On the Electrodynamics of Moving Bodies

In this article, I will explain Einstein's paper On the Electrodynamics of Moving Bodies.

My goal is not to do a historical investigation of exactly what Einstein thought. My goal is to justify special relativity

The problem

Einstein begins by noting the following problem: Maxwell's equations have a different interpretation in some reference frames than they do in others.

He gives the following example, involving a magnet and a hoop of wire which are in motion to relative to one another, and the induced current in the hoop that results.

In the reference frame in which the hoop is moving and the magnet is stationary, we would describe the situation as follows: the charged particles inside the hoop are moving in the presence of a constant magnetic field, so by the Lorentz force law $\mathbf {F} =q\mathbf {v} \times \mathbf {B}$, some force is applied to those particles which causes them to start moving---i.e. which causes there to be a current.

In the reference frame in which the magnet is moving and the hoop is stationary, we would describe the situation as follows: the magnetic field $\mathbf {B} _{\text{magnet}}$ is changing, and so by Faraday's law ${\textstyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}}$ it gives rise to an electric field $\mathbf {E} _{\text{magnet}}$. The electric field then applies a force to the charges in the hoop according to $\mathbf {F} =q\mathbf {E}$, which causes there to be a current.

He also mentions "the unsuccessful attempts to discover any motion of the earth relatively to the 'light medium'," where he was likely alluding to the Michelson–Morley experiment.

Einstein's two "postulates"

Einstein puts forward two postulates

1. That Maxwell's theory holds true in "all frames of reference for which the equations of mechanics hold good," i.e. in all inertial reference frames.
2. That light always propagates in space with a constant speed c, independent of the velocity of the body which emitted the light.

and says he will proceed to deduce some consequences from them.

Despite what Einstein may have believed, these postulates were far from arbitrary. They probably could have been proved, based only on knowledge that was known in 1905 (I only say "probably" because I do not have a complete theory of induction---i.e. I do not have an explicit standard of when a generalization is or is not to be considered proved). At the very least, they were plausible hypotheses. Specifically, my stance is that the first postulate was a plausible hypothesis, and that the second postulate could have been deduced from the first.

That postulate 1 was a plausible hypothesis comes from the observation that Maxwell's theory holds true in all inertial frames of reference in which it had been tested. See the Michelson-Morley experiment [TODO].

Now, let's examine postulate 2. From Maxwell's equations,

$${\begin{aligned}\nabla \cdot \mathbf {E} &=\rho /\epsilon _{0}\\\nabla \cdot \mathbf {B} &=0\\\nabla \times \mathbf {E} &=-{\frac {\partial \mathbf {B} }{\partial t}}\\\nabla \times \mathbf {B} &=\mu _{0}\mathbf {J} +\mu _{0}\epsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\end{aligned}},$$

the phenomenon of light can be explained: it is a plane wave solution to these equations. This plane wave is found to move with speed $c=1/{\sqrt {\mu _{0}\epsilon _{0}}}$. That is, Maxwell's theory predicts that the speed of light is given by constants $\mu _{0}$ and $\epsilon _{0}$ that appear in Maxwell's equations. So if Maxwell's equations are valid in all reference frames, it follows that the speed of light is the same in all reference frames. Thus postulate 1 logically implies postulate 2.

Simultaneity

Einstein notes that when we sync up clocks, ...