Gravity Wave Speed: Deriving Constancy from Maxwell Eqs.

[Narasoma]

We can derive the constancy of the speed of light from Maxwell equations. My questions are: 1. Why it is then need to postulate it when we can obtain it from Maxwell equations?
2. It is stated in many books that gravity wave also propagates with the same speed, c. How do we conclude that? Is there pre-Einstein equations like Maxwell's, from which we can derive the speed of gravity wave?

 

[phyzguy]

For 1., I think Einstein just said, let's take the constancy of the speed of light as given by Maxwell's equations, and postulate that it is a maximum speed which is always true in all reference frames. This allowed him to extend this idea to things like the motion of objects which do not involve electromagnetism.
As for 2. the speed of gravitational waves is easily derived from the Einstein field equations of General Relativity. I don't think a "pre-Einstein" derivation exists. Why are you asking for one? In Newtonian gravity, gravitation propagates at infinite speed.

 

[PeterDonis]

We can derive the constancy of the speed of light from Maxwell equations.

We can derive the constancy of the speed of electromagnetic radiation from Maxwell's Equations. But "the speed of light" as that term is used in relativity is not limited to EM radiation. It is an absolute limiting speed that applies to everything. That much more general application is what is standardly assumed in special relativity.

Is there pre-Einstein equations like Maxwell's, from which we can derive the speed of gravity wave?

No. But we can derive it from the Einstein Field Equation, which is, for this purpose, the analogue of Maxwell's Equations for gravity. The EFE says that gravitational waves propagate at the speed of light, just like EM waves.

 

[Dale]

We can derive the constancy of the speed of light from Maxwell equations. My questions are: 1. Why it is then need to postulate it when we can obtain it from Maxwell equations?

In addition to what was said above, at the time that Einstein formulated his postulates it was widely believed that Maxwell’s equations were not correct, or rather that they were only correct in one reference frame. So the first postulate by itself, which says that the laws of physics are invariant, was not taken at that time to include Maxwell’s equations.

 

[Ibix]

1. Why it is then need to postulate it when we can obtain it from Maxwell equations?

You can make a weaker postulate: there exists a finite speed, $k$, that is the same in all inertial frames. That leads you to the Lorentz transforms, but with an unknown velocity $k$ in place of the usual $c$. From there you can derive Newtonian mechanics as a low speed limit if you assert that $k$ is much greater than any speed you've tested experimentally. And you can also show that Maxwell's equations are consistent if $k=c$.

That was really the point of special relativity, at least in a historical context. It provided a single viewpoint in which Maxwell's equations were valid and Newtonian physics was correct to a very good approximation in the regime where it had been tested.

 

[Narasoma]

Okay. Well noted. Thanks all.

 

[Meir Achuz]

"I think Einstein just said...". Einstein said, "The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.''

"'the speed of light' as that term is used in relativity [by whom] is not limited to EM radiation."
E said, "light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body", referring to the speed of light. The question asked was about "constancy of the speed of light", not anything else.

"was not taken at that time to include Maxwell’s equations." Not taken by whom? What could E have meant by " the laws of electrodynamics"?

 

[Meir Achuz]

"We can derive the constancy of the speed of light from Maxwell equations. My questions are: 1. Why it is then need to postulate it when we can obtain it from Maxwell equations?"
It is not needed. "The laws of Electrodynamics" do include Maxwell's equations (at least by today).
"will be valid for all frames of reference" means that the parameter c in Maxwell's equations equations must be the same in any frame of reference. The "speed of light" (no other speed) can be derived from Maxwell's equations to equal c. That means that "the speed of light" is the same in any frame of reference, and is not needed as a second postulate.

 

[Dale]

You really need to use the quote feature. It is very hard to see what you might or might not be seeking a response to

 

[berkeman]

You really need to use the quote feature. It is very hard to see what you might or might not be seeking a response to

I see what you did there...

 

[Meir Achuz]

For 1., I think Einstein just said

Einstein said, "The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.'

the speed of light" as that term is used in relativity is not limited to EM radiation.

E said, "light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body", referring to the speed of light. The question asked was about "constancy of the speed of light", not anything else.

it was widely believed that Maxwell’s equations were not correct,

Not believed by whom? What could E have meant by " the laws of electrodynamics"?

 

[PeterDonis]

The question asked was about "constancy of the speed of light", not anything else.

This is obviously false based on the plain language of the OP. The OP asked why relativity needs to "postulate" anything about "the speed of light", and also asked about gravitational radiation. My response that you quoted was aimed precisely at those concerns.

Not believed by whom?

Most physicists at the time believed that Maxwell's Equations were only precisely valid in one particular inertial frame (the "ether rest frame" for those who favored the ether hypothesis), and would have to be modified for frames moving with high enough velocity relative to that frame, and that Newton's laws were the ones that were truly frame invariant. Or, to put it another way, they believed that the correct laws of physics would be Galilean invariant, which Newton's laws were, but Maxwell's Equations were not.

Einstein reversed that: he proposed that the correct laws of physics should be Lorentz invariant, not Galilean invariant, so Maxwell's Equations would be valid in all inertial frames, and Newton's laws were the ones that would have to be modified.

 

[Dale]

Not believed by whom?

By the scientific community as a whole. The prevailing opinion of the time is that there was a luminiferous aether and that Maxwell’s equations only held in the rest frame of that aether. In other frames Maxwell’s equations were generally assumed to not hold exactly. The Michelson Morley experiment was designed to confirm this extant belief.

 

[Sagittarius A-Star]

Not believed by whom?

Maxwell and Special Relativity
...
Maxwell does not appear to have crisply drawn the above conclusion, that the speed of light is independent of the velocity of the observer, but he did make arguments in Arts. 599-600 and 770 of [55] that correspond to the low-velocity approximation to special relativity, as pointed out in sec. 5 of [86]

Source:
http://kirkmcd.princeton.edu/examples/maxwell_rel.pdf

via:
http://kirkmcd.princeton.edu/examples/

 

[vanhees71]

As far as I remember, in the late 19th century and the beginning 20th century, there was indeed some doubt about the validity of various theories about electromagnetism. Naturally in England there were many proponents of Maxwell's theory, while on the continent there were still strong proponents of other theories, based on an action-at-a-distance paradigm (particularly a theory by Weber). Among others it was Helmholtz, who initiated a research program to empirically figure out, which theory was right, triggering particularly the discovery of the electromagnetic waves by H. Hertz, which I think finally convinced a majority of physicists about the validity of Maxwell's equations.

Now indeed the problem with Maxwell's equations was that they cannot be made in any way Galilei covariant, i.e., one cannot find transformation laws for the fields and charge and current densities under Galilei transformations that keep the Maxwell equations forminvariant when changing from one inertial frame to another. The interpretation was that finally a means to determine Newton's absolute space and absolute time was possible using electromagnetic phenomena, and this preferred inertial reference frame was identified as the rest frame of the aether, which was thought to be a substance whose vibrations are the electromagnetic waves (light) like sound waves are the vibrations of air. This lead to theories describing how the Maxwell equations have to be modified when described in an inertial frame which is not this preferred aether rest frame (among others one famous was by Hertz).

The experimental challenge was that the corresponding phenomenology worked out nicely at the first glance, and it was pretty soon clear that one needed experiments sensitive at the order $v^2/c^2$, where $v$ is the velocity of an inertial frame relative to the aether restframe. Famously the Michelson Morley experiment was such an experiment, and the null result was very unexpected, but still most physicists tried to stick with the aether model and adapting it with additional "mechanisms" like FitzGerald and Lorentz who introduced the idea of "FitzGerald-Lorentz contraction", according to which rigid bodies contract in the direction of their velocity wrt. the aether rest frame.

That explains, why Einstein titled his famous paper "On the Electrodynamics of Moving Bodies". Amazingly Einstein didn't argue with the Michelson-Morley experiment, which is not even explicitly mentioned in the paper, but with a symmetry argument, i.e., the interpretation of the induction of a current in a loop moving in the field of a permanent magnet vs. the same situation in a reference frame, where the loop is at rest and the magnet moving. Einstein just states that the asymmetry stated by the usual interpretations of Maxwell's equations are simply not there and the phenomena should indeed not depend on the motion wrt. some preferred reference frame or an aether-rest frame but just on the relative motion between the loop and the magnet. He also mentioned "unsuccessful attempts to discover any motion of the Earth relatively to the “light medium"" without specifying which he had in mind.

In any case Einstein's breakthrough was to assume the validity of the 1st Newtonian Law (special principle of relativity) and make it compatible with Maxwell's equations. Amazingly in the first part of the paper he just uses the most simple implication of this assumption, i.e., that the speed of light must be independent of the velocity of the light source wrt. any inertial reference frame, and from this he could derive the Lorentz transformations for space and time, i.e., the substitute for the Galilei transformations between inertial reference frames. Then he also gave the transformation rules for the electromagnetic field etc. The point was that one had not preferred inertial frame, which could be identified with Newtons absolute space and time or the rest frame of some aether, but that the laws of mechanics had to be adopted to the new spacetime model with the Lorentz (Poincare) transformations being the correct symmetry transformations rather than the Galilei transformations.

 

[Meir Achuz]

The prevailing opinion of the time is that there was a luminiferous aether and that Maxwell’s equations only held in the rest frame of that aether

We are in agreement on that. That is why I was surprised at his sentence that included, "it was widely believed that Maxwell’s equations were not correct,". The "or rather" did mitigate that, but I was responding to the "not correct part".

 

[Meir Achuz]

the laws of physics are invariant, was not taken at that time to include Maxwell’s equations.

Are you saying that Einstein did not include Maxwell's equations in 'the laws of physics'?

 

[PeterDonis]

Are you saying that Einstein did not include Maxwell's equations in 'the laws of physics'?

Einstein did--more precisely, he included them in what he believed to be the correct laws of physics, valid in every inertial frame. Other physicists did not: they thought Newton's laws were valid in every inertial frame, but not Maxwell's Equations. But, as has already been pointed out, Einstein did not think Newton's laws, at least not as they were then formulated, were valid in every inertial frame. So he did not include Newton's laws in "the laws of physics".

 

[DrGreg]

Are you saying that Einstein did not include Maxwell's equations in 'the laws of physics'?

I think, maybe, that when Einstein referred to "the laws of physics" he meant "the true laws of physics, whatever they turn out to be" rather than "what we humans have currently formulated the laws of physics to be".

 

[PeterDonis]

I think, maybe, that when Einstein referred to "the laws of physics" he meant "the true laws of physics, whatever they turn out to be" rather than "what we humans have currently formulated the laws of physics to be".

Einstein may have, but it should be made clear that the usage of the term "the laws of physics" that the poster you responded to has been asking about is not from Einstein. It's from other posts in this thread, which are talking about what other physicists besides Einstein believed around the time when Einstein published his 1905 papers.

 

[Meir Achuz]

Einstein meant "The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good."

 

[Meir Achuz]

My questions are: 1. Why it is then need to postulate it when we can obtain it from Maxwell equations?

Does anyone else want to answer this question?

 

[vanhees71]

I thought I answered this from the historical perspective in #15.

It was just an ingenious way to get at the resolution of the problem that Maxwell's equations are not Galilei invariant, which apparently contradicts the postulate that Maxwell's equations should hold in any inertial reference frame.

The resolution must be that the description of space and time must be changed in such a way that the transformation from one inertial frame of reference to the other is such that the Maxwell equations indeed hold in any inertial reference frame.

The conclusion is that all of physics has to be adapted to the new spacetime description, and thus one should derive how the time and the position vector have to be transformed from one inertial frame to another moving with a given relative velocity against this inertial frame. For this you don't need to consider the Maxwell equations but just the one thing that is related to purely kinematic properties of space and time, and that's the speed of light, i.e., if Maxwell's equations should hold in any inertial frame, and you consider a situation where in one such frame a light source is at rest and the Maxwell equations describing the light from this source are the same in another inertial frame, where this light source moves with constant velocity, the speed of light must be independent of the velocity of the light source. This enabled Einstein to first think about the simpler problem of how should the time and space variables transform when going from one inertial frame to another.

Einstein put this much less formal in his paper, i.e., from a more instrumental point of view about how space and time measurements must be defined (!) given the "constancy of the speed of light", and he came up with the analysis described in his paper, leading to the Lorentz transformation for the time and space variables, given the corresponding synchronization convention (!) for clocks at rest within any fixed inertial reference frame.

This is of course not sufficient to resolve the issue with the apparent violation of the special principle of relativity by the Maxwell equations, because now given the "new" transformations between space and time coordinates of different inertial reference frame, you must also find out, how to transform the electric and magnetic fields as well as the charge and current densities, which he also derives in his 1905 paper, and this shows that indeed within the spacetime description derived just from the special principle of relativity and the postulate of the "constancy of the speed of light" is compatible with the form invariance of Maxwell's equations when using the appropriate transformation rules for the fields and charge-current distributions.

 

[Nugatory]

Does anyone else want to answer this question [why the constancy postulate is needed]?

It really comes down to the historical context. In 1905 the argument that Maxwell's equations plus the principle of relativity implied light-speed invariance was not compelling, in large part because no one was considering that the principle of relativity did not necessarily mean Galilean relativity. Thus Einstein framed his argument as "suppose light-speed is invariant - where does that take us?"; we could place our tongues solidly in our cheeks and restate the second postulate as "and I really mean the first postulate, even for electromagnetism".

It is now the twenty-first century and we are far more willing to abandon Galilean relativity, so the second postulate feels unnecessary. (Of course we still need the isotropy assumption, but that one is sufficiently generally accepted that it often goes unmentioned).

 

[vanhees71]

It's not unnecessary, because from the 1st postulate alone + the usual symmetry arguments you get, of course, both Galilei-Newton as well as Einstein-Minkowski spacetime with the corresponding symmetry groups being the Galilei and the Poincare group, respectively. Minkowski spacetime implies the existence of a "limiting speed", which distinguishes it from Galilei-Newton spacetime which doesn't have such a thing. Empirically the "limiting speed" is with very high accuracy the phase velocity of electromagnetic waves in vacuo.

 

[Dale]

Are you saying that Einstein did not include Maxwell's equations in 'the laws of physics'?

Yes. I believe that Einstein believed that Maxwell’s were one of laws of physics as written. But he was writing to his peers and deliberately chose the two separate postulates with that meaning in order to have an acceptable foundation for them. So I believe that Einstein’s intended meaning was to not include Maxwell’s equations (a priori) as a law of physics.

 

[Dale]

I think, maybe, that when Einstein referred to "the laws of physics" he meant "the true laws of physics, whatever they turn out to be" rather than "what we humans have currently formulated the laws of physics to be".

Yes, this is my take too. It is a statement restricting possible laws of physics rather than a statement regarding any specific laws

 

[Meir Achuz]

I thought I answered this from the historical perspective in #15.

Why is the second postulate needed today?

 

[Meir Achuz]

Dale, Sorry, my thumbs up was due to a faulty mouse. I think Einstein meant "The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good."

 

[PeterDonis]

I think Einstein meant "The same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good."

Yes, he did mean that. But he also meant that "the equations of mechanics" would have to be modified from the then current formulation of Newton's laws, so that the equations of mechanics were Lorentz invariant instead of Galilean invariant.

Other physicists would have agreed with Einstein's statement as you quoted it, but would not have agreed that the equations of mechanics should be Lorentz invariant; they thought instead that the correct "laws of electrodynamics and optics" would be Galilean invariant, and therefore that Maxwell's Equations were not the correct "laws of electrodynamics and optics", but only approximations to those laws valid for small enough speeds relative to the ether rest frame.

 

[Meir Achuz]

he also meant that "the equations of mechanics" would have to be modified

Why didn't he say what "he meant"?

 

[Sagittarius A-Star]

Why is the second postulate needed today?

I think the opposite direction would be nicer. Derive Minkowski spacetime from postulate 1 (the laws of physics are the same in all inertial frames) and postulate 2 (there exists an invariant speed) + assumed linearity. Then derive a theory of electromagnetism from assumed Coulomb’s law +SR.

There exist approaches in this direction (I did not read it because of the paywall):

ABSTRACT
Maxwell’s equations are obtained by generalizing the laws of electrostatics, which follow from Coulomb’s law and the principle of superposition, so that they are consistent with special relativity.

Source:
https://aapt.scitation.org/doi/abs/10.1119/1.14521

 

[PeterDonis]

Why didn't he say what "he meant"?

If "he" means "Einstein", he did. That's the entire point of his 1905 paper "On the Electrodynamics of Moving Bodies", which is where the phrase you quote appears: to show how "the equations of mechanics" must transform in order to be Lorentz invariant, and what the implications of this are.

 

[vanhees71]

Why is the second postulate needed today?

With the 1st postulate alone and the other symmetry assumptions you get Galilei-Newton spacetime or Einstein-Minkowski spacetime. To decide, which one is the correct (of rather the better) description of the spacetime structure you need empirical input. That empirical input at the time when Einstein wrote his paper were Maxwell's equations, summarizing all then known phenomena concerning electromagnetism including its application to optics, and from a purely kinematic point of view the one conclusion from the assumption that Maxwell's equations must look the same in any inertial reference frame is the "constancy of the speed of light", i.e., the independence of the phase velocity of electromagnetic waves in a vacuum of the velocity of the light source wrt. any inertial reference frame.

 

[vanhees71]

Why didn't he say what "he meant"?

But he did! In his 1905 paper he gave the modified equations of motion for point particles, though in a very deformed way from the modern point of view, introducing various kinds of "relativistic mass". This was clarified already in 1906 by Planck and finally by Minkowski in 1908, revealing the mathematical structure of Einstein's special-relativistic spacetime model.