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I am reading a book titled "Relativity Demystified --- A self-teaching guide by David McMahon".

He explains the derivation of electromagnetic wave equation. $$ \nabla^2 \, \begin{cases}\vec{E}\\\vec{B}\end{cases} =\mu_0\epsilon_0\,\frac{\partial^2}{\partial t^2}\,\begin{cases}\vec{E}\\\vec{B}\end{cases} $$

He then compares it with

$$ \nabla^2 \, f =\frac{1}{v^2}\,\frac{\partial^2 f}{\partial t^2} $$

and finally find

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

where $c$ is nothing more than the speed of light.

The key insight to gain from this derivation is that electromagnetic waves (light) always travel at one and the same speed in vacuum. It does not matter who you are or what your state of motion is, this is the speed you are going to find.

Now it is my confusion. The nabla operator $\nabla$ is defined with respect to a certain coordinate system, for example, $(x,y,z)$. So the result $v=c$ must be the speed with respect to $(x,y,z)$ coordinate system. If another observer attached to $(x',y',z')$ moving uniformly with respect to $(x,y,z)$ then there must be a transformation that relates both coordinate systems. As a result, they must observe different speed of light.

Questions

Let's put aside the null result of Michelson and Morley experiments because they came several decades after Maxwell discovered his electromagnetic wave derivation.

I don't know the history of whether Maxwell also concluded that the speed of light is invariant under inertial frame of reference. If yes, then which part of his derivation was used to base this conclusion?

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    $\begingroup$ You seem to be asking about Maxwell and his peers opinion of the invariability of the speed of light in their own period, which might be more suitable for the History of Science and Mathematics $\endgroup$
    – StephenG - Help Ukraine
    Jun 27, 2020 at 9:13
  • $\begingroup$ @StephenG: So the quoted sentences from the book are based on the result of Einstein and Lorentz? If yes, the author did not seem to arrange the derivation and conclusion chronologically. It made me confused. $\endgroup$
    – Display Name
    Jun 27, 2020 at 9:17
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    $\begingroup$ Does this answer your question? What is the complete proof that the speed of light in vacuum is constant in relativistic mechanics? $\endgroup$
    – Michael Seifert
    Jun 28, 2020 at 14:09
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    $\begingroup$ It looks like Mr McMahon is talking nonsense. The speed of light obviously can't be constant in all rest frames, according to the Galilean/Newtonian view of the universe that prevailed before Einstein's annus mirabilis. Without the background of Einstein's Minkowskian space-time geometry, such a notion is logically impossible. $\endgroup$
    – TonyK
    Jun 28, 2020 at 21:17

6 Answers 6

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Your question is an excellent one and you are right about the $\nabla$ operator. And you are also right about the insufficiency of the argument you report in the book you are reading.

To make the argument more carefully, there are two options. The first would be to work out how the Maxwell equations themselves change as you go to another inertial frame. That would take a lot of calculating if you start from first principles. (And by the way, they don't change---you get back the same equations but now in terms of ${\bf E}', {\bf B}', \rho', {\bf j}', {\bf \nabla}', \partial/\partial t'$).

A second option, mathematically easier but still requiring some work if you are not familiar with it, is to show that the $\nabla$ operator and the $\partial/\partial t$ operator have a special property: when you combine them in the combination $$ \nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t^2} $$ then their effect is the same as $$ \nabla'^2 - \frac{1}{c^2} \frac{\partial^2}{\partial t'^2} $$ All the changes when moving from unprimed to primed coordinates cancel out. If you are familiar with partial differentiation then you could try checking this. When you learn the subject more fully, it becomes an example that can be handled more easily using the language of 4-vectors.

I think that McMahon might possibly have not thought carefully enough about what he was deriving and what he was assuming in his argument. He might for example have been taking it for granted that the Maxwell equations themselves take the same form in all inertial frames. But if he did not first prove that in his book then he ought not to claim that the derivation of waves of given speed from them proves that the wave speed will be independent of the motion of the source.

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  • $\begingroup$ Sorry, I have another related question here. $\endgroup$
    – Display Name
    Jun 27, 2020 at 11:34
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    $\begingroup$ I think it should be emphasized that in order to derive the fact that the Maxwell equations are independent of the inertial frame, you have to choose to use the Lorentz transformation - IOW, you're assuming that special relativity is correct in order to derive that the speed of light is independent of the frame. If you use Galilean transformations instead, you get a different result, as per this answer. (The OPs textbook appears to be claiming that Maxwell's equations, by themselves, prove that special relativity is correct.) $\endgroup$
    – Harry Johnston
    Jun 27, 2020 at 22:38
  • $\begingroup$ In your final sentence do you meant "observer" rather than "source"? Surely the Maxwell equations do unambiguously show that the speed of light is independent of the speed of the source, regardless of whether you're using classical or relativistic mechanics? $\endgroup$
    – Harry Johnston
    Jun 28, 2020 at 4:32
  • $\begingroup$ @HarryJohnston Good point. But the notion of 'ether' arose because it was thought a moving body could disturb the background. I suppose this amounts to saying one is not sure what to make of $\epsilon_0$ and $\mu_0$ until one has the full understanding---one might not be sure if a moving source could modify them, near to itself at least. $\endgroup$
    – Andrew Steane
    Jun 28, 2020 at 9:33
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    $\begingroup$ @ShirishKulhari Folks were not determined that the speed of light is constant; they thought it was not, but observations suggested it was. This was part of Einstein's contribution: to take the hint (not the proof) offered by Maxwell's equations, assert $c=$const, and be prepared to accept the implications for time and simultaneity. (The theoreticians had already figured out which transformation left Max. eq. unchanged, and that length contraction was implied). $\endgroup$
    – Andrew Steane
    Jun 29, 2020 at 17:19
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If Maxwell's equations have the same form in all frames of reference, then the wave speed is defined by the product of two physical constants, irrespective of coordinate system. i.e. Your book just implicitly assumes that, but of course it requires experimental testing - i.e. Michelson-Morley etc.

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  • $\begingroup$ +1 This is what I was taught in college. Maxwell's Equations show the speed of light is equal to two constants multiplied. End of proof that the speed of light is the same for all observers. @Andrew Steane's accepted answer says that's lacking but I don't see how. $\endgroup$
    – user1717828
    Jun 27, 2020 at 19:03
  • $\begingroup$ @user1717828, there are two options for the speed of light being constant under Maxwell's equations: either every observer sees the speed of light being the same (relativity), or every observer sees a privileged reference frame in which the speed of light is constant (luminiferous aether). $\endgroup$
    – Mark
    Jun 28, 2020 at 2:23
  • $\begingroup$ @Mark in which case Maxwell's equations would be different in different frames of reference. $\endgroup$
    – ProfRob
    Jun 28, 2020 at 4:33
  • $\begingroup$ That's easy to say with hindsight but it took 150 years before that conclusion was reached. $\endgroup$
    – Rodney
    Jun 28, 2020 at 12:44
  • $\begingroup$ @Rodney where do you get the 150 years number from? The equations date from 1861/62. The statement written above is true without any hindsight required. It's just maths. About 45 years was required to understand the full implications of their covariant nature (i.e. Special Relativity). $\endgroup$
    – ProfRob
    Jun 29, 2020 at 16:48
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Your observation is correct, Maxwell's equations alone do not imply an invariant speed of light. One can make a Galilean transformation and get an observer-dependent speed of light as shown in the answer to this question. However, the derivation of Maxwell's equations makes no assumption of a privileged reference frame: $\varepsilon_0$ and $\mu_0$ are assumed to be properties of the vacuum. Yes, a coordinate system must be chosen, but from the point of view of derivation of the equations this is totally arbitrary. In order to keep a non-constant speed of light one would have to retroactively make the assumption after the fact that the coordinates chosen happened to be stationary coordinates with respect to the aether.

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    $\begingroup$ But... nobody proved that the vacuum properties don't change with respect to the frame of reference... So you can't even say that the equation can only be applied to the 'aether'-based coordinates if we want non-constant speed of light. $\endgroup$
    – user21820
    Jun 27, 2020 at 17:53
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Without experimental evidence the constancy of the speed of light cannot be concluded. If space contained a medium, the aether, for electromagnetic waves one would expect the velocity of light in wrt to the aether to be constant. The aether theory was disproved by the experiment of Michelson and Morley. That left special relativity as an alternative.

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    $\begingroup$ I don't think you mean ether drag, I think you just mean that light waves would travel at a constant speed relative to the ether, which could be detected. This is what Michelson-Morley disproved. AFAIK, "ether drag" is a theory that was invented to explain Michelson and Morley's results - the idea is that the Earth drags the ether with it, so the ether always matches the Earth's velocity. You need additional experiments, beyond Michelson-Morley, to disprove that. (Otherwise a good answer.) $\endgroup$
    – N. Virgo
    Jun 28, 2020 at 11:12
  • $\begingroup$ @Nathaniel You are correct $\endgroup$
    – my2cts
    Jun 28, 2020 at 14:37
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Maxwell originally assumed that the speed of light would vary depending on the frame of reference. This would imply that Maxwell's equations only hold with respect to some kind of universal coordinate system. When experiments (like Michelson-Morley) indicated that the speed of light did not vary between inertial reference frames, physicists like Hendrik Lorentz figured out how to transform Maxwell's equations in a way that would keep the speed of light constant when moving from one reference frame to another. This required all kinds of weird concepts like length contraction and time dilatation. In 1905, Einstein demonstrated that these weird ideas could be derived in a very natural way by throwing away old ideas about space and time being absolute, and starting with the assumption that the laws of physics (including Maxwell's equations) are equally valid in all inertial reference frames. Your book apparently just adopts this view from the start. There is certainly an aesthetic argument to be made for adopting this point of view, but obviously, any scientific idea needs to be backed up by experimental evidence. Therefore, any book that tries to "derive" scientific laws without reference to experiment is really just feeding you misleading arguments like this one.

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A related point, which seems very little known, is that Maxwell's electromagnetic theory does not imply the speed of light is $c$ in all directions! It is only because we implicitly input that assumption (of isotropic $c$) when formulating the equations, that it pops out at the end. Anderson, Vetharanium, & Stedman (1998) $\S2.3.3$ formulate "Electromagnetism in a more general synchronization" (that is, a different simultaneity convention). Another paper which does this is Rizzi, Ruggiero, & Serafini (2004) $\S A2$.

Having said this, it still seems the most natural choice that $c$ is the same in all directions, for all observers. It's just that Maxwell doesn't prove this, nor does any other theory or experiment prove the one-way speed of light.

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