways to convince oneself that the postulates lead to contradictions that anyway cannot be proved precisely due to the ambivalence of the semantics of the postulates and the concept of distant simultaneity.
What contradictions could possibly be imagined with b)?

#### vanhees71

Gold Member
My answer would be that I don't know, because the question refers to a particular formulation of an obsolete axiomatization, and I don't think anyone in the year 2015 should be memorizing that kind of historical trivia (what's postulate #1, what's postulate #2, etc.). I think it's unfortunate if people are still teaching their students SR using Einstein's postulates, because they reinforce various misconceptions, such as the belief that c has something to do with the speed of light, or that light plays some fundamental role in relativity. Since Einstein himself had a view of SR that, looking back from 2015, seems to have been in many ways hazy and incorrect, why would it be of interest to anyone other than historians of science to try to figure out exactly what he had in mind?
In general I agree with you that teaching physics in the "historical" way is not always the best choice, also some knowledge about the history of physics is also good to gain understanding of the meaning of concepts, despite the fact that history is also an interesting subject in itself.

On the other hand, in this case, I'm not so sure. Physics is, after all, an empirical science and based on observations and the attempt to find some fundamental observations that can be used to build mathematical models in a bit like an axiomatic approach (although I don't think that we have a sharp axiomatic system of all of contemporary physics).

It is no surprise that relativity was discovered by thinking about electromagnetic phenomena as they were summarized brillantly in form of Maxwell's equations and a bit later in form of Lorntz's "elctron theory". The speed of light has been part of Maxwell's equations and was found to be the phase velocity of electromagnetic waves, which finally were experimentally discovered and investigated in detail by H. Hertz. This is still the most convincing empirical manifestation of the relativistic space-time model.

Of course, it is important to stress that relativity is comprehensive and not limited to electromagnetic/optical phenomena. You can derive the special-relativistic space-time structure just from the special principle of relativity and symmetry assumptions (which boils down to postulate euclidicity of space for any inertial observer). Then you get two space-time models in terms of the corresponding symmetry groups, i.e., the Galilei-Newton spacetime and Einstein-Minkowski spacetime. It is, of course, still an empirical question to decide which one describes nature better, and then again the most convincing arguments come from electromagnetism and optics. Last but not least the question, whether the limiting universal speed of Einstein-Minkowski spacetime is the phase velocity of light or not, is still just really empirical. There is no fundamental law in contemporary physics, which can be used as an argument that electromagnetic waves are strictly massless vector fields (or in QFT language that the photons are really exactly massless). There is of course overwhelming empirical evidence that this is true.

Of course, nowadays another strong argument for the correctness of the relativistic spacetime models is high-energy particle physics. To construct the accelerators to do experiments with particles and nuclei at high energies, the use of relativistic dynamics is mandatory. You couldn't use Newtonian physics to plan an accelerator like the LHC. Note that this part is also very much (classical) electrodynamics and relativistic mechanics of (bunches of) charged particles. Then the tremendous success of the Standard Model is of course further strong evidence for the relativistic spacetime model which is the most important building block of the underlying local microcausal QFT paradigm (which to a large part is representation theory of the Poincare group, which strongly hints towards the usefulness of gauge theories).

But a modern physics didactics does not wait to introduce relativity at the end of the undergrad level, and in my opinion, right so. You can not start early enough to introduce special relativity at least in the theory curriculum (in Frankfurt that's usualy already done at the end of the very 1st freshmen semester). Then you cannot use high-brow group theory or field theory to establish relativity, but the traditional approach a la Einstein is very valuable. It comes with a minimum of assumptions, and you can do relativistic mechanics of point particles in this way, which is nice as a conclusion for the 1st theory semester and a good starting point for E+M in the 3rd (in the 2nd semester usually you have analytical mechanics (Hamilton's action principle etc), where usually also the relativistic case is treated again).

#### vanhees71

Gold Member
The essential feature of Maxwell's electrodynamics for the derivation is contained in the second postulate, and Einstein clarified (1907):

"It is by no means self-evident that the assumption made here, which we will call "the principle of the constancy of the velocity of light," is actually realized in nature, but -at least for a coordinate system in a certain state of motion- it is made plausible by the confirmation of the Lorentz theory [Lorentz1895], which is based on the assumption of an ether that is absolutely at rest, through experiment [Fizeau].

Classical relativity remains a possible solution without the second postulate (and with Occam's razor, it can be argued to be the most plausible one).
If you have the Maxwell equations at hand, you can introduce relativity very easily by just looking for the spacetime symmetry under which the Maxwell equations are covariant. Of course, that there is a universal constant with the dimension of a speed involved is evident from the equations to begin with, although nowadays hidden in the complicated SI units ;-). So it is immediately clear, without much deeper mathematics, that the Galileo symmetry of Newtonian mechanics cannot be the right thing, because there no such universal constant is seen.

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