# Maxwell's Equations and Special Relativity [closed]

A doubt has arisen regarding the following question:

Are Maxwell's Equations necessary to prove the postulates of special relativity?

What I mean to say is, Einstein assumed two postulates, namely

1. All inertial frames are equivalent.

2. The speed of light is a constant in all inertial frames.

Does the requirement of 2 come from Maxwell's equations? Or can it be argued without referring to them?

If the former, then how can we prove it, since Maxwell's equations do not put a velocity bound? If the latter, how?

P.S. I ask this question because, even if it's an assumption that works pretty well, we just cannot take it because it's got no logical footing. Why only photon? Why not anything else? People generally answer by referring to massless particles, bt the concept of massless particles itself comes from special relativity, so all we have done is trace back a circle.

• Consider a model in which there is a primitive concept called light (but no electromagnetic field) subject to your 1) and 2). Clearly there are such models, and clearly they do not entail Maxwell's equations (nor do they even allow you to formulate Maxwell's equations). Jun 16, 2017 at 4:21
• That means... Maxwell's equations are not necessary... And that special relativity can be formulated without ever mentioning them...! Right? Jun 16, 2017 at 4:24
• Yes, that is exactly what it means. I'm a little surprised this needed to be said. Jun 16, 2017 at 4:36
• Actually... I wasn't able to confirm it from anywhere... But your comment made my day... A big thanks to you... Maybe you can write this as an answer... And elaborate it a bit... Jun 16, 2017 at 4:37
• All those dots...are a little bit annoying... Jun 16, 2017 at 11:52

Either you can say the second postulate is an experimental fact - people measure light's speed in different frames directly or indirectly and get that the speed of light is the same in all the frames. Or, you can derive this fact theoretically from Maxwell's equations with a little aid of the first postulate.

Maxwell's equations read as following in vacuum:

$$\nabla \cdot \vec{E} = 0$$ $$\nabla \times \vec{E} = -\dfrac{\partial \vec{B}}{\partial t}$$ $$\nabla \cdot \vec{B} = 0$$ $$\nabla \times \vec{B} = \epsilon_0 \mu_0\dfrac{\partial \vec{E}}{\partial t}$$

Therefore, $$\nabla \times (\nabla \times \vec{B}) = \epsilon_0 \mu_0 \dfrac{\partial}{\partial t}(\nabla \times \vec{E})$$ Or, $$\nabla^2\vec{B} = \epsilon_0\mu_0\dfrac{\partial^2\vec{B}}{\partial t^2} \tag{1}$$

Similarly, $$\nabla \times (\nabla \times \vec{E}) = -\dfrac{\partial}{\partial t}(\nabla \times \vec{B})$$ Or, $$\nabla^2\vec{E} = \epsilon_0\mu_0\dfrac{\partial^2\vec{E}}{\partial t^2} \tag{2}$$

Equations $(1), (2)$ are simply the equations of waves of $\vec{E}$ and $\vec{B}$ propagating with a speed $\sqrt{\dfrac{1}{\epsilon_0\mu_0}}$ - which is completely the same in every frame the Maxwell's equations are valid. And now, from the first postulate, we can say they are valid in all the inertial frames (or you can call it an experimental fact if you wish). This is how we conclude that the speed of electromagnetic waves (which is light) is the same in all the inertial frames.

But we call it a postulate in the sense of postulating that Maxwell's equations are true laws of Physics and are, thus, according to the first postulate are valid in all the inertial frames.

Edit Regarding the relation of the second postulate with masslessness of particles: As such the second postulate doesn't refer to any particle at all. It (as illustrated above in this answer) refers to waves and their speed of propagation in the vacuum. So, in order to get to the second postulate, we don't need to know anything about massless particles. But, once we have these postulates, laws of momentum conservation and energy conservation forces upon us that the energy of a particle must be $\dfrac{E_0}{c^2\sqrt{1-\dfrac{v^2}{c^2}}}$ and that its momentum must be $\dfrac{E_0 v}{c^2\sqrt{1-\dfrac{v^2}{c^2}}}$ (where $v$ is the speed of the particle and and $E_0$ is the energy of the particle in its rest frame). From these formulae, it becomes clear if $v=c$ then the energy and momentum would become infinitely large unless $E_0=0$. This is to say that the only possible way a particle can travel at the speed of light is for the particle to be massless (i.e., $E_0 = 0$). On the other hand, the only way for a massless particle to have finite energy and momentum is to travel at the speed of light. So, if a particle is massless then it has to go at the speed of light as well as if a particle goes at the speed of light then it has to be massless. But this is something we derive from Special Relativity - not something we postulate to derive Special Relativity.

• That's a nice answer... And yes I get it that the Maxwell's Equations can derive the second postulate( with the help of the first postulate)... But is their anyway we can just argue it without referring to the equations? Or is this the only way out? Jun 16, 2017 at 6:20
• Theoretically, it is the only way out. Well, you can argue on the basis of only homogeneity, isotropy, and principle of relativity (the first postulate) that there ought to be a frame invariant speed $\sqrt{\dfrac{1}{K}}$, with $K$ being a constant. But you can't determine whether $K$ is zero or non-zero. In order to really nail down the value of $K$, you ought to invoke Maxwell's principles - either theoretically or as experimental facts.
– ACat
Jun 16, 2017 at 7:13
• Reference related to my comment above: arxiv.org/abs/physics/0302045
– ACat
Jun 16, 2017 at 7:19
• Your answer is historically quite inaccurate. Maxwell equations were first though to be true only in the frame where the Ether was at rest. In any other frame, there would be effects from the Ether wind. That's the Michelson-Morley experiment that forced Lorentz to introduce extra hypothesis that, for all practical purposes, made the Maxwell equations correct in any frame.
– user154997
Jun 16, 2017 at 12:17
• Yes, I realize that. But I have presented the thoughts that goes into the logical structure of the postulates of special relativity. I know that validity of Maxwell's equations in all frames isn't something people took for granted and thus, I have explicitly mentioned that the crux of the second postulate is that we postulate the validity of Maxwell's equations in all inertial frames. And as I have mentioned, this postulate is motivated by the first postulate as well as by the experiments, of course.
– ACat
Jun 16, 2017 at 12:20

The Michelson-Morley experiment showed the speed of light is constant. Special relativity was developed to explain how that could be.

• That's an experiment... And that's historically how the upper bound of velocity was confirmed... But I asked for a theoretical answer that incorporates the experiment as well.... Moreover the experiment did not show it... It was postulated so that the the null result of the experiment could be explained... Jun 16, 2017 at 3:50
• Well, if we invoke historical context then Einstein didn't really care much about Michelson-Morley experiment, he developed Special Relativity out of the theoretical troubles with the compatibility of Maxwell's theory and Newton (and Galileo)'s theory.
– ACat
Jun 16, 2017 at 7:16
• @Dvij - You are right. But there were moves toward special relativity before Einstein. In particular the Lorentz contraction hypothesis was an attempt to explain the Michelson-Morley experiment. Jun 16, 2017 at 13:41

I think that the article in wikipedia on history of the Lorenz transformation clarifies how it was derived.

Lorentz (1892–1904) and Larmor (1897–1900), who believed the luminiferous ether hypothesis, also looked for the transformation under which Maxwell's equations are invariant when transformed from the ether to a moving frame. They extended the FitzGerald–Lorentz contraction hypothesis and found out that the time coordinate has to be modified as well ("local time").

So it is the interplay between data ( Michelson Morley experinent) and the theoretical model that established the Lorenz transformations which result in an invariant velocity for the electromagnetic waves predicted by Maxwell's equations.

That is the way physics has progressed to the point we are now: model with known data, predict, check for consistency with data, remodel.