Physicists at the beginning of the 20th century were thus faced with a conundrum. They had Newton’s theory of mechanics, working in perfect harmony with Galileo’s principle of relativity, tested over two centuries and never once found wanting. Maxwell’s theory of electrodynamics was, by comparison, the new kid on the block. But its experimental confirmation, particularly thanks to the work of Heinrich Hertz (1857-1894) proving the existence of electromagnetic waves travelling at the speed of light, was compelling. Maxwell’s theory pointed to the future of physics, it signalled a radical departure from the ‘action at a distance’ concept implicit in the classical interpretation of the interaction of physical bodies. But what was to be made of its inconsistency with Galilean relativity and the null result of Michelson-Morley?

Both Hendrik Lorentz (1853-1928) and Henri Poincaré (1854-1912) made significant contributions to the solution of this puzzle but it was Albert Einstein (1879-1955) in his 1905 paper “On the Electrodynamics of Moving Bodies” who had the clarity and audacity of vision to see that what was required was nothing less than a radically new understanding of the relationship between space and time. His solution was as simple as it was bold. He declared that the laws of physics, including Maxwell’s equations, are indeed valid in all inertial frames of reference. In particular, this means that no matter how fast a light source is travelling, the light always travels at the same speed \(c\). The Galilean transformations between inertial frames were no longer tenable but, thanks to Lorentz, their replacement, the Lorentz transformations, were already known. They were part of a theory, “Lorentz Aether Theory”, which Einstein’s bold insight swept aside. Einstein was able to show that the Lorentz transformations were a natural consequence of the fundamental principle of the constancy of the speed of light. The aether was now redundant.

The Galilean transformations assume that time is absolute, that observers in uniform motion relative to one another agree on the rate at which time passes and so always agree on the time interval between two given events. It was this assumption of an absolute time, such a deeply intuitive notion, which Einstein had the brilliance to dispense with. Subsequent notes in this series will discuss in more detail the derivation and remarkable consequences of the Lorentz transformations but its worth having a look at them now to get a qualitative sense of their departure from the Galilean paradigm.
\begin{align*}
x’&=\frac{x-vt}{\sqrt{1-(v/c)^2}}\\
y’&=y\\
z’&=z\\
t’&=\frac{t-vx/c}{\sqrt{1-(v/c)^2}}
\end{align*}Notice how the spatial and time coordinates have become intertwined. Notice also that in the limit \(c\to\infty\) the Lorentz transformations become the Galilean transformations. Over the course of the next few notes we’ll come to appreciate the speed of light as Nature’s speed limit. We’ll also see how Newtonian mechanics had to be modified to become consistent with this new principle of relativity.

To this day special relativity is regarded as the correct geometric setting for all of physics except gravity. Three of the four fundamental forces known, the electromagnetic and strong and weak nuclear forces are understood in terms of quantum field theories, a framework in which quantum mechanics and special relativity are successfully reconciled. Gravity though is specifically excluded in special relativity. Being an action at a distance theory, Newtonian gravity had no place in the new framework and it took Einstein 11 years to complete his monumental general theory of relativity which established gravity as curvature of spacetime. En route, in 1907, Einstein made a crucial observation regarding its nature. As I’ve already mentioned, gravity is a somewhat peculiar force in that it accelerates all masses equally. This led Einstein to the realisation that in free fall gravity is no longer perceptible. Nowadays this effect is familiar to us from footage of astronauts floating weightless in the International Space Station (ISS). Note that the space station isn’t in some sort of zero-gravity environment. On the contrary, the earth’s pull up there is only about 5% less than we experience it on the ground. The ISS is simply falling. It is in free-fall, but doesn’t come crashing down to earth since it has just the required velocity perpendicular to ‘down’ to ensure that as fast as it’s falling, the earth is curving away from it so it maintains its orbit. In fact, though pretty thin, the atmosphere in the space station’s orbit creates a drag which requires periodic re-boosts to maintain this crucial balance. During these, the ISS is not in free-fall, an effect vividly demonstrated by its crew members in this video:

The boosts ‘turn on’ gravity momentarily. This is in fact the crucial point. If you are in a windowless spaceship, there is no way to tell the difference between the spaceship being at rest on earth or being in deep space, far from any massive gravitation-inducing bodies, with its boosters on to provide an acceleration equal to that induced by earth’s gravity. The weightfulness will feel identical in both cases, just as the weighlessness experienced in free fall is no different from that which would be experienced in deep empty space. These are both examples of Einstein’s principle of equivalence upon which he based the general theory of relativity.

Though that is the beginning of a story for another day, we should now recall our definition of an inertial frame of reference as one in which a free test particle would have a constant velocity. We previously brushed over the issue of gravity. Now we see that something like the ISS is an excellent approximation to an inertial frame of reference. In fact, to be precise we should restrict attention to local reference frames. That is, windowless spaceships small enough so that tidal effects due to the non-uniformity of the gravitational pull are not perceptible ¹ With respect to a local frame of reference in free-fall such free test particles really do exist! Thus, real world inertial frames of reference, those to which Einstein’s special relativity applies, are local free-fall frames, sometimes called free-float frames.

But if inertial frames of reference are really free-fall frames then where does that leave our earth-bound ‘inertial’ frames. In particular, are we entitled to use special relativity in analysing particle trajectories at the LHC? Fortunately the answer is yes. Since we are in any case interested in understanding the behaviour of objects moving at or near light speed, over the relevant time scales gravity isn’t an issue. To see this we note that in a laboratory on earth in a time \(t\) a particle falls a distance \((1/2)gt^2\), where \(g\approx10\text{ms}^{-2}\) is the acceleration due to gravity. So if the smallest displacement we can detect is of the order of a micrometer, \(10^{-6}\text{m}\), (the best spatial resolution of the tracking devices at the LHC), then that corresponds to a falling time of the order of \(10^{-4}\text{s}\). That doesn’t sound like long but near light speed particles can cover distances of the order of \(10\text{km}\) in that time so no deviation from inertial, straight line, motion could be detected in a realistically proportioned earth-bound laboratory.

To summarise then, we can say that the known laws of physics are invariant under Lorentz transformations and these transformations relate inertial frames which are best understood as local free-fall frames in uniform relative motion with respect to one another. An earth-bound laboratory is a reasonable approximation of a free fall frame when considering motion at or near light speed since over sensible distances the relevant time frames are so short that gravity may reasonably be ignored. Newtonian physics is invariant under Galilean transformations. That physics and those transformations are the low speed approximations of relativistic mechanics and Lorentz transformations respectively. In either context gravity doesn’t have to be excluded and earth-bound laboratories are reasonable approximations of inertial frames of reference when earth’s rotational motion is irrelevant.

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.