Why does light have invarient speed?

[mdeng]

If the theory is based on the first and second postulates, then the speed of light 'c' is guaranteed, but the abstract mathematical manipulations will not reveal the 'how' or 'why'. The purpose of the theory is to produce numbers that agree with measurements/perceptions by the observer. To explain 'why', you have to analyze the behaviour in terms of physical processes. There is an answer to this, just as there is for time dilation.

Right, I have no issues with the revealing math results or their accuracy, but I am curious about any insights on how nature does 'c' and what this insight may tell us over and above SR. BTW, did you mean "there will be an answer" or there is one already?

 

[rbj]

I tend to agree with you on this point. I have often thought that the first postulate was sufficient and that the second postulate is simply a corolary to the first one.

i'm glad to think it wasn't just i that was going crazy. like we're in Opposite World where we get to switch who is in a subset of what. are the quantitative parameters of a law part of the law?

I have a similar "wrong but not strong" opinion about Newton's first and second laws.

as if an acceleration rate of zero is a subset of the second law. why would you think such an heretical thing?

 

[lightarrow]

I meant v_1=-v_2 and |v_1| = c. I guess it still holds.

In that case R itself is not defined because artgh(1) is not defined.
Note that the case v1 = c cannot however studied in SR because v1 is the speed of the moving ref. frame S' with respect to the stationary ref. frame S (v2 is the speed of the object with respect to S') and we know that no ref. frame with that speed can exist.

 

[Xeinstein]

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

Thanks,
- Ming

It's a matter of time.
The solution, Einstein explained, lay in a reconception of the idea of time.

Einstein lifted the idea that the speed of light is constant intact from electromagnetic theory, devised forty years earlier by the Scottish-born physicist James Clerk Maxwell. Part of Einstein's larger ambition was to reconcile electromagnetism with Galilean relativity. Then one night in May 1905, after discussing the problem with his longtime friend Michele Besso, Einstein saw how to do so.

Thank you!" Einstein greeted Besso the following morning. I have completely solved the problem."

The solution, he explained, lay in a reconception of the idea of time. Any velocity is simply distance divided by time. In the case of light, though, the velocity isn't just 186,282 miles per second; according to Einstein's postulate, it's always 186,282 miles per second. It's a constant. It's on one side of the equal sign, humming along at its imperturbable rate. On the other side of the equal sign are distance and time, which become, by default, variables. They can undergo as many changes in value as you can imagine, as long as they continue to divide in such a way that the result is 186,282 miles per second. Change the distance, and you have to change the time.

You can solve the problem too.

 

[Xeinstein]

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

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

I disagree.
I'd say the underlying properties of space-time is a by-product of the invariance of the speed of light which is a postulate of SR

 

[Doc Al]

I disagree.
I'd say the underlying properties of space-time is a by-product of the invariance of the speed of light which is a postulate of SR

Don't confuse how we deduce the consequences of relativity from the usual postulates with how we interpret "why" things are the way they are. I'd agree with belliott4488 that it's space and time itself that is structured in such a way as to make anything moving with speed c have an invariant speed with respect to any frame. It's interesting that light has such a property, but not fundamental.

 

[Xeinstein]

Don't confuse how we deduce the consequences of relativity from the usual postulates with how we interpret "why" things are the way they are. I'd agree with belliott4488 that it's space and time itself that is structured in such a way as to make anything moving with speed c have an invariant speed with respect to any frame. It's interesting that light has such a property, but not fundamental.

I would say Einstein postulated the invariant speed of light in his 1905 paper first.
It was Minkowski who pointed out how important the geometry of spacetime was.
Einstein himself did not at first seem to think geometrically about spacetime.

 

[Doc Al]

Einstein used the invariant speed of light to deduce how space and time behaved. (It's not just a "trick of light".) That's his huge contribution. True, the full modern view of the geometry of spacetime came later.

 

[rbj]

Don't confuse how we deduce the consequences of relativity from the usual postulates with how we interpret "why" things are the way they are. I'd agree with belliott4488 that it's space and time itself that is structured in such a way as to make anything moving with speed c have an invariant speed with respect to any frame. It's interesting that light has such a property, but not fundamental.

Doc, i think that it is fundamental (perhaps not yet verified experimentally) that any of these fundamental interactions, EM, gravity, weak, strong, all ostensibly "instantaneous", are all believed to have a delayed effect on a distant object when viewed by an observer that is equi-distant from the source of the action and the object affected by the action. it's not that light just happens to propagate at a speed of c. it's that light is EM and EM is one of these fundamental interactions and all of these fundamental interactions have effect that propagate at the same finite speed.

 

[Doc Al]

Doc, i think that it is fundamental (perhaps not yet verified experimentally) that any of these fundamental interactions, EM, gravity, weak, strong, all ostensibly "instantaneous", are all believed to have a delayed effect on a distant object when viewed by an observer that is equi-distant from the source of the action and the object affected by the action. it's not that light just happens to propagate at a speed of c. it's that light is EM and EM is one of these fundamental interactions and all of these fundamental interactions have effect that propagate at the same finite speed.

I agree with you. Light doesn't just "happen" to have a speed equal to the apparent "speed limit" of the universe. Something more fundamental is going on.

I don't think I expressed myself very well before. My point was that relativity itself is more fundamental than just a strange consequence of the behavior of light. (Some folks argue that relativistic effects are just illusions due to the strange nature of light. They are wrong.)

 

[Xeinstein]

Another question I had, as posted in the Quantum group, what happens to non-photon particles that are moving at a speed close to c and are moving against each other? Is the relative speed of the two particle beams (whose sum is > c) capped by c? Relativity theory says yes. But what would be the mechanics behind this phenomenon? And would this be called "invarance of upbound of relative speed"?

Yes, the relative speed of the two particle beams (whose sum is > c) capped by c.
This is a result of time dilation. The faster you travel in space, the slower you travel in time. Nevertheless, the length of 4-velocity of any inertial observer is always c

 

[JesseM]

Another question I had, as posted in the Quantum group, what happens to non-photon particles that are moving at a speed close to c and are moving against each other? Is the relative speed of the two particle beams (whose sum is > c) capped by c? Relativity theory says yes. But what would be the mechanics behind this phenomenon? And would this be called "invarance of upbound of relative speed"?

Two particles moving at .999c and -999c relative to one observer still see a relative speed less than c in their own frames.

Yes, the relative speed of the two particle beams (whose sum is > c) capped by c.
This is a result of time dilation. The faster you travel in space, the slower you travel in time. Nevertheless, the length of 4-velocity of any inertial observer is always c

It's worth distinguishing two types of "relative motion" here. In the frame where both particles are moving in opposite directions, each one is moving at less than c, but the distance between them can increase at a rate greater than 1 light-year per year. But in each particle's own rest frame, using rulers and clocks at rest in that frame to measure the distance covered by the other particle in a given time, the other particle's speed in this frame will be less than c.

 

[rbj]

I agree with you. Light doesn't just "happen" to have a speed equal to the apparent "speed limit" of the universe. Something more fundamental is going on. My point was that relativity itself is more fundamental than just a strange consequence of the behavior of light.

i'm glad we agree. i wish i could say the same about Wikipedia when i try to check a little POV over there (i got to do it anonymously now, since they kicked me out).

anyway, i think that the fundamental reason that there is a "speed limit" is because the fundamental interactions are all "instantaneous" in the same way: some cause changes over here and some effect is notices over there. it's not merely the speed limit of such interaction, it's the speed of propagation of the interaction, and since nothing pushes or interacts with anything else, except by way of these fundamental interactions, how can information or any other causal phenomena propagate any faster?

whether the cause and effect are EM, nuclear, or gravitational, it doesn't matter. for an observer that is equidistant from the thing that is the "causal agent" and the other thing that is affected by it, that observer will count some non-zero time between the perturbor and perturbed. that means that this "c" is finite (and real and positive), not infinite, which is the salient physics. it doesn't matter what that finite speed is, whatever it is, our scaling would adapt to it. indeed the scaling of things in the universe depends directly on c, G, and h (as we measure such quantities with our meter, kilogram, and second) and they could be whatever finite, real, and positive values they choose to be and nothing would be perceived to be different on our part. the tick marks of Nature's ruler, clock, and weighing scale would adjust and the quantitative properties of all of the things in Nature would change proportionally with it (lest some dimensionless parameter change which is something that would make a difference) and things would appear the same to us as before. there really is no operational meaning to any particular values for c, G, and h as long as they are real, finite, and postive.

so it's not just c that is invariant. and, if i understand Einstein's sentiment correctly, Nature had little other choice. i don't know how he would have taken it if the M-M experiment came out differently than it had.

 

[JesseM]

so it's not just c that is invariant. and, if i understand Einstein's sentiment correctly, Nature had little other choice. i don't know how he would have taken it if the M-M experiment came out differently than it had.

Why do you think nature had little other choice? There doesn't seem to be anything inherently inconsistent about the Newtonian universe which allows arbitrarily large velocities.

 

[rbj]

so it's not just c that is invariant. and, if i understand Einstein's sentiment correctly, Nature had little other choice. i don't know how he would have taken it if the M-M experiment came out differently than it had.

Why do you think nature had little other choice? There doesn't seem to be anything inherently inconsistent about the Newtonian universe which allows arbitrarily large velocities.

well, i was trying to reflect Einstein's sentiment. here is the quote of Einstein that leads me to believe that he thought that nature had little other choice:

All I have tried to do in my life is ask a few questions. Could God have created the universe in any other way, or had he no choice? And how would I have made the universe if I had the chance?

when i consider that along with other quotes of Einstein, regarding whom or what Albert is referring to when he uses the word "God"

I believe in Spinoza's God, who reveals Himself in the lawful harmony of the world, not in a God Who concerns Himself with the fate and the doings of mankind...I do not believe in a personal God and I have never denied this but have expressed it clearly.

when i put those two together, i think that it's Nature that Einstein means when he referred to "God" in the first quote above. i think that Einstein thought that a universe where the laws of nature were different for two inertial observers was a universe that did not make sense, could not make sense, and to anthropomorphize, that choice of a universe was simply not in the cards. if every inertial observer must have the same laws of nature, every inertial observer must have the same c. and, with the same c, we all know what the consequences of that would be.

i guess this doesn't answer your question about what is inherently wrong with a "universe which allows arbitrarily large velocities". i don't have a good answer for that other than that would mean that the fundamental interactions would have to have instantaneous effect over any arbitrarily large distance (as Newton or Coulomb had modeled for gravity or electrostatics). there is nothing inherently wrong with it that i am aware of, it's only that the physics is different. if c were infinite, there would be no observed magnetic effects. there would only be electrostatics. don't know if there could even be a Planck Length or a Planck Time. don't know how reality could be.

at the very least, things would be qualitatively different with an infinite c vs. a finite c. but once the physics is determined that c is finite, it doesn't matter what finite value it is. we all would adjust to it. may as well call it "1".

 

[mdeng]

I like the way JesseM presents it.

I think the argument of "the laws of nature were different for two inertial observers was a universe that did not make sense" is strong (as it fits what we observed, but remember Newtonian laws were also observed as fits for hundreds of years). However, RBJ's earlier argument assumed that this was sufficient to derive SR. I am glad RBJ adds back the constancy of C to the postulate. I am very curiously though, why without it Einstein's SR would fall apart even though he already had Maxwell's equation? In other words, what's wrong to replace Einstein's 2nd postulate with Maxwell's conclusion of C (which is a theory, not a postulate, though I am not sure what Maxwell's postulate was)?

I keep thinking about the relation between constancy of C and locality principle. Does anyone have an intuitive (or even better, theoretical) explanation why this principle would imply a speed limit (instead of just *finite* speed, but unbounded) and this this limit has to be constancy? And what impact would quantum physics observation of instantaneity would have on locality principle (at a large scale) and SR (and consequently GR)?

 

[rbj]

I think the argument of "[a universe where] the laws of nature were different for two inertial observers was a universe that did not make sense [to Einstein]" is strong (as it fits what we observed, but remember Newtonian laws were also observed as fits for hundreds of years). However, RBJ's earlier argument assumed that this was sufficient to derive SR. I am glad RBJ adds back the constancy of C to the postulate.

be careful how you represent what other people say (or type). i haven't changed my position. the constancy of c (for various inertial observers) is because of the constancy of the laws of nature (for the same inertial observers).

I am very curiously though, why without it Einstein's SR would fall apart even though he already had Maxwell's equation? In other words, what's wrong to replace Einstein's 2nd postulate with Maxwell's conclusion of C (which is a theory, not a postulate, though I am not sure what Maxwell's postulate was)?

i didn't think that Maxwell concluded that c was constant for different inertial observers. if he did, i would like to know of the record of that. i thought that Maxwell (as well as Faraday) understood that c was in reference with the aether. what Maxwell concluded was that

c = \sqrt{\frac{1}{\epsilon_0 \mu_0}}

for the propagation of electromagnetic waves. and then he figures out that this c that he calculates from the electric and magnetic constants is the very same as the speed of light. then that pretty well nailed down the fact (that was already suspected) that the same visible light we see with, is nothing other than an electromagnetic wave.

 

[mdeng]

be careful how you represent what other people say (or type). i haven't changed my position. the constancy of c (for various inertial observers) is because of the constancy of the laws of nature (for the same inertial observers).

Well, you did add the constancy of C when you mentioned Einstein's view above, which made me think you agreed with him on that postulate. But technically, you did not.

i didn't think that Maxwell concluded that c was constant for different inertial observers. if he did, i would like to know of the record of that. i thought that Maxwell (as well as Faraday) understood that c was in reference with the aether. what Maxwell concluded was that

c = \sqrt{\frac{1}{\epsilon_0 \mu_0}}

What I meant to say is, why did Einstein have to make the 2nd postulate. All he seemed to have to do was to argue that Maxwell's equation must be true for all inertia systems (I think that was your line of arguments earlier). I believe he could not do that. But I don't truly understand why he could not. IOW, why would he be wrong if he did not have his 2nd postulate?

 

[rbj]

What I meant to say is, why did Einstein have to make the 2nd postulate? All he seemed to have to do was to argue that Maxwell's equation must be true for all inertia systems (I think that was your line of arguments earlier). I believe he could not do that. But I don't truly understand why he could not. IOW, why would he be wrong if he did not have his 2nd postulate?

well, you're basically proving my point. i think the 2^nd postulate was there for clarity and was not strictly needed. acceptance of the 1^st postulate forces one to accept the 2^nd. there is no way for the laws of nature to be precisely the same for every inertial observer yet somehow they have different quantitative values for c. with our meter sticks and clocks, the quantitative value of c is part of the laws of nature. now, if reality were different, if the M-M experiment measured a difference in c (in orthogonal directions) at different times of the year, indicating that there might be an aether, then the 1^st postulate of SR could not be proposed without an immediate refutation. if there is an aether, then inertial observers sharing the same frame of reference with the aether would measure the speed of light the same in all directions, whereas someone moving through the aether at a sufficiently fast speed, would measure the speed of light to be slower in the frontward direction than they would measure in the rearward direction. (edit: actually it would have to be frontward vs. sideward directions, since we would have to measure the speed of light in a round trip., frontward and rearward would be the same.)

but making it a 2^nd and explicit postulate helps nail the coffin shut for argument sake, and if anyone complains about it in 1905, one can point to the M-M experiment which did precede it.

i don't know what it is that you believe Einstein could not do. was that extrapolate the 2^nd postulate from the 1^st?

 

[JesseM]

well, you're basically proving my point. i think the 2^nd postulate was there for clarity and was not strictly needed. acceptance of the 1^st postulate forces one to accept the 2^nd. there is no way for the laws of nature to be precisely the same for every inertial observer yet somehow they have different quantitative values for c.

With quantum field theory I think this is true, but back when the only law of nature involving c was Maxwell's laws, couldn't one have argued that Maxwell's laws are not really fundamental, but just a description of the behavior of a certain physical medium filling space (the aether)? After all, no one says that since observers at rest relative to the atmosphere measure sound waves in air to have the same speed in all directions, then this must imply by the first postulate that sound waves in air must have the same speed in all directions in every frame (even in a universe where all of space was filled by such an atmosphere).

 

[rbj]

With quantum field theory I think this is true, but back when the only law of nature involving c was Maxwell's laws, couldn't one have argued that Maxwell's laws are not really fundamental, but just a description of the behavior of a certain physical medium filling space (the aether)?

sure, i s'pose. but the expression of this law or description would be different for different inertial observers. i am not saying that the first postulate naturally follows from nothing. the first postulate is a postulate that one has to sort of axiomatically accept (or accept it on the basis that no experiment could show any difference in how the physical reality was different for different inertial frames of reference). i s'pose that they could say that now, that Maxwell's laws are not really fundamental.

After all, no one says that since observers at rest relative to the atmosphere measure sound waves in air to have the same speed in all directions, then this must imply by the first postulate that sound waves in air must have the same speed in all directions in every frame (even in a universe where all of space was filled by such an atmosphere).

but we have a different experience with sound and air. sound is not as fundamental as a perturbation of a fundamental force like EM, gravity, or nuclear. sound and propagating vibrations in matter requires matter as a medium. there ain't no sound in a vacuum, but there is light (E&M), or gravity waves (if we could only measure them, we better be ready to the next time a supernova that is decently close occurs) in a vacuum. so the basic question is a vacuum devoid of everything, or was this aether left in it, even after we suck all the air molecules out of the jar? so, for sound in air, we are saying that the laws that govern the propagation of sound (i think i can derive the wave equation from continuity, Newton's second law, and the gas law for adiabatic compression) are different for people stationary w.r.t. the air vs. those who are moving through it. we have to do that with wave phenomena that has a medium. it's different when you are moving through a medium than when you aren't.

i don't know even a quarter of the physics you do, Jesse. for me, i am applying epistemology to the physics that i do know (what they teach us Neanderthals in an ABET accredited engineering curriclum). i just do not see how, semantically, the second postulate of SR can be false if the first postulate is true. If the first postulate said that "all laws of physics, with the exception of those that govern E&M, are precisely the same for all inertial observers", then the second postulate would be necessary to go on with SR. but with the broader and simpler expression of the first postulate, i don't see why it would be necessary to add: "just in case you forget, when we say all laws of physics, we mean all laws and every qualitative and quantitative aspect of those laws."

 

[DrGreg]

i think the 2^nd postulate was there for clarity and was not strictly needed.

The important part of the 2nd postulate is that the speed of light does not depend on the speed of the emitter. This might seem "obvious" nowadays, but it was not always so. You could formulate a theory, consistent with the first postulate, in which two light sources moving at different speeds would emit light at different speeds (both speeds relative to a single observer). The 2nd postulate can be paraphrased by saying that it's impossible for any photon to overtake another photon traveling in the same direction. This does not automatically follow from the 1st postulate alone.

Once you've accepted light's independence from the motion of its emitter, the fact that all observers calculate the same numeric speed is a consequence of the 1st postulate. (And you also need to explicitly state that the speed of light is not infinite, otherwise Newtonian theory would satisfy both postulates.)

 

[rbj]

The important part of the 2nd postulate is that the speed of light does not depend on the speed of the emitter.

i remember that (but forgot it).

This might seem "obvious" nowadays, but it was not always so. You could formulate a theory, consistent with the first postulate, in which two light sources moving at different speeds would emit light at different speeds (both speeds relative to a single observer). The 2nd postulate can be paraphrased by saying that it's impossible for any photon to overtake another photon traveling in the same direction.

as if the guy with a flashlight that is whizzing by an observer at 0.9c, that somehow he can give his beam of light a little boost (from the POV of the observer) resulting in a beam of light at 1.9c.

thanks for reminding me of the precise language.

 

[mdeng]

With quantum field theory I think this is true

What are the postulates of QF from which constancy of c is derived?

but back when the only law of nature involving c was Maxwell's laws, couldn't one have argued that Maxwell's laws are not really fundamental, but just a description of the behavior of a certain physical medium filling space (the aether)? After all, no one says that since observers at rest relative to the atmosphere measure sound waves in air to have the same speed in all directions, then this must imply by the first postulate that sound waves in air must have the same speed in all directions in every frame (even in a universe where all of space was filled by such an atmosphere).

Jesse, I am not sure whether your analogy is correct. I believe that Maxwell's law did not say or require the observer must be stationary with respect to the vacuum. So if I factor that into your analogy, I would have said "no one says that since observers at rest or *moving* relative to the atmosphere measure sound waves in air to have the same speed in all directions..." and we know this "since" part is not true.

 

[mdeng]

The important part of the 2nd postulate is that the speed of light does not depend on the speed of the emitter. This might seem "obvious" nowadays, but it was not always so. You could formulate a theory, consistent with the first postulate, in which two light sources moving at different speeds would emit light at different speeds (both speeds relative to a single observer). The 2nd postulate can be paraphrased by saying that it's impossible for any photon to overtake another photon traveling in the same direction. This does not automatically follow from the 1st postulate alone.

Once you've accepted light's independence from the motion of its emitter, the fact that all observers calculate the same numeric speed is a consequence of the 1st postulate. (And you also need to explicitly state that the speed of light is not infinite, otherwise Newtonian theory would satisfy both postulates.)

I am still not clear. Didn't Maxwell's equation say (implicitly, and made explicit by Einstein) that light speed c is regardless whether the observer is flying toward the light or, stated it in another way, the emitter is flying toward us? Therefore, neither overtaking or undertaking ever would happen.

Perhaps Maxwell assumed there was aether and perhaps that was his basis of the equation. However, I believe that his law does not really require this. Perhaps this point can't be proven and thus Einstein just made the 2nd postulate to avoid this sticky issue?

 

[DrGreg]

I am still not clear. Didn't Maxwell's equation say (implicitly, and made explicit by Einstein) that light speed c is regardless whether the observer is flying toward the light or, stated it in another way, the emitter is flying toward us? Therefore, neither overtaking or undertaking ever would happen.

You are right that the second postulate follows from Maxwell's equations. The point is that you don't need the whole of Maxwell's theory to logically develop the theory of relativity; the second postulate (together with the first) is sufficient.

The postulates of any theory are a set of assumptions from which the rest of the theory can be proved without any further assumptions. The development of the theory from the assumptions is a process of logic which doesn't actually depend on any experimental verification.

It's desirable to make the assumptions as simple as possible. You could replace Einstein's second postulate by a postulate that Maxwell's equations are valid in some reference frame (and therefore in all reference frames by the 1st postulate). But Einstein's version is simpler and is all that is necessary for the logic.

If you formulate the 1st postulate as "the laws of physics are the same in every inertial frame", its weakness, from a rigorous mathematical point of view, is that it doesn't actually specify what the laws of physics are. But, ironically, that is its very strength from physical point of view. It is a general framework that you can attempt to apply to any set of physics theories (there aren't really any "laws"); there is no logical requirement that you have to include Maxwell's equations amongst your "laws", so long as nothing breaks the two postulates.

Perhaps Maxwell assumed there was aether and perhaps that was his basis of the equation.

Indeed that was the case when he first formulated them; I believe everything in his equations was measured relative to a postulated aether. However, experiments performed in the years leading up to the formulation of Relativity indicated that Maxwell's equations appeared to be true in moving frames too, which led Lorentz to formulate the Lorentz transformation and later led to Einstein's Special Theory.

The point is, to understand relativity, you don't need to understand Maxwell's equations (partial differential equations which require a moderately advanced knowledge of calculus). It's sufficient to understand the two postulates (no calculus required, until you get to acceleration and gravity).

In essence I'm agreeing with rbj that "2nd postulate was there for clarity and was not strictly needed" provided you accept Maxwell's equations; but if you include the 2nd postulate then you don't need to bring Maxwell into it at all.

 

[DrGreg]

To ressurrect a question implied long ago in this thread (post #24, 5 January):

In calculating this "speed", you're taking the distance traveled as measured in one reference frame, and dividing it by the time as measured in another reference frame. This number is definitely of practical significance to space-travelers, but I think most physicists would resist calling it "speed" because of this mixing of reference frames. Unfortunately, I can't think of a good word to use instead of "speed" here.

This way of measuring motion is called by various authors "proper speed" or "celerity". The celerity of light is infinite. And, for massive objects, momentum = invariant mass * celerity. And celerity = c * sinh(rapidity).

 

[mdeng]

Indeed that was the case when he first formulated them; I believe everything in his equations was measured relative to a postulated aether. However, experiments performed in the years leading up to the formulation of Relativity indicated that Maxwell's equations appeared to be true in moving frames too, which led Lorentz to formulate the Lorentz transformation and later led to Einstein's Special Theory.

Maybe this is it. While Maxwell's equation did not mention motion, we can't just take that as proof that it applies to moving frame as well. The absence of moving does not automatically mean it would hold when motion is involved.

The point is, to understand relativity, you don't need to understand Maxwell's equations (partial differential equations which require a moderately advanced knowledge of calculus). It's sufficient to understand the two postulates (no calculus required, until you get to acceleration and gravity).

In essence I'm agreeing with rbj that "2nd postulate was there for clarity and was not strictly needed" provided you accept Maxwell's equations; but if you include the 2nd postulate then you don't need to bring Maxwell into it at all.

This could be an explanation but I feel it's weak. Einstein himself stated repeatedly that he advocates simplicity. I would speculate that if he could remove a postulate, even if it means more complexity SR must rest upon without it, he would have done that. This is especially so about postulate. We don't make them lightly, they are so fundamental. They should be made not for clarity but for absolute necessity.

 

[rbj]

While Maxwell's equation did not mention motion,

whatever gave you that idea? of course Maxwell's equations (as well as anything describing the magnetic field or force) mention motion.

 

[rbj]

in fact, i knew i mentioned this recently before:\oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 \iint_S \mathbf{J} \cdot \mathrm{d}\mathbf{S} = \mu_0 \iint_S \rho \mathbf{v} \cdot \mathrm{d}\mathbf{S}

d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{(\mathbf{J}\, dV) \times \mathbf{\hat r}}{r^2} = \frac{\mu_0}{4\pi} \frac{(\rho \mathbf{v}\, dV) \times \mathbf{\hat r}}{r^2}

\mathbf{F} = q \cdot(\mathbf{E} + \mathbf{v} \times \mathbf{B})

what do you think they mean by \mathbf{v}? what did they measure it against?\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}

\nabla \cdot \mathbf{B} = 0

\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}

\nabla \times \mathbf{B} = \frac{1}{c^2} \left( \frac{\partial \mathbf{E}} {\partial t} + \frac{\rho}{\epsilon_0} \mathbf{v} \right)more mentions of motion.