Towards the end of the 19th century, it had become generally accepted that Maxwell’s equations, as presented by James Clerk Maxwell (1831-1879) in his 1865 paper “A dynamical theory of the electromagnetic field”, were the correct and unifying description of the physics of electricity and magnetism. Light was by then understood to be electromagnetic waves, with Maxwell’s equations specifying their speed in vacuum to be a universal constant of nature, \(c=299,792,458\text{ms}^{-1}\approx3\times10^8\text{ms}^{-1}\). The equations make it clear that the speed of light does not depend on the speed of the source. It was therefore assumed that light waves must propagate through some kind of material medium, a ‘luminiferous aether’, just as sound waves propagate, independent of the speed of their source, through air. Consistent with this belief, Maxwell’s equations are not invariant under Galilean transformations. The presumption was that they held only in those frames which happen to be at rest with respect to the mysterious aether — only in such a preferred frame would light travel in all directions at speed \(c\). But this state of affairs should then present an opportunity to detect the relative motion between earth and the aether. The most famous such attempt was the Michelson-Morley experiment of 1887. ¹ Here is a schematic of the optical interferometer used in their experiment.

Sodium light was split into two beams travelling at right-angles to one another. After travelling (approximately equal) distances \(L=11\text{m}\) each beam is reflected back to the beam splitter where they are recombined and directed towards a detector ready to observe interference fringes. The apparatus was mounted on a bed of mercury allowing it to be smoothly rotated. If by some miracle (the earth’s velocity relative to the sun is \(30\text{kms}^{-1}\) and \(200\text{kms}^{-1}\) relative to the centre of the milky way) the apparatus was at rest in the aether, then no shift in the observed interference fringes would be expected as the apparatus is rotated. Considering the more likely scenario of the interferometer travelling with some velocity \(v\) relative to the aether’s rest frame and aligned at an angle \(\theta\) to this direction we consider the following schematic.

We can write down the following pairs of equations for the round trip paths taken by each of the pair of beams.
\begin{align*}
c^2{t_1}^2&=(L-vt_1\sin\theta)^2+v^2{t_1}^2\cos^2\theta,\\
c^2{t_2}^2&=(L+vt_2\sin\theta)^2+v^2{t_2}^2\cos^2\theta,
\end{align*}
and
\begin{align*}
c^2{T_1}^2&=(L+vT_1\cos\theta)^2+v^2{T_1}^2\sin^2\theta,\\
c^2{T_2}^2&=(L-vT_2\cos\theta)^2+v^2{T_2}^2\sin^2\theta.
\end{align*}

From which we calculate(for example)
\begin{equation*}
(c^2-v^2){t_1}^2+2Lv\sin\theta t_1-L^2=0
\end{equation*}
so that
\begin{equation*}
t_1=\frac{-2Lv\sin\theta+2L\sqrt{v^2\sin^2\theta+(c^2-v^2)}}{2(c^2-v^2)}
\end{equation*}
and then
\begin{equation*}
t_1=\frac{-2Lv\sin\theta+2L\sqrt{c^2-v^2\cos^2\theta}}{2(c^2-v^2)}
\end{equation*}etc… the respective total round trip paths to be
\begin{equation*}
c(t_1+t_2)=\frac{2L\sqrt{1-(v\cos\theta/c)^2}}{1-(v/c)^2},
\end{equation*}and
\begin{equation*}
c(T_1+T_2)=\frac{2L\sqrt{1-(v\sin\theta/c)^2}}{1-(v/c)^2},
\end{equation*}and the path difference, which we’ll call \(\Delta(\theta)\), to be
\begin{equation*}
\Delta(\theta)=\frac{2L}{1-(v/c)^2}\left(\sqrt{1-(v\sin\theta/c)^2}-\sqrt{1-(v\cos\theta/c)^2}\right).
\end{equation*}If the apparatus is rotated through \(90^\circ\) then the path difference is \(-\Delta(\theta)\) so the expected fringe shift between the two orientations will be a fraction \(2\Delta(\theta)/\lambda\) of a wavelength where \(\lambda=589\times10^{-9}\text{m}\) is the wavelength of sodium light. Assuming the apparatus starts off with an orientation of \(\theta=45^\circ\) to the direction of relative motion, in which case \(\Delta(\pi/4)=0\), and assuming the aether is at rest relative to the sun so the relative velocity is \(v=30\text{kms}^{-1}\) we can plot the expected shifts.

The greatest shift is expected to occur between two alignments in which one arm is parallel and the other perpendicular to the direction of motion. In this case we have a fringe shift of approximately \(2Lv^2/c^2\lambda\approx0.37\). In fact Michelson and Morley found nothing of the sort, observing fringe shifts no bigger than 0.01 of a wavelength which translate to a relative velocity of less than \(5\text{kms}^{-1}\) ². The extraordinarily slim chance that the ether and earth frames happened to be comoving with the same velocity relative to the sun was eliminated by repeating the experiment at three month intervals. The result was the same.

A somewhat bizarre but theoretically possible explanation for the Michelson-Morley result, that the earth somehow drags the aether with it, is ruled out by the well established phenomenon of stellar aberration. The gist of the issue here is familiar to anyone who’s noticed that when cycling through falling snow, the snowflakes seem to fall towards us from somewhere in the sky in front of us rather than, as we observe when stationary, straight down. If somehow the clouds producing the snow were moving with us this apparent shift in the source of the snow wouldn’t occur. Analogously, due to earth’s motion in orbit about the sun, the apparent positions of stars in the sky is shifted. This is stellar aberration and, if the aether were dragged along with earth, it wouldn’t be observed — but it is.