# Do Maxwell's equations independently impose constraints on the speed of light?

My question is about the relations and equations that makes us to impose constraints on the velocity at which electromagnetic waves propagate.

• Do Maxwell's equations independently impose constraints on the speed of EM waves?

• Are these equations compatible with the two special relativity principles with no need to consider some constraints?

• Does exceeding the limitation for speed of light violate the implications of Maxwell's equations?

• Does considering unequal constant values of velocity of light for different inertial references violate what Maxwell's equations imply?

• Who imposed such a limitation theoretically at first? What motivated him/her to suppose there is a limitation for the group velocity of electromagnetic waves?

We know Lorentz transformations are constructed on the assumption of constant speed of light in moving frames. What made him consider such an assumption, if Einstein was not the first one to consider the second postulate of special relativity (i.e. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light)?

Einstein assumed "constancy of the speed of light for all observers of all moving frames" to derive and use his "Lorentz transformation" like transformation! Then he constructed his special theory of relativity based on two principles which we all heard about. Did I get this right?

Considering the transformation for a moving frame along the $x$ axis for a frame moving at the speed of $v$ you get $${x_2} = {{{x_1} - {v_1}{t_1}} \over {\sqrt {1 - {{({v \over c})}^2}} }}.$$ This transformation mathematically implies that no frame is allowed to move at a speed higher than $c$. So this assumption puts some constraints on the speed of any moving frame too!

Summing all these up, say:

A) "the maximum speed of light has an upper bound which is called $c$"

B) "nothing travels faster than light"

C) "the speed of light is measured to be the same by all observers"

From those, some questions arise:

1. Considering the definition of a frame and the observer which could be at a quite arbitrary conditions of velocity and so on, what motivated him to accept such a limitation for the speed of frames? I mean such a precise assumption can't come from nowhere! Particularly when its consequences seems unbelievable!

2. Why light? How was he so sure that nothing else could be faster than light? Was there any evidence of light being the fastest thing ever exists?

3. Where does the role of Maxwell's equations rise up in the creation story of this assumption?

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I meant the bunch of them, individually. Independent from any other assumption or relation like Lorentz transformation or any other theorem.I mean if Maxwell's contributions on Electromagnetism can lead us to derive such a constraint ? And you are right, "independently" is a better choice. Thank you! –  2physics Sep 17 '13 at 20:51

• The speed of EM waves is a consequence of Maxwell equations alone. However, they do not impose constraints individually but as a collective. They let you derive a wave equation which contains the (phase) velocity as a parameter.
• Electrodynamics (as described by Maxwell's equations) is what we call a covariant theory, i.e. it is in compliance with special relativity. E.g. when you have a static charge density and you switch to a moving frame, there will also be a current density due to the moving charge density. This is exactly the same as in relativistic mechanics where time and position mix in the Lorentz-transform. In fact, the transformation is the same. There is even a ('covariant') way of rewriting Maxwell's equations such that they won't change form under Lorentz-transforms.
• Historically, electrodynamics was what motivated Einstein to pursue the idea of having the Lorentz-transform govern mechanics as well. Indeed, the original paper in which he proposed special relativity was titled 'Über die Elektrodynamik bewegter Körper' ('On the electrodynamics of moving bodies'). So in a way, no extra work was required on electrodynamics to make it relativity-ready. It was Newtonian mechanics which was flawed and needed to be fixed up by Einstein.
• It is relatively easy to show that the Lorentz-transform is as it is when you assume that the speed of light is the upper bound on velocity. It is also possible to show that given that there is an upper bound on velocity, it has to be the speed of light, but it's more difficult. I think it's hard to make an accurate statement on the importance of Maxwell's equations here. The impossibility of breaking the speed of light is a consequence of the Lorentz-transform which is motivated from electrodynamics. But it needed the genius of Albert Einstein to realize that you could also apply the Lorentz-transform on mechanics which you have to when you want to make a statement on moving bodies and their velocities.
• As to the history of the Lorentz transform, I know that it was know before Einstein published his theory of special relativity. (That's why it's Lorentz transform after Hendrik Antoon Lorentz, not Einstein-transform). But people didn't realize that is was the 'true' nature of space and time. Some thought it was an effect due to the motion of the aether, but that has been disproven experimentally by Michaelson & Morely.

Hope that helps a little.

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Bunch of Maxwell's equations lead to a relation for group velocity in terms of permittivity and permeability.You know Maxwell's equations have no sense about what "electrical permittivity" or "magnetic permeability" is. By "constraints" I mean if Maxwell's equations together impose limitations on how great (e.g. infinitely!) this relation could/could not be? i.e. suppose we can take negative or even very small values for permittivity and permeability in to account. –  2physics Sep 17 '13 at 17:37
My question is exactly about how to prove and show "there is an upper bound on velocity? And why "it has to be the speed of light"? . If it's just an assumption and a principal that Einstein assumed? or there is a proof leading us to it. If it's just an assumption with no need to be proved So where did it come from?? Violating this assumption should violate some preceding principals and relations. I'm looking for them. –  2physics Sep 17 '13 at 18:38
I edited my question and add some things to it. please have a look at it. Thank you for helping me to learn physics :) –  2physics Sep 17 '13 at 20:22
The answer to your three questions is basically given in my answer. Electrodynamics is covariant, i.e. transforms according to Lorentz, so this was the motivation to assume that mechanics should also be Lorentz-covariant. This is the motivation for special relativity. –  Jonas Sep 17 '13 at 20:39
Thank you for your help and your answer. But my question is not especially about how Einstein affected mechanics using Lorentz transformations.My question is about the hypothesis of constant velocity of light that affected physics. And I asked this supposition that made by Einstein, and I am curious about its origins and where it rose from. Yes, you are right! Electrodynamics is covariant! But many other scientists knew that it's covariant. You can't say light speed in vaccum is constant because electrodynamics is covariant.I've found the answer .. Thanks anyway. –  2physics Sep 26 '13 at 20:09

Jonas's answer is nice. I'll just say a few additional things.

The special role of light in relativity is purely historical. Physicists today do not think of the $c$ in relativity as the speed of light but rather as a kind of conversion factor between space and time. See Would it be possible to develop special relativity without knowing about light? .

Does exceeding the limitation for speed of light violate what Maxwell's equations imply?

As Jonas's answer explains, the first steps are to observe that the Lorentz transformation is the symmetry underlying Maxwell's equations, and then to think of that as applying more generally. We now think of the Lorentz transformation as not just applying to light and to mechanics but to space and time themselves. Once you do that, you get the full theory of SR as a logical consequence. SR does not forbid velocities greater than $c$, but it does put some very strong constraints on such motion: http://physics.stackexchange.com/a/61129/4552

Does considering unequal constant values of velocity of light for different inertial references violate what Maxwell's equations try to imply?

Yes, in the sense that Maxwell's equations require that you represent boosts by Lorentz boosts if you want the form of the equations to be the same in all frames. Under a Lorentz boost, there is only one invariant speed, which is what we call $c$.

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I'm confused! that's (the second part) obvious! Because Lorentz transformation is based on considering a constant (C) value for light speed for all observers in all moving frames.So any transformation performed by using Lotrentz boosts shouldn't violate this assumption! But my question is about the individual role of maxwell's equations.Thanks. –  2physics Sep 17 '13 at 18:20
@2physics: What do you mean by "individually?" Do you mean, e.g., whether Gauss's law all by itself forces $c$ to be constant or to be some value? –  Ben Crowell Sep 17 '13 at 19:30
:No no! I'm talking about whole equations together. I mean considering all four equations simoltaneously ( e.g. with no consideration of Lorentz transformation and it's assumptions or any other principle , law and etc). By "individually" I mean "all Maxwell's work" individually with no help from relativity or any other theorem.("independently" is a better choice.)Thanks again –  2physics Sep 17 '13 at 21:52
@2physics: Relativity is implicit in Maxwell's equations. It doesn't make sense to try to separate them. Doing Maxwell's equations with, say, Galilean relativity would be like trying to do Newton's laws with an Aristotelian view of motion. –  Ben Crowell Sep 18 '13 at 15:38
I was trying to find the view at which Einstein looked at the issues of electrodynamics. By that time theory of relativity didn't exist and he postulated some principles and theorized the relativity theorem. Maxwell's equations beside experiments of measuring the speed of light together result in relativity.Thank you –  2physics Sep 26 '13 at 23:35

Maxwell's equations and special relativity are equivalent to each other, in that it is possible to derive one from the other. Special relativity has the speed of light being constant in all reference frames as one of its axioms, while it is possible to derive the constancy of speed of light from Maxwell's equations by using the electric and magnetic permeability constants.

Travelling faster than the speed of light does not violate Maxwell's equations. In fact it is possible to predict what would happen to charged particles if they traveled at superluminal speeds by using Maxwell's equations. For example if you had two electrons travelling in the same direction at faster than the speed of light, then they would actually be attracted to each other instead of repelled, as if they were travelling backwards in time. However, I think it can also be derived from Maxwell's equations that it is impossible to actually accelerate a charged particle to superluminal speeds, because the only way to accelerate a charged particle is to use photons, which only travel at the speed of light, so as a particle moves faster it becomes harder for photons to "catch up" to the particle in order to accelerate it.

Historically Maxwell's equations came before special relativity because they were derived from experimental results, while special relativity came from thought experiments where a connection was made between some conclusions of Maxwell's equations, such as two electrons being more weakly repelled from each other the faster they moved relative to an observer, and the flow of time.

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1 vote for first part of the second paragraph. specially first sentence :) –  2physics Sep 26 '13 at 20:28

Like what Thomas said, I think travelling faster than the speed of light does not violate Maxwell's equations. Which means Maxwell's equations may result in a constant value for the speed of light but this constant is not necessarily the speed at which light propagates in vaccum. These equations just reflect the results of some experiments and they can't be used to generalize global rules and laws of nature. They say some thing but definitely not every thing enough to interpret this issue of light speed.

The fact is that the assumption of a global constant velocity for light which made by Einstein was just a good assumption caused only by his ingenuity and his strong imagination ability. Put yourself in his place. Maxwell's equations are covariant ,considering the principles of Galileo the speed of light shouldn't be constant. But experiments show its constancy.. You have two choices: 1. interpret existence of aether 2.to Consider time and distance as relative variables and no more absolute!(which needs the constancy of light speed for all observers). Opposite of all his other contemporary scientists, he decided to think about the second probability which supposed to be impossible and unbelievable at that time! And this is his most important characteristic which made him different from others.

Maybe that's why he says: “Imagination is more important than knowledge. For knowledge is limited to all we now know and understand, while imagination embraces the entire world, and all there ever will be to know and understand.”

The assumption of the constancy of light speed results in a theory which describes our world phenomena better than all other existing theories of his age. Traveling faster than light doesn't violate nature laws necessarily. It violates laws of a theory in which our world is best described. This theory is adapted to physical phenomena. and there is no explicit evidence of violation of this theory so we have accepted it as the general and most precise theory of describing various phenomena of our world. There could be something that it can't explain. And of course there could be a more precise theory which works better to explain and this current theory will be replaced by the newer one which is more powerful.

I found this in "ABC of Relativity" by Bertrand Russell (1872 - 1970):

"Einstein, in the special theory of relativity, set to work to show how electromagnetic phenomena could be unaffected by uniform motion through the aether - if there be an aether. This was a more difficult problem, which could not be solved by merely adhering to the principles of Galileo. The really difficult effort required for solving this problem was in regard to time. It was necessary to introduce the notion of 'proper' time which we have already considered, and to abandon the old belief in one universal time. The quantitative laws of electromagnetic phenomena are expressed in Maxwell's equations and these equations are found to be true for all observers, however they may be moving. It is a straightforward mathematical problem to find out what differences there must be between the measures applied by one observer and the measures applied by another, if, in spite of their relative motion, they are to find the same equations verified. The answer is contained in the 'Lorentz transformation', found as a formula by Lorentz, but interpreted and made intelligible by Einstein. Our solution of this problem has to satisfy certain conditions. It has to bring out the result that the velocity of light is the same for all observers, however they may be moving. And it has to make physical phenomena - in particular, those of electromagnetism - obey the same laws for different observers, however they may find their measures of distances and times affected by their motion. And it has to make all such effects on measurement reciprocal. That is to say, if you are in a train and your motion affects your estimate of distances outside the train, there must be an exactly similar change in the estimate which people outside the train make of distances inside it. These conditions are sufficient to determine the solution of the problem."

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