Mathematical Field Notes

Is the speed of light constant?

I recently came across a beautiful argument due to De Sitter (1913), which gave the (first?) experimental evidence that light moves with a constant speed.

Constancy of the speed of light is one of those things that always bothered me, and I spent a couple of days recently trying to unbother myself. De Sitter’s argument is what finally satisfied me. Below, I’m going to explain the background, then I’ll explain De Sitter’s argument. The De Sitter paper is only a couple of paragraphs long and is available via Wikisource, so if you don’t need the introductory remarks in the blogpost below, just follow the link above and read it.

THE ASSUMPTIONS BEHIND RELATIVITY

Einstein’s theory of relativity is based on two fundamental assumptions:

1. (Constancy of the speed of light) That light moves with a constant speed c, independently of how it is produced. For example, a lamp at rest in your living room will emit light travelling at speed c, just as a lamp on a distant spaceship moving at 3 million miles per hour will emit light that travels at speed c.

2. (Principle of relativity) That the results of experiments performed in inertial frame (no external forces acting, like gravity) will always be the same independently of the frame chosen. In other words, if two experimenters are drifting past each other in empty space, it’s impossible to do an experiment to tell which of them is actually moving (only that they’re moving relative to one another).

This second assumption always seemed completely reasonable to me, and the first assumption always seemed kind of strange. To see why these two assumptions give interesting consequences, we consider an experiment first performed by Michelson and Morley in the 1880s.

THE MICHELSON-MORLEY EXPERIMENT

Set up an experiment as follows. Send a beam of light in direction A, then split it into two beams moving at right angles (one still going in direction A, the other in direction B perpendicular to A). Place two mirrors a fixed distance $d$ from the beam splitter so that the two beams reflect back and recombine where they started (“the origin”). Because the two beams travel the same distance ($2d$) in the same time, they are “in phase” when they recombine, which means that there is no interference between the beams (by constrast, if one of the mirrors were closer than distance $d$ then one of the beams will arrive back sooner, so the beams will be out of phase and will usually interfere with one another).

Now give the whole apparatus a kick in the A-direction so that it starts moving with velocity $v$ in the A-direction. The light beam moving in the A-direction has slightly further to travel on the journey out because the mirror is moving away from it, but it has less distance to travel on the journey back because the origin is moving towards it. Overall, it travels the same distance as before. By contrast, the light beam moving in the B-direction now has to travel further! It needs to go from the origin to the B-mirror and back (which is the same perpendicular distance as before) but it also needs to move forwards to keep up with the moving apparatus. Suppose it takes the B-lightbeam a time $t$ to reach the B-mirror; in this time, the apparatus moves forward a distance $vt$, so (by Pythagoras’s theorem) the light has travelled a distance $\sqrt{d^2+v^2t^2}$. Since light always moves with speed $c$, this distance is equal to $ct$, so

$ct=\sqrt{d^2+v^2t^2},$

which means that

$t=d/\sqrt{c^2-v^2}.$

In total, then, the light has to move a distance $2dc/\sqrt{c^2-v^2}=2d/\sqrt{1-v^2/c^2}$, which is strictly further than the distance $2d$ travelled by beam A. The two beams will surely arrive back out of phase and will interfere with one another.

However, Michelson and Morley observed no such interference. Indeed, if they had observed interference, it would have contradicted Einstein’s second assumption (the principle of relativity), because they could have figured out their speed v by measuring the interference! Relativity explains how to modify the above analysis so that the beams arrive back in phase: you incorporate the factor of $1/\sqrt{1-v^2/c^2}$ into the coordinate change between the moving frame and the rest frame, which we can think of as a “length contraction” of the experimental apparatus observed by the observer at rest (but not by the moving experimenter).

You see: once we make these two assumptions (constancy of light speed and the principle of relativity) we soon run into relativistic corrections to our classical Newtonian intuition.

AN ALTERNATIVE ASSUMPTION

Now here’s what bothered me. Let’s drop the first assumption (constancy of the speed of light) and replace it by the following at-first-sight-equally-plausible assumption:

1′. Light produced by a source moving with velocity v will travel at velocity v+c.

Here c could be any vector whose length is the usual speed of light. For example, we imagine that the lamp on the spaceship (velocity v) is hurling out photons in all directions: those going forwards (in the direction of the spaceship) travel at speed c+v miles per hour, those emitted backwards travel at speed c-v, those emitted sideways travel at speed c. This, of course, is completely consistent with the principle of relativity: someone sitting on the spaceship next to the lamp will see these photons behaving like usual photons, someone outside the spaceship (at rest) will notice the photons moving with different speeds, but that will simply tell them that they are moving relative to the source of the photons.

This theory was actually posited by Ritz in 1908 – three years after Einstein’s. On the face of it, assumption 1′ seems much more natural than assumption 1. Moreover, it’s completely consistent with the outcome of the Michelson-Morley experiment: it’s a nice exercise in velocity vector addition to check that if your light beams behave the way I just described then they will arrive back at the origin in phase without needing to invoke some crazy relativistic corrections.

SO WHY IS THE SPEED OF LIGHT CONSTANT?

When I realised this, I wanted to re-convince myself that constancy of the speed of light is a sensible assumption. Here are three different ways to convince yourself of this.

1. In Maxwell’s theory of electromagnetism, the components of the electromagnetic field satisfy the wave equation with speed c. Since light is supposed to be an electromagnetic wave, this means it will move with speed c. If you know some vector calculus and you want to see a quick derivation of the wave equation starting from Maxwell’s equations, Wikipedia has it.

This is the textbook explanation, and is ultimately the best theoretical reason to believe the assumption. Indeed, Einstein’s work was mostly motivated by theoretical considerations underpinning Maxwell’s equations and their interpretation. The influence of Maxwell’s theory on his thought is clear; indeed, his original paper is called “Zur Elektrodynamik bewegter Körper” (“On the electrodynamics of moving bodies”) and the opening paragraph explains the particular paradox which bothered Einstein: namely, that if you have a conductor moving in a magnetic field, there is an induced current (according to Maxwell’s theory), but that the source of this current is either an induced electric field or a Lorentz force exerted by the magnetic field depending on whether you’re looking at the problem from the point of view of the conductor or the magnet.

So it’s safe to say that light moves at a constant speed because Maxwell tells you so. However, this explanation is less convincing if you don’t know about the wave equation. Indeed, if you were really skeptical about relativistic effects like length contraction and time dilation, you might be tempted to say “Maybe the relativistic corrections should be made to Maxwell’s theory, rather than to Newtonian mechanics?”. So I want to offer two more explanations: one is a heuristic about waves (essentially saying that Maxwell’s predictions are what you should have expected as a reasonable, fin-de-siècle physicist) and one is De Sitter’s experiment.

2. Light exhibits many behaviours which are analogous to sound waves or water waves. For example, it refracts when it passes through different media, it diffracts when it passes through a grating, and it can be made to interfere with itself (no giggling). Sound waves in air travel at 768 miles per hour, regardless of whether they’re produced by someone singing in the bath or by a passing low-flying jet (I say low-flying because the speed of sound does depend on the air pressure, which is lower at high altitudes). This is precisely the assumption we’re making about light! But because the speed of sound is so much slower than that of light, it’s easier to check this constancy in experiments. Indeed, sound is so slow that it’s possible for a very fast aeroplane to fly faster than 768mph, in which case it is travelling faster than the sound it is producing, giving rise to a massive concentration of sound wave-fronts moving along behind the jet (heard as a “sonic boom”). So we really do expect that waves satisfy assumption 1! Given that light behaves in many ways like a wave, it’s a reasonable assumption to make of light. Of course, explanation 2 is just explanation 1 in disguise, without the mathematical dressing.

3. So much for theory and expectation: can we do an experiment to determine whether the speed of light is constant or if it depends on the velocity of the light source as Ritz conjectured? De Sitter (1913) explains a very simple experiment which supports the hypothesis that light moves at a constant speed.

Consider a distant binary star system (two stars, A and B, orbiting each other) and assume that the orbits of the stars obey Kepler’s laws (so they trace out ellipses). Assume Ritz’s theory so that the speed of light depends on the velocity of the star. When star A is moving towards us (at speed v) in its orbit, the light it emits in our direction will be moving at speed c+v. When it is moving away from us (also at speed v), the light it emits in our direction will be moving slower, with speed c-v. Therefore the motion will appear very nonuniform: the star will seem to speed up as it comes towards us and slow down as it moves away. This is not consistent with what we observe in practice, which is uniform Keplerian motion. This means that Ritz’s theory cannot be an accurate description of the motion of light.

This is an incredibly simple explanation. There are many ways in which you could criticise this argument (maybe the motion we see is the result of Ritz’s theory and highly eccentric elliptical orbits?) and he gives a more detailed argument with reference to specific binary systems in a follow-up paper. In this follow-up paper he phrases the conclusion slightly more conservatively: one can put an upper bound on the dependence of the speed of light on the velocity of its source using astronomical observations of binary systems according to the argument sketched above.

Of course, there are many other experiments which confirm the predictions of special relativity, but this one has the advantage that it only requires you to have a good telescope rather than some complicated configuration of interferometers. It’s also the simplest imaginable experiment you could design to directly test the constancy of the speed of light: essentially racing lightbeams against one another!

REFERENCES

De Sitter, W. “A proof of the constancy of the velocity of light”, Proceedings of the Royal Netherlands Academy of Arts and Sciences, 1913, 15 II: 1297-1298, https://en.wikisource.org/wiki/A_proof_of_the_constancy_of_the_velocity_of_light

De Sitter , W.”On the constancy of the velocity of light”, Proceedings of the Royal Netherlands Academy of Arts and Sciences, 1913, 16 I: 395-396, https://en.wikisource.org/wiki/On_the_constancy_of_the_velocity_of_light

Einstein, A. “Zur Elektrodynamik bewegter Körper”, Annalen der Physik (1905) 322 (10): 891–921. A 1920 English translation by Meghnad Saha is available at https://en.wikisource.org/wiki/On_the_Electrodynamics_of_Moving_Bodies_(1920_edition)

Michelson, A. A.; Morley, E. W. “On the Relative Motion of the Earth and the Luminiferous Ether”. American Journal of Science (1887) 34: 333–345, https://en.m.wikisource.org/wiki/On_the_Relative_Motion_of_the_Earth_and_the_Luminiferous_Ether

Ritz, W. “Recherches critiques sur l’Électrodynamique Générale”, Annales de Chimie et de Physique (1908) 13: 145–275. http://gallica.bnf.fr/ark:/12148/bpt6k349439.image.f143