Historical motivations for special relativity

This post was originally written to be the first part of a text companion for a series of introductory lessons on the basics of special relativity. Ultimately I decided to write a different introduction and the lesson series didn't eventuate anyway, so I thought I might as well post it here since I'm not currently working on a blog post. Keep in mind it's not 100% perfect, I haven't fully edited it or added any illustrations, but if I did that I might as well write a new post altogether. So anyway, enjoy!

Let's begin by examining non-relativistic classical mechanics. This is sometimes referred to as Newtonian mechanics because its foundations are given by Newton's three laws of motion. For our discussion, the relevant laws are the closely-related first and second laws. As a refresher, they are given by:

First Law
Unless acted upon by an external force, an object will remain in constant motion.

Second Law
$\mathbf{F}_{\text{net}}=m\mathbf{a}$

An object in constant motion is one whose velocity (speed and direction) is not changing, or one that is not moving at all. The first law thus defines the domain of validity of Newton's laws, saying that they only hold in inertial reference frames. For our purposes, a reference frame is a coordinate system with an associated clock, and an inertial reference frame is such a reference frame which is not accelerating with respect to any other inertial reference frame. $\mathbf{F}_{\text{net}}$ is the net force, or the sum of all forces acting on an object (equivalently the resultant or overall force).

The difference between inertial and non-inertial reference frames is easy to understand given a simple example; two friends are on a straight road, with one standing and the other riding a rocket-powered skateboard towards him. If the standing friend asks the skating friend whether she feels a force pushing her closer to him, she will most certainly answer that she does. Thus the standing friend, who observes from an inertial reference frame, correctly associates force with acceleration (Newton's second law). However, if the skating friend asks the standing friend if he feels any force, he will answer that he does not, even though he appears to be accelerating towards her. This is because the skating friend occupies a non-inertial reference frame, and so Newton's second law no longer applies.

Any two inertial reference frames can be put into so-called 'standard configuration' by choosing a Cartesian coordinate system for both and, when their origins overlap, aligning the spatial coordinate axes (ensuring that the relative velocity is in the $x$-direction) and setting both clocks to 0. Having frames in standard configuration will prove very useful for simplifying calculations, as we will now see. We can generally choose to put inertial frames in standard configuration because our coordinatisation of space should not affect the physics of a given situation, but merely describe it.

Mathematically speaking, it is natural at this point to ask how we get from one inertial reference frame to another. In the context of Newtonian mechanics, we want to be able to transform spatial coordinates (and clocks, represented by a time coordinate) in such a way that Newton's second law holds, that is, the force on an object measured in one inertial frame is identical to that measured by any other inertial frame. The most intuitively obvious transformation laws we might consider are the Galileo transformations, which for two frames in standard configuration (a frame $S$ and a ''prime' frame $S'$) are given by
\begin{align*}
t'&=t\\
x'&=x-vt\\
y'&=y\\
z'&=z\\
\end{align*}
where $v$ is the relative velocity between the two frames and the primes denote the coordinates belonging to the $S'$ frame. These coordinate transformations immediately lead to a velocity transformation (for an object travelling in the $x$-direction):
\begin{align*}
u'&=\frac{\mathrm{d}x'}{\mathrm{d}t'}\\
&=\frac{\mathrm{d}}{\mathrm{d}t}(x-vt)\\
&=\frac{\mathrm{d}x}{\mathrm{d}t}-v\\
&=u-v\\
\end{align*}
where we have used the fact that $t'=t$. It is now not difficult to see how Newton's second law behaves under Galileo transformations. Let's consider an object accelerated parallel to the relative velocity of two frames in standard configuration (it is not difficult to generalise this case to acceleration in an arbitrary direction):
\begin{align*}
F'&=ma' \\
&=m\frac{\mathrm{d}u'}{\mathrm{d}t'} \\
&=m\frac{\mathrm{d}}{\mathrm{d}t'}(u-v) \\
&=m\frac{\mathrm{d}u}{\mathrm{d}t} \\
&=ma=F
\end{align*}
where we have assumed that masses are unchanged in different frames and used the fact that $v$ is constant. That Newton's second law is unchanged by Galileo transformations is what makes Newtonian mechanics 'Galileo-invariant'.

Let us now make a slight diversion into electrodynamics. Electrodynamics, naively the study of electricity and magnetism, is governed by Maxwell's equations. The equation for electromagnetic force can be derived from these equations, and is known as the Lorentz force law:
\begin{equation*}
\mathbf{F}_{\textrm{em}}=q(\mathbf{E}+\mathbf{u}\times\mathbf{B}).
\end{equation*}
It is immediately obvious that unlike Newton's second law, the Lorentz force law is not Galileo-invariant, as the velocity term $\mathbf{u}$ will be transformed by the Galileo transformations to $\mathbf{u}'=\mathbf{u}-\mathbf{v}$. Another issue with Maxwell's equations is that they permit two wave-like solutions, one for the electric and one for the magnetic field. These solutions are given as
\begin{align*}
\nabla^2\mathbf{E}&=\epsilon_0\mu_0\frac{\partial^2\mathbf{E}}{\partial t^2} \\
\nabla^2\mathbf{B}&=\epsilon_0\mu_0\frac{\partial^2\mathbf{B}}{\partial t^2}
\end{align*}
where $\mathbf{E}$ and $\mathbf{B}$ are the electric and magnetic fields respectively, and $\epsilon_0$ and $\mu_0$ are physical constants known as the vacuum permittivity and the vacuum permeability respectively. To compare, the wave equation for an arbitrary function $\phi$ is given by
\begin{equation*}
\nabla^2\phi=\frac{1}{c^2}\frac{\partial^2\phi}{\partial t^2}
\end{equation*}
where $c$ is a constant interpreted as the wave speed. We can therefore infer that for the wave-like solutions of Maxwell's equations, the wave speeds of the electric and magnetic waves in a vacuum are both given by $1/\sqrt{\epsilon_0\mu_0}=c$. As $\epsilon_0$ and $\mu_0$ are physical constants, so too must $c$ be. Physical constants cannot change value depending on your reference frame, so just as the Lorentz force law is not preserved under Galileo transformations, clearly neither are these wave-like solutions.

The resolution to this issue that was most favoured before the advent of relativity was that of the aether. It was postulated that Maxwell's equations were only valid in the reference frame of the luminiferous ("light-bearing") aether, a strange rigid fluid that permeated all space. The properties of the aether needed to be tweaked over time to make them come into line with experimental observations, but this could be managed -- at the price of a large number of assumptions about this apparently unobservable fluid, sometimes seemingly ad hoc. Special relativity offered an alternative which not only matched the data just as well, but was simpler, more predictive and much more elegant.