Was Einstein's postulate for the speed of light a consequence of Maxwell's equations?

Maybe it should be stated that the term "aether frame" wasn't well defined at this time period:

1) Fresnel in 1818 proposed an aether almost completely at rest, as necessary to explain the aberration of light. Here, light is a transverse wave in the aether. However, he had do include "partial aether dragging" to explain the negative outcome of the Arago experiment. So the speed of light is constant in vacuum, but variable within matter in accordance with Fresnel's dragging coefficient (later confirmed in the Fizeau experiment in 1851).

2) Stokes (1844) proposed that the aether is completely dragged by Earth. He had to invent certain auxiliary hypothesis to derive Fresnel's dragging coefficient and the aberration of light (which wasn't very convincing).

3) Maxwell (1865) derived his equations (actually twenty of them). But he made no comment on the state of motion of the luminiferous aether in matter or its vicinity. Later (shortly before his death) he argued that if there is an aether wind, then it is of second order in v/c when two-way measurements were made. (See Maxwell's own Ether entry in the encyclopedia britannica, section "Relative motion of the aether", 1878).
Encyclopædia Britannica, Ninth Edition/Ether.

4) Michelson (1881) and Michelson&Morly (1887) followed Maxwell's suggestion and executed their famous second order experiment - no aether wind was found. A contradicting situation arose: The Fizeau experiment and the aberration of light "proved" an almost resting aether, while the Michelson-Morley experiment seemed to imply complete aether dragging.

5) Heaviside and Hertz (1890) brought Maxwell's (now four) equations into their modern form - but Hertz believed in complete aether dragging (even though Hertz knew that this was at variance with the Fizeau experiment).

6) Lorentz was the first (1892 and 1895) to argue that the aether is completely motionless and thus totally unaffected by the motion of matter. Now we can talk about one (and only one) "aether frame", in which the speed of light is the same in all directions. To avoid "aether wind" effects, he (together with Larmor and Poincaré) had to include "local time" and "length contraction", and later the complete Lorentz transformation (1904).

7) Einstein used the Maxwell-Lorentz theory of 1895, in which the speed of light was constant in only one frame. Per the relativity principle, he argued that this constancy and Maxwell's equations must apply to all reference frames, none of them should be called "aether" any more. Special relativity was born.
Very nice summary!

[..] But how exactly were the time/space dilation effects proved so as to keep electromagnetism consistent with relativity? I have seen the weird-looking formulas involving square roots of 1 - (v/c)^2 but I've never understood how they came up with precisely this expression...
It cannot be proved in a strong sense; instead a theory is a creative solution or invention that is meant to explain observations. However, in this case there is not much room for alternative solutions, so to that extent it can be considered "proved".
Do you understand why according to Lorentz and Einstein a "moving" interferometer such as that of Michelson and Morley must "be" contracted in length by that factor? If not, that's a good point to start with. Next there is a similar example with a "light clock". You can "google" for those - and you may find hits of this forum.
Also, how do you make something "invariant" with respect to something else?
I mean, answers mention a lot the "invariance" of the equations, but I don't really get what it means. How did they translate the fact that the equations should work in all reference frames mathematically?
That's exactly what is meant with "invariant"! The speed of a light ray or wave is invariant in the sense that it is "observed" as going at c by means of any inertial reference frame that has been set up in a certain way. And to understand how it works, it may be good to start with the things that I mentioned just here above.

Ibix

Also, how do you make something "invariant" with respect to something else?
I mean, answers mention a lot the "invariance" of the equations, but I don't really get what it means. How did they translate the fact that the equations should work in all reference frames mathematically?
Following Newton:

You are standing in a stationary train which is on a track parallel to the x axis and I am on the platform outside. We are each holding a ball, and we drop them simultaneously at time t=0. The initial positions of the balls are $(x_0,y_0)$ (there is a z-coordinate perpendicular to the track which would have different values, but I am ignoring here because it doesn't matter). After time t, the positions are both $(x_0,y_0-gt^2/2)$.

Now, we repeat the experiment but with your train passing through the station at speed v in the x direction. We drop the balls at time t=0, as we pass one another. My ball still follows the same path: $(x_0,y_0-gt^2/2)$. But your ball obviously won't. The principle of relativity states, though, that you are entitled to consider yourself at rest, so you must see the same path you did when you were at rest. If you are to see $(x_0,y_0-gt^2/2)$, but from my perspective you are moving at speed v in the x direction then the path I see must be $(x_0+vt,y_0-gt^2/2)$. You can make the exact same argument (except that to you, I am moving at -v): your ball follows $(x_0,y_0-gt^2/2)$; I am entitled to believe I am stationary so my ball must (to you) follow $(x_0-vt,y_0-gt^2/2)$ so that I can continue to believe that.

That is invariance; the equations are the same for you and for me. Nothing different happens just because we are moving with respect to one another. The maths I use to describe me are the same you use to describe you; the maths I use to describe you are the same as the maths you use to describe me. I leveraged that observation to derive something useful: if I want to know the position of a ball according to somebody else, all I need to do is take my observations and add the current position ($(vt,0)$ in the example above) of the other person.

What we are discussing on this thread is a fifty-year investigation that lead to the conclusion that our watches won't agree (although, again, my complaint about your watch will be the same as your complaint about mine). Since I have assumed above that we will always agree on the t coordinate, what I have written is subtly wrong. This would be obvious to everybody if we typically moved at 10% of lightspeed, or thereabouts, but we move a lot slower than that so we never notice. Maxwell's equations, describing things that do move at light speed, noticed.

Does that kind of make sense?

Yes this makes perfect sense. But then what bothers me is that when the speed of the train approaches c, then you can no longer add v*t in the equation, and I don't know how to reason it out.

Say you are in a spaceship moving with speed v. Next to you comes a ray of light in vacuum with speed c. Now normal intuition would suggest that you will see the ray racing against you at speed c - v. Relativity theory says this is impossible.

So somehow, we need to change another parameter so that you end up seeing light moving with speed c no matter how fast you move. Suppose you and the light ray started at x = 0 in Jupiter. On the clock in your spaceship, you measure t seconds, and find that you are at position v*t next to planet Mars, and light at position x, next to Earth. Then the speed of light as measured by you is c = (x - vt)/t.

From an observer at rest in Jupiter, when you reach Mars, light is at position c*t'. (t' is another time parameter). Now I think that when you reach Mars. both you and the observer see the light ray next to Earth, at position x, am I right? So I can say that x = c*t'.
So c = (c*t' - vt)/t = c*(t'/t) - v so t' = t * (v+c)/c = t*(1+v/c).

This would mean that the observer at rest has lived 1 + v/c times longer than you (from your perspective) when you reached Mars (time dilation). But this is not the true factor which is used in relativity theory, so I get confused even more...

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But how exactly were the time/space dilation effects proved so as to keep electromagnetism consistent with relativity? I have seen the weird-looking formulas involving square roots of 1 - (v/c)^2 but I've never understood how they came up with precisely this expression...
See the attached excerpt from the Feynman Lectures on Physics, specifically the last two pages.

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Actually now that I think about it, its the other way around. I'm confused, now, because since no matter at what speed you move, you obviously will measure the same electric and magnetic constants, and so light ought to have the same speed. But as you move faster, your speed is added to that of the light and so you should see light moving faster. Since we know this is false, it would follow that the e/m constants are not, in fact, constants???
Can someone clarify that to me?
If Newton's Laws as written in Principia were precisely true, applying to both ordinary matter and this hypothetical ether scientists believed in, the electric and magnetic constants could be constant only for an observer in the inertial frame where the ether is stationary. There would be a one specific frame where Maxwell's equations could be written in the simplest form. It didn't matter whether or not there really is an ether. By the logical argument that you just gave, Maxwell's equations could be written in a simple form in only one frame. I will call this hypothetical unique frame the "ether frame", even though there may be no ether. If Newton's Laws are precisely valid, the constants are not invariant because the change for observers other than the ether frame.
This also assumes that the measuring instruments are unaffected by the motion relative to the ether frame. This si consistent with Principia. However, Maxwell's equations may not be consistent with Principia. More on that later.
Maxwell's equations would not be valid as currently written in any other frame except one. An experimenter moving at a nonzero constant velocity with respect to that one frame would find different electric and magnetic constants from those of that one "proper frame". In fact, he would probably have to replace those constant numbers by constant matrices.
There was no experiment that showed this wouldn't be true. However, what bothered Einstein is the idea that the "constants" would be different for two observers moving at a constant velocity with respect to each other.
To visualize the problem, imaging that you moving at the speed of light with reference to the "ether frame". Suppose you hold up a mirror that relative to your eyes is in the direction of motion. You couldn't see your reflection! You also probably drop dead since your body has evolved for the electromagnetic constants being what they are on earth.
So this bothered Einstein. However, there is another reason to doubt the Principia. The reason is that the forces that hold bodies together are largely electromagnetic. Therefore, their dimensions and even their movements have to be affected by this speed relative to the ether!
Imagine that extended bodies are held together by a combination of electromagnetic and nonelectromagnetic forces. Our bodies contain a number of electrically charged particles. Using the laws of electromagnetic forces, it is easy to see that when in motion one particle will apply a force to the other particle. You can use the Law of Biot and Savart, the Lortentz force law, and anything else you want. Motion with respect to the ether frame will cause a stress on the extent body that will affect both its dimensions and its motion. For instance, an electrical coil moving with respect to the ether frame should experience a torque. Furthermore, a material that is not birefringen should show birefringence if it is in motion relative to the ether frame.
Efforts were made to measure this stress. There were two scientists, Raleight and Brace, that tried to measure the force on an electrical coil caused by the earths motion. This was an experiment not as well known as the Michaelson-Morley experiment. They had a null result that was just as important.
Experimental results indicated that Principia couldn't hold for electrical measurements.The motion of an electrical coil doesn't cause a torque on itself! A nonbirefringent material is nonbirefringent at all speeds!
I got the references to this from a paper by Lorentz. However, I have never found a copy of these experiments. They are just as important as the Michaelson Morley experiment. Yet, I can't find a copy. Maybe you will have better luck.
Null result on measuring the birefringence of an isotropic material in motion.
1) Rayleigh, Phil. Mag. 6(4), 678 (1902).
2) Brace, Phil. Mag. 6(7), 317 (1904)
Null result on measuring the torque on an electrical coil:
3) Troughton and Noble, Roy. Soc. Trans. A 202, 165 (1903).
Experiments 1-3 showed clearly that Principia was faulty.
The problem comes about because measuring instruments are extended bodies! Suppose and observer calibrates his measuring instruments, both rulers and clocks, in the ether frame. Some other observer moving at a constant velocity relative to the first observer grabs those measuring instruments, and accelerates them to his velocity. The internal forces will distort those measuring instruments!
He may try to calibrate them by a series of timing experiments. However, most timing experiments require synchronization of clocks using electrically charged particles. Thus, it appears impossible to calibrate measuring instruments that contain electrically charged particles or electromagnetic radiation.This includes every measuring instrument ever invented, then and now! The presence of nonelectromagnetic forces doesn't change things, because some distortion remains as long as any electromagnetic force is used.
Furthermore, there is a contradiction if both Principia and Maxwell's equations are used. The third law of motion is written in present tense. So Principia implies that the action and the reaction have to occur simultaneously. Maxwell's equations implies a delay in electromagnetic forces consistent with the speed of light. So they both can't be true under all conditions. Either Principia or Maxwell's equations or both are a little bit wrong.
H. A. Lorentz did a complete analysis on how the electromagnetic stress of motion would effect rulers and clocks. It turned out that the rulers and clocks would behave just as Einstein described in special relativity.
However, I will just write down the names of these articles where H. A. Lorentz put down his theory of electrons.
4) "Electromagnetic phenomena in a system moving with any velocity smaller than that of light," by Hendrik Lorentz. Proceedings of the Royal Netherlands Academy of Arts and Sciences, 809-831 (1904).
5) "The Theory of Electrons" by Hendrik Lorentz (1915).

There are copies of 4 and 5 free on the Internet. Someone on this forum listed them.
Lorentz only varied from Principia in one regard. He assumed that the forces holding things together have a minimum delay given by the speed of light. It is this delay that makes the measuring instruments act in a "relativistic" way.

In any case, your conundrum is resolved when you consider the stress in a measuring instrument caused by the motion of the measuring instrument. The atomic components of clocks and rulers interact with each other to distort the measurements in a way consistent with relativity.
If all forces are Lorentz invariant, then the changes in the dynamics of the particles that comprise a measuring instrument causes the measurements to determine the same electromagnetic constants in any frame. Furthermore, the speed of light delay in the forces makes it impossible to determine which of the frames is an ether frame.

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Ibix

'Simultaineous' is a hairy concept in relativity. No, the Jupiter and ship observers do not agree that light reaches Earth at the same time that the ship reaches Mars. That's why your maths is wrong.

The usual way to show this is a train of length 2L passing through a station. You are on the train in the exact middle; I am on the platform again. At the instant we pass, lightning strikes the front and back of the train. I see the light from that at time L/c and am able to calculate backwards and determine that the flashes occurred simultaneously. I can also see that you, moving at speed v, saw the flash from the front at time L/(v+c) and the one from the back at time L/(c-v). Not simultaneous. The same thing would occur with the sound, of course, but the special thing about light is that c is constant in all frames. You may consider yourself at rest. Since the flashes occurred equal distances away but you did not see them at the same time, they must not have been simultaneous in your frame.

I am not sure off the top of my head if you can derive the Lorentz transforms directly from this argument - possibly. I also cannot view Lugita's attachment on this phone, but I guess that will explain it fully.

Yes this makes perfect sense. But then what bothers me is that when the speed of the train approaches c, then you can no longer add v*t in the equation, and I don't know how to reason it out.
You can (and must!) if you use the reference frame in which the train is moving; there is nothing to "reason out".
Say you are in a spaceship moving with speed v. Next to you comes a ray of light in vacuum with speed c. Now normal intuition would suggest that you will see the ray racing against you at speed c - v. Relativity theory says this is impossible.
That's a common confusion which stems from the fact that in Newtonian mechanics it's harmless to confound a switch of reference system (= a transformation) with a difference of velocities (= a subtraction). This is because the so-called "Galilean transformation" is numerically identical to calculating a velocity difference. In SR that is not the case and so you have to distinguish those.

Thus we have to rephrase your question (which immediately answers it!) as either:

Say you are in a spaceship moving with speed v relative to Earth, but consider yourself in rest - you set up a "ship reference system" S'. Next to you comes a ray of light in vacuum with speed c according to S'. Now normal intuition would suggest that you will see the ray racing against you at speed c, as Relativity theory also says.

or:

Say you are in a spaceship moving with speed v according to reference system S. Next to you comes a ray of light in vacuum with speed c according to S. Now normal intuition would suggest that you will see the ray racing against you at speed c - v. Relativity theory says this is what you will "observe" if you stick to using reference system S.

PS the topic "constant speed of light" is in discussion in a parallel thread, see my last posting there:

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well this gets pretty hard, I guess... I suppose I'll have to study more electricity and special relativity to understand all that stuff better :P For the moment it's beyond my head, without a solid background.

Thanks for the replies, though! They are very informative.

But then Poincare demonstrated that the Lorentz transformations relate not only the aether frame to a moving frame, but also related moving frames to each other. (Specifically, he showed that the composition of LT's is an LT, and the inverse of an LT is an LT.)
So, as a newbie, is this saying Lorenz transforms form a group? With the operator being Lorenz composition? The corollary would be any Lorenz transform must have c as the speed limit.

"It is, in fact, possible to derive the Lorentz transformations from the principle of relativity alone and obtain the constancy of the speed of light as a consequence."
http://en.wikipedia.org/wiki/Principle_of_relativity

This is interesting. Can somebody point me to a proof?

bcrowell
Staff Emeritus
Gold Member

"It is, in fact, possible to derive the Lorentz transformations from the principle of relativity alone and obtain the constancy of the speed of light as a consequence."
http://en.wikipedia.org/wiki/Principle_of_relativity

This is interesting. Can somebody point me to a proof?
The statement in WP is footnoted to a 2004 paper by Friedman, which I don't have access to. But I think they are probably referring to an argument that, in various forms, dates back to 1911:

W.v.Ignatowsky, Phys. Zeits. 11 (1911) 972
Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51
Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008, Appendix I
Palash B. Pal, "Nothing but Relativity," http://arxiv.org/abs/physics/0302045v1
http://www.lightandmatter.com/html_books/0sn/ch07/ch07.html [Broken] (my own presentation)

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OK I can see that there has to be a universal speed limit, but why does this have to be the speed of light (or other electromagnetic waves)? What implies that light must travel at a constant speed anyway (waves in water can have various speeds)?

bcrowell
Staff Emeritus
Gold Member

OK I can see that there has to be a universal speed limit, but why does this have to be the speed of light (or other electromagnetic waves)? What implies that light must travel at a constant speed anyway (waves in water can have various speeds)?
Water waves have a speed relative to the water. The speed of light in a vacuum can't be relative to anything, because there isn't any medium for it to be relative to.

"It is, in fact, possible to derive the Lorentz transformations from the principle of relativity alone and obtain the constancy of the speed of light as a consequence."
http://en.wikipedia.org/wiki/Principle_of_relativity

This is interesting. Can somebody point me to a proof?
Instead I can point to a counter claim in that same encyclopedia:
http://en.wikipedia.org/wiki/Histor...rentz_transformation_without_second_postulate
In fact, that should be obvious: Classical mechanics has the PoR but with the Galilean transformations.

OK I can see that there has to be a universal speed limit, but why does this have to be the speed of light (or other electromagnetic waves)? What implies that light must travel at a constant speed anyway (waves in water can have various speeds)?
That was based on observation combined with Maxwell's theory which models light as a wave with constant speed, similar to the speed of sound in a homogeneous medium.

Water waves have a speed relative to the water. The speed of light in a vacuum can't be relative to anything, because there isn't any medium for it to be relative to.
Instead I can point to a counter claim in that same encyclopedia:
http://en.wikipedia.org/wiki/Histor...rentz_transformation_without_second_postulate
In fact, that should be obvious: Classical mechanics has the PoR but with the Galilean transformations.

So the implication here is that if we have 'something' that doesnt travel relative to anything, this 'something' has to travel at the 'universal speed limit' (this is probably provable). Light in a vacuum is an example of this 'something'; are there any others (gravitons?)

Also the speed of light in a vacuum is an axiom, rather than being able to be derivable; even Pauli thought so.

So the implication here is that if we have 'something' that doesnt travel relative to anything [..]
Instead, and sticking with the topic, the model that is used is that of Maxwell, according to which light propagates at speed c relative to a "stationary" frame, or "space":

"light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body"
"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."

- Einstein 1905

"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.
- Einstein 1907

Einstein explained it as follows in 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 [1895], which is based on the assumption of an ether that is absolutely at rest, through experiment". [footnote refers to Fizeau's experiment]

And with the PoR this model can be used for any inertial frame:
"the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest"
- Einstein 1905

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bcrowell
Staff Emeritus
Gold Member

Also the speed of light in a vacuum is an axiom, rather than being able to be derivable; even Pauli thought so.
Whether it's a postulate or a theorem depends on what system of axioms you pick. We have a FAQ about this: https://www.physicsforums.com/showthread.php?t=534862 [Broken]

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