3.5: Waves and Aether

A Bit About Waves

The first postulate seems innocuous enough, and perhaps it even seems intuitive. But this business about every observer measuring the same speed for a beam of light didn't sit well with many physicists at the time Einstein proposed it. To see why, and to fully understand the implications of this being true, we need to review a little bit about waves. We will avoid a detailed discussion about waves other than to briefly summarize some of the properties most important to the subject at hand.

Effect of medium

Waves are phenomena which transfer energy through space by the means of a self-propagating disturbance of a medium (the stuff that is "waving"). A wave on a string carries energy along because a piece of the string displaces, which pulls on an adjacent piece of string, displacing it, and so on. Surface waves on water and sound waves through air work in a similar manner. What these all have in common is that medium itself doesn't travel with the energy (it moves with the disturbance), and the speed with which adjacent particles in the medium interact with each other is a property of the medium. Put another way, the speed of a wave is exclusively a function of the properties of the medium. A wave on a string travels faster when the string is more taut, and slower when the string is made is more dense, for example.

Superposition

When two waves traveling through the same medium encounter each other, the effects they have on the medium are additive at that the same point in the medium. So if one wave displaces a string by \(y_1\) at some position \(x\) on the string, and another wave displaces the string by \(y_2\) at the same position, then the total displacement of the string with the two waves present at \(x\) at the same time is \(y_1+y_2\). This additivity property is called superposition, and it has the particularly interesting feature that two waves can actually cancel each other entirely (a phenomenon called destructive interference), if the two waves happen to be displacing the medium equal amounts in opposite directions.

If a wave happens to be harmonic (the medium displacement as a function of position and time is a sine wave), then this destructive interference occurs when two identical waves are out of phase by \(\pi\).

Figure 1.1.2 – Destructive Interference of Two Harmonic Waves

The Michelson-Morley Experiment

The crux of the problem for those originally opposed to Einstein's assertion that the speed of light is the same when measured by any observer is that this is not true of other waves. If one moves through the air into an oncoming sound wave, that sound wave is moving faster relative to that person than relative to someone stationary in the air. The point is that, as stated above, the speed of a wave is entirely determined by the medium, and this speed is relative to that medium. So if an observer moves relative to a medium, then the relative speed of waves propagating through that medium can change.

So we are left with the question: What medium is disturbed as a light wave passes through it, and can't we see a change in the speed of light if we just move relative to this medium? This was an open question at the time of Einstein, and was very puzzling for a couple reasons. The first is that the theory of light did not require the existence of any medium at all.

But far more puzzling than this was the fact that no one could seem to come up with any experimental evidence of the existence of a medium for light (which they referred to as the luminiferous aether). The most convincing null result of such a search was performed by two American physicists Albert Michelson and Edward Morley, who employed a device called an interferometer, the basics of which are still used today for countless applications.

The basics of an interferometer work like this: A single light beam is split into two separate beams. As they came from the same source, they are in phase with each other, but if we send them on separate journeys, and then bring them back together, they may no longer be in phase. For example, one of the beams may travel farther than the other. If the difference in their phases when they get back together is just right, they will destructively interfere with each other.

[For the two beams to behave this way, the original beam needs to be somewhat coherent, meaning that parts of the light near each other are in phase.  Nowadays we easily achieve this artificially with lasers (sunlight is also quite coherent), but fortunately the degree of coherence achievable in a lab in the late 1800's was sufficient to successfully perform this experiment.]

The ingenuity of the MM interferometer is that it splits the beam of light and sends the two pieces on journeys that are equal distances, but are at right angles to each other. If the laboratory is stationary in the aether, then the beams come back to the detector in phase, and do not cancel each other out:

Figure 1.1.3 – Laboratory Stationary in Aether

If this device is moving through aether (let's say in a direction parallel to one of these directions), then one of the arms of the interferometer moves parallel to the direction of the light during the journey, while the other moves perpendicular to it. These two journeys are not equal in distance, which means the beams can enter the detector out of phase:

Figure 1.1.4 – Laboratory Moving through Aether

We can do a bit of math to determine the difference in the distance between these two journeys. The only difference between the paths of the two beams comes after they encounter the half-silvered mirror the first time, and before they encounter it the second time, so we focus on this portion of the process for both of them:

Figure 1.1.5 – Distance Traveled by the Transverse Beam

The transverse beam (the one perpendicular to the laboratory's motion) follows a diagonal path in both directions, the distance of which can be determined from the Pythagorean Theorem:

\[s = 2\sqrt{L^2+\left(v\Delta t\right)^2}\]

But the distance traveled by the beam also equals the speed of light through the aether \(c\) multiplied by the time elapsed, so we get:

\[s = 2c\Delta t \;\;\;\Rightarrow \;\;\; s = 2\sqrt{L^2+\left(v\dfrac{s}{2c}\right)^2} \;\;\;\Rightarrow \;\;\; s = \dfrac{2L}{\sqrt{1-\frac{v^2}{c^2}}}\]

Figure 1.1.6 – Distance Traveled by the Longitudinal Beam

The longitudinal beam (the one parallel to the laboratory's motion) stays on a straight line, but the time it spends going in each direction is not the same, so we compute these times separately then use them along with the speed of light to get the full distance this beam traverses during this part of the process:

\[\left. \begin{array}{l} s_1 = L+v\Delta t_1 = c\Delta t_1 && \Rightarrow && \Delta t_1 = \dfrac{L}{c-v} \\ s_2 = L-v\Delta t_2 = c\Delta t_2 && \Rightarrow && \Delta t_2 = \dfrac{L}{c+v} \end{array} \right\} \;\;\; \Delta t_1 + \Delta t_2= \dfrac{2LC}{c^2-v^2} \;\;\;\Rightarrow\;\;\; s = c\left(\Delta t_1 + \Delta t_2\right) = \dfrac{2L}{1-\frac{v^2}{c^2}}\]

We can see from these two results that the two beams travel different distances. The denominator for each case is a fraction that is less than 1, so the square root is the larger denominator.  This means that the longitudinal path (with the smaller denominator) is longer than the transverse path. The difference between these paths can be expressed by a multiplicative factor that depends upon the speed of the laboratory through the aether and the speed of light:

\[ s_{longitudinal} = \dfrac{1}{\sqrt{1-\frac{v^2}{c^2}}} s_{transverse} \]

We therefore find that if the laboratory is moving through the aether at just the right speed, then the difference in distances traveled will equal half the length of a wave, and the two beams will cancel at the detector, resulting in darkness. In fact, this is not the only case that will give this result. If the difference in distance traveled is \(\frac{3}{2}\) of a full wave, then again the troughs and crests of the waves will match up, making destructive interference. Moreover, even if there is not complete destructive interference, every offset of the waves will result in some variation in the brightness of the light.

So given that the earth is moving through space, and is also rotating, one would expect that it is not difficult to find some evidence that the aether exists, and that we are moving through it. But try as they might, Michelson and Morley found no such thing. Many explanations were offered for the problem (many before the experiment was even performed), such as the aether being "dragged" by the earth, so that it was stationary around us here on the surface (logical arguments based on starlight observations and additional experiments on starlight proved this to be false). Einstein alone showed the courage to just discard the existence of the aether altogether, no matter how nonsensical the consequences might seem to be at first.