While we wait, it is worth noting that in June 0f 2011 a group of researchers performed an experiment to see if light itself could move faster than light!  In particular, the scientists used a little known optical phenomenon known as an optical precursor to see if individual photons might travel faster than while propagating in a material.  In the end, the experiment suggests that these single photons did not in fact violate Einstein’s speed limit, though the results still got a significant amount of press.

The response of many physicists to the news was a collective, “Well, duh!”  The prevailing attitude seems to have been: “What’s so interesting about proving something we already knew?”  In this post I’d like to explore that question a little bit, and explain how some uncertainty remains about the behavior of light in materials.  Along the way, we’ll introduce the fascinating phenomenon of precursors, and see how they can be used both to probe the nature of matter as well as the nature of light.

To start, we need to understand a bit about how light and matter interact with one another, and how this relates to the speed of light in matter.  This is a surprisingly tricky subject, which I’ve covered several times previously on this blog; we’ll review the important aspects, with a few additional details.

Let’s talk about how we measure the speed of an object first.  If we’re looking at the motion of a rigid object, like a speeding car or a thrown baseball, the speed can be determined simply by measuring how much time it takes for an object to travel a distance .  The speed is simply the distance divided by the time:

.

There’s a small subtlety to this definition: cars and baseballs are extended objects!  To accurately measure an object’s speed, we have to be consistent in how we define its position.   For a car driving down the road, for instance, we should do all measurements of its position from a fixed position, such as the front bumper, to measure the speed.

But what do we do when the object doesn’t have a fixed position on it?  For example, what is the best way to measure the speed of a hurled bucketful of water?

Water is, of course, not a rigid body, and the volume of water changes shape and spreads out as it travels.  It is strange to realize, after some thought, that a blob of water like this doesn’t really have a well-defined speed.  We arrive at the same problem with a pulse of light traveling in matter: in general, a light pulse has no “fixed point” upon it, and it can change shape as it travels.  There isn’t a single definition for the speed of light in matter that is useful in all cases.

There are three different descriptions of the speed* of light used in physics, each with its own significance and usefulness.  We can introduce each of these by looking at similar definitions for our bucketful of water.

We imagine that our volume of water may be broken up into a collection of well-defined droplets.  The most straightforward and thorough thing to do is to measure the speed of each individual droplet: the velocity of each droplet will be referred to as its phase velocity.  If there is only one or a handful or drops, this set of numbers will be a good description, but for a large number of droplets this collection of numbers will not necessarily tell us anything about the motion of the entire bulk of the water.

A better option is to define the speed of the body of water by its center of mass.  This definition, referred to as the group velocity, works quite well in many cases, but can also be misleading.  If we’re interested, for instance, in the time it takes for water to first reach its target (say a fire), there may be droplets well ahead of the center of mass and the use of the group velocity can overestimate the time.

We may also choose to define the speed of the body of water as the speed of the fastest drop in the collection; this definition is called the signal velocity.  One problem with the signal velocity is the opposite of that of the group velocity: if we need to know when the bulk of a body of water reaches a fire, the signal velocity will underestimate the time.

Physicists use roughly analogous definitions to characterize the speed of light in matter.  Instead of “droplets”, a pulse of light can be decomposed into a collection of waves of different frequencies (colors).  It can be shown that a narrow pulse of light necessarily consists of a very broad spectrum (range of colors), while a very broad, slowly varying pulse has a narrow spectrum.  Also, and significant for our later discussion, a pulse with a very sharp edge necessarily has a very broad spectrum, as well.  This idea is illustrated crudely below:

Atoms respond differently to light of different frequencies, in a phenomenon known as dispersion.  The result of this is that some frequencies of light are absorbed more than others, and in general every frequency of light travels at a different speed.  The term phase velocity is used to refer to the different speeds of each frequency of light.  For a pulse narrow in frequency (like the upper one above), this can sometimes serve as an approximation to the overall speed of light.  However, in many circumstances this phase velocity can be faster than the vacuum speed of light , and therefore does not accurately represent the speed of the pulse.

Because of difficulties with the phase velocity, the group velocity is the standard method of defining the velocity of a pulse in a medium.  It may be considered, in essence, the speed of the “center of mass” of the pulse.  However, if the shape of the pulse is highly distorted as it propagates, the group velocity can lose all useful meaning.  Also, it has been shown that the group velocity can be greater than the vacuum speed of light, a result that generated quite a bit of controversy at first.

The absolute speed of light in a medium is the signal velocity, which is loosely defined as the speed at which information can be conveyed in a medium, or the fastest possible speed that the front of a light signal can travel.  This is generally thought to be no faster than the vacuum speed of light , in agreement with Einstein’s special theory of relativity.

But is the fastest speed of light in matter?  It seems likely, but a few gaps in our knowledge leave it a partially open question.  The quantum-mechanical nature of light-matter interactions leaves open the possibility that some “quantum weirdness” allows Einstein’s speed limit to be broken in a subtle way.

A more concrete concern involves our understanding of how light propagates in matter.  It is well-known that the “causality” of a light signal — and the absolute speed of light — is built into the frequency-by-frequency response of light to matter.  If one knows the speed of light, and the absorption properties of light, for every frequency, one can determine whether or not is the top speed.

But we don’t know these properties for every frequency!  In particular, we don’t know exactly what happens to light in matter for arbitrary high frequencies (frequency approaches infinity) or for arbitrary low frequencies (frequency approaches zero).  There is always the possibility that some small violation of relativity is “hiding” in these extreme frequency ranges.

This is where the idea of optical precursors becomes useful.  Let’s consider the temporal and spectral properties of another pulse, with a square envelope:

This pulse has a large central peak in frequency, but also has long frequency “tails” that stretch out, to some degree, to arbitrarily high frequencies (to infinity) and arbitrarily low frequencies (to zero).  These tails arise due to the instantaneous start and end of the pulse in time: the rapid rise and fall that give the “square pulse” its name.

At the beginning of the 20th century, physicists Arnold Sommerfeld and Léon Brillouin independently performed theoretical work to investigate what happens to such sharp-edged pulses as they propagate a long distance through matter.  In 1914, they each published results** showing that the main body of the pulse (traveling at the group velocity) is preceded by faster-moving waves, now known as “precursors”.  The precursors are general broken into two types: the Sommerfeld precursor, which actually travels at the vacuum speed of light , and the Brillouin precursor, which travels at the speed of light , where is the refractive index of the medium at zero frequency.  A crude illustration of the arrangement of precursors is shown below:

The most remarkable thing about the precursor fields is that they are very weakly absorbed by matter, much less so than the main body of the pulse.  If we send our original square pulse through a very thick piece of absorbing material, the main body will be almost completely absorbed while the precursors will be absorbed only weakly.  This fact has led to some researchers suggesting that precursors could be used to “see” through normally opaque objects, like clouds.

It took quite a few years for Sommerfeld and Brillouin’s theoretical work to be confirmed.  The first observation of precursors was performed in 1969, using microwaves***.  Since then, they have been observed for a variety of wavelengths, and have been observed even in “mundane” materials such as water****.

What is the origin of these precursors?  I wasn’t able to find a simple explanation for them in the literature, perhaps not surprising in terms of their mathematical complexity.  I would like to introduce a (hopefully accurate) description of them, in what I will refer to as the “pendulum slapping” model!

We have all been taught, at some point in our education, the planetary model of the atom.  In this model, an atom consists of a positively-charged nucleus surrounded by one or more orbiting negatively-charged electrons, much like the planets in the solar system orbit the Sun:

With our modern understanding of quantum mechanics, we know that this model isn’t a terribly good one: electrons act much more like “clouds” of negative charge centered on the nucleus, and don’t “orbit” in a well-defined sense.  However, the planetary model does get one thing right: electrons have a characteristic frequency associated with their motion, much like the planets each orbit the Sun with a characteristic frequency (the Earth orbits the Sun with a frequency of 1/365 days).  In a very loose sense, we can picture atoms (and molecules) as collections of oscillating electron pendulums, each vibrating with a characteristic frequency:

What happens when we apply an oscillating electric field (i.e. a light wave) to our pendulum?  The electric field applies a force to the electron, driving its oscillation.

The effectiveness of the electric field in moving the pendulum depends on how close its frequency is to the pendulum frequency.  The pendulum will have the largest swing when the electric field matches its own frequency; this is like a child pumping his legs on a swing to increase his motion.  Any other frequency of the electric field will be less effective in moving the pendulum, but pretty much every frequency will force some oscillation in it.  The energy that is imparted to the pendulum is lost by the electric field, and consequently the light wave; this is the origin of the absorption of light by matter.

There are two extreme limits of frequency in which no energy is transferred to the pendulum: the limit of infinite frequency and the limit of zero frequency.  We can visualize what happens in these cases by imaging that we are “slapping” a pendulum with our hands on either side (or you can try it yourself, if you have a pendulum). When we slap a pendulum at a frequency much higher than its characteristic frequency, a push on the left is almost immediately canceled by a push on the right — the pendulum doesn’t move, and we’ve transferred no energy to it!  In the limit of very low frequency, the pendulum moves, but doesn’t oscillate: we “lift” it to the right with our left hand, then slowly lower it back to its rest position and then “lift” it to the left with our right hand.  At no point does the pendulum oscillate freely, and therefore we have transferred no energy to it.

These limits, in which no energy is transferred to the pendulum, are the origins of the precursors.  We have seen that a square wave has components of arbitrarily high frequency and arbitrarily low frequency, and these components are only very weakly absorbed by the medium.  The high frequency components of the wave result in the Sommerfeld precursor, and the low frequency components of the wave result in the Brillouin precursor.

For tests of the absolute speed of light, the Sommerfeld precursor is of particular interest, because (a) it tests the whether a violation of relativity is “hiding” at high frequencies, and (b) the speed of a precursor is theoretically supposed to be equal to , making “faster than light” violations relatively easy to spot.

All of this brings us back at last to the June paper in Physical Review Letters on the “Optical precursor of a single photon”, by a research group in Hong Kong.  A photon is a single quantum particle of light; all of the discussion of precursors up to this point have concerned pulses consisting of many, many photons.  Two questions arise when considering precursors and single photons:

(1) Do precursors even exist for a single photon?  One would naturally be inclined to say “yes”, but it may be that, on a quantum level, precursors inherently involve the interaction of many photons at once.

(2) Can single photons travel faster than ?  One would be inclined to say “no” in this case, but again the behavior of single photon precursors (if they exist) might be subtly different than a group of many photons.

The Hong Kong researchers investigated these possibilities by producing coupled pairs of photons using the following experimental configuration (adapted and simplified from the article):

The fundamental components of the system are a pair of magneto-optical traps (MOT) containing Rubidium atoms.  Magneto-optical traps use light and magnetic fields to localize a group of atoms and cool them to a low temperature, forming a low-density medium.  The first MOT is excited by a pair of lasers — a pump beam and a coupling beam — and through a complicated light-matter interaction pairs of photons are produced.  The physics of this production is too complicated to discuss here, but the result is a higher frequency “Stokes photon” and a lower-frequency “anti-Stokes photon”.  These photons are produced at the same time and are therefore correlated in time.

The Stokes photon is picked up by detector 1 and sends a signal to a function generator, which triggers an electro-optic modulator that the anti-Stokes photon passes through.  This modulator “chops off” the front end of the anti-Stokes photon signal, producing a sharp-edged pulse necessary for precursor generation*****.  This sharp-edged pulse propagates into the second MOT, in which its waveform is expected to take a precursor shape.

It is essential for this experiment that a single anti-Stokes photon be measured at a time; after the second MOT, the light signal is split by a beam-splitter and sent to a pair of detectors, detectors 2 and 3.  Because a single photon can only arrive at a single detector, the simultaneous tripping of both detectors implies the presence of more than one photon and the event is thrown out.  The time of arrival of the single anti-Stokes photons can be tallied, producing a profile of the average behavior of the photons.

The coupling laser illuminating the second MOT could modify the optical properties of the Rubidium atoms into one of two configurations.  In the first configuration, the absorption properties of the medium can be completely suppressed, in a technique called electromagnetic induced transparency (EIT).  With EIT, both the main body of the pulse and the precursor signal make it through the MOT and can be measured by the detectors.   Furthermore, the main signal is slowed considerably in speed and is separated in time from the optical precursor.  In the second configuration, the group velocity of the main signal is faster than , i.e. the group velocity is “superluminal”.  However, this main signal is highly absorbed.

What were the results of the experiments?  The researchers measured the speed of the main body of the pulse and the precursor for both MOT configurations.  The precursor was clearly visible in both cases, and its speed was found to be exactly , regardless of the MOT behavior.  Therefore no true “faster than light” behavior was observed, even when the group velocity was greater than the vacuum speed of light.

An important aspect of this result is a partial answer to a debate in quantum information theory — how fast does a single photon transmit information?  As we’ve noted, there are multiple definitions of the speed of light in matter, and it has remained a bit of a mini-mystery (at least to some) which of these described the speed at which information is transmitted.  The observation that a photon does have a precursor signal suggests that single photons can travel at the vacuum speed of light in matter, at least under the right conditions.

The result isn’t as earth-shattering as the possible discovery of superluminal neutrinos, but it does highlight a number of unusual aspects of optics, and further strengthens our understanding of both relativity and quantum mechanics!

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* In physics, “velocity” is used to refer to the vectorial motion of an object: not only how fast it is going, but in what direction.  ”Speed” is used to refer to the magnitude of velocity, or simply how fast the object is going.  We use the terms interchangeably here in the sense of “speed”.

** A. Sommerfeld, Ann. Phys. (Leipzig) 349 (1914), 177.  L. Brillouin, Ann. Phys. (Leipzig) 349 (1914), 203.

*** P. Pleshko and I. Palócz, “Experimental observation of Sommerfeld and Brillouin precursors in the microwave domain,” Phys. Rev. Lett. 22 (1969), 1201.

**** S-H. Choi and U. Österberg, “Observation of optical precursors in water,” Phys. Rev. Lett. 92 (2004), 193903.

***** Actually, the anti-Stokes photon will already have a precursor signal due to its propagation in MOT 1!  This precursor is also chopped off by the EOM.

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Zhang, S., Chen, J., Liu, C., Loy, M., Wong, G., & Du, S. (2011). Optical Precursor of a Single Photon Physical Review Letters, 106 (24) DOI: 10.1103/PhysRevLett.106.243602

A great case study of this is Einstein’s special theory of relativity, introduced in 1905.  Einstein’s groundbreaking work transformed mankind’s perceptions of space and time, provided answers to puzzling problems and led directly to other major discoveries, including the harnessing of nuclear energy.  However, Einstein’s revelations were preceded by some twenty years of gradual progress, during which time researchers put forth tantalizing hypotheses that came closer and closer to the truth.

One such discovery was made in 1889 by George FitzGerald.  To explain a seemingly incomprehensible experimental result, he suggested that objects in motion shrink along their direction of travel.  In this post, we will discuss what is now known as the FitzGerald-Lorentz length contraction and explain how FitzGerald fell short of the astonishing ideas that would be conceived by Einstein.

The history behind the confusion and FitzGerald’s solution stretches all the way back to the beginning of the 19th century.  Since the publication of Newton’s Opticks in 1704, scientists had been convinced that light consisted of a stream of particles, referred to as corpuscles.  In the early 1800s, however, Thomas Young demonstrated that light possesses wavelike properties with his famous double-slit experiment, and within the next decade most researchers had come around to idea that light is a wave.

This wave nature of light introduced its own problems, however.  All waves known at that time were vibrations of some sort of material medium — water waves are oscillations of water, sound waves are oscillations of air, and waves on a string are oscillations of the string.  What were light waves oscillations of?  A hypothetical substance known as the “aether” was introduced: invisible, non-tangible and only detectable through the light waves excited in it.  This aether model seemed to explain a number of curious observations regarding the propagation of light, and by the late 1800s it was an unchallenged part of physics.

The supposed existence of the aether became more and more vexing as time went on, however.  If there was this material, permeating all space and all matter, it should be detectable somehow, right?  Some scientists attempted to deduce the properties of the aether through theoretical reasoning, but these attempts also felt unsatisfying.

The best strategy that arose for at least indirectly observing the aether was to measure changes in the speed of light due to the motion of the Earth.  The speed of light was presumably constant with respect to the fixed aether, but an observer moving in the aether could potentially see a faster and slower speed.  The effect would be similar to measuring car speeds while driving down the freeway: cars driving in the same direction are moving relatively slower, while cars driving in the opposite direction are moving relatively faster.  Similarly, light moving parallel or anti-parallel to the Earth’s motion would, according to the aether theory, be measured as moving slower or faster than expected, respectively.

The Earth was already known to move at 30 kilometers/second in its path around the Sun. It would have to be moving, at some point in its orbit, at least at this speed relative to the aether. Even though nobody knew how to directly measure the aether, one could presumably indirectly detect it by measuring changes in the speed of light as the Earth moves.  These changes would be very small, however, considering the speed of light is 3,000,000 kilometers/second, and would require a very sensitive measurement apparatus.

Such a device was built and employed by Albert Michelson and Edward Morley in 1887, in what would quickly become known simply as the Michelson-Morley experiment.  The experiment uses the interference of light waves to measure changes in the speed of light, and the setup is illustrated below:

A light beam is incident from the left. It hits a half-silvered mirror which is inclined at a 45 degree angle, which splits the beam into two parts: one which is reflected, one of which is transmitted. Each fraction of the beam travels to a mirror and back, and at the half-silvered mirror they are recombined and their sum is projected onto a screen. When the light recombines from each arm of the interferometer, we get an interference pattern on the screen.

To understand how this allows us to measure the motion of the Earth in the hypothetical aether, we consider a simple analogy of boats in a river.*  In this case, the boats represent the light traveling through the two arms of the interferometer, and the flowing river represents the aether moving relative to the experiment:

Each boat has a top speed of c, and the river flows to the right with velocity v. Boat 1 travels a distance d from point A to point B, and then returns, while boat 2 travels a distance d from point A to point C, and then returns. How long does it take each boat to return to the starting point?  Boat 2 moves very fast with the current on the way to C, but moves very slow against it on the way back.  Boat 1 must steer partly upstream in order to cancel the motion of the river, and its speed across the river is somewhat reduced:

The detailed mathematics is presented at the end of this post, but the net result is that boat 2 actually takes a longer time to make its round trip than boat 1.  This difference can be measured using the interference of light waves.

The problem, as it seemed to be at the time, is that Michelson and Morley could detect no difference in travel times between boats… ahem… light beams!   According to their experiment, the speed of light seemed to have a fixed value, independent of the relative motion of the Earth with respect to the aether.  As you might imagine, this presented a problem for the already somewhat ailing aether theory, and researchers scrambled to explain what was going on.

The first good guess came from George FitzGerald (1851-1901), an Irish professor of natural science at Trinity College, Dublin.  In 1889, he published a short paper in Science titled “The Ether and the Earth’s Atmosphere” which came up with a very straightforward explanation of the Michelson-Morley experiment: he suggested that the experimental apparatus might be shrinking in the direction of motion!

The full text of FitzGerald’s letter**, which is very short, is reproduced below:

I have read with much interest Messrs. Michelson and Morley’s wonderfully delicate experiment attempting to decide the important question as to how far the ether is carried along by the earth.  Their result seems opposed to other experiments showing that the ether in the air can be carried along only to an inappreciable extent.  I would suggest that almost the only hypothesis that can reconcile this opposition is that the length of material bodies changes, according as they are moving through the ether or across it, by an amount depending on the square of the ratio of their velocity to that of light.  We know that electric forces are affected by the motion of the electrified bodies relative to the ether, and it seems a not improbable supposition that the molecular forces are affected by the motion , and that the size of a body alters consequently.  It would be very important if secular experiments on electrical attraction between permanently electrified bodies, such as in a very delicate quadrant electrometer, were instituted in some of the equatorial parts of the earth to observe whether there is any diurnal and annual variation of attraction, — diurnal due to the rotation of the earth being added and subtracted from its orbital velocity; and annual similarly for its orbital velocity and the motion of the solar system.

We can illustrate FitzGerald’s idea quite readily.  Boat 2 takes longer than boat 1 to make its round-trip; suppose, however, that the force of the moving water scrunches up the shoreline, making the distance between A and C shorter:

So the marker buoy at C ends up getting pushed to the point C’, and boat 2 has to move a shorter distance d’ instead of distance d.  If d’ is just the right value, the two boats will take the same amount of time to perform their round trips.

This is the idea behind FitzGerald’s length contraction.  Since the aether was (in 1800s physics) the medium that communicated electromagnetic waves, and atoms and molecules are held together by electricity, it was reasonable to assume that moving through the aether might apply an electric pressure of sorts, causing an object to shrink along the direction it is moving.  A full-size car, driving fast enough through the aether, would shrink to a compact!

Obviously this shrinkage would be generally very small, only becoming appreciable when one’s speed with respect to the aether approaches the speed of light.

FitzGerald even suggests a way to test his hypothesis.  If electrical forces depend on one’s motion through the aether, it should be possible to see daily (diurnal) variations due to the Earth’s rotation: during half of the day, the Earth’s rotational motion will add to its orbital motion, and during the other half of day, it will subtract.  Also, on a yearly basis, during half of the year the Earth’s orbital motion will add to the motion of the solar system through the aether, and during the other half of the year it will subtract.

Nobody seems to have taken up FitzGerald’s experimental suggestions, though Lord Rayleigh attempted to demonstrate this shrinkage using another clever optical technique – and failed.  In 1895, Hendrik Antoon Lorentz developed the idea of length contraction even further, developing a detailed set of equations that describe the properties of an aether that shows no change in the speed of light; the idea of length contraction is therefore known today as FitzGerald-Lorentz length contraction.  Lorentz, at least at first, viewed length contraction as an inevitable consequence of Michelson and Morley’s work (translation from Larmor’s Aether and Matter, 1900):

However extraordinary this hypothesis may appear at first sight, it must be admitted that it is by no means gratuitous, if we assume that the intermolecular forces act through the mediation of the aether in a manner similar to that which we know to be the case in regard to electric and magnetic forces.  If that is so, the translation of the matter will most likely alter the action between two molecules or atoms in a manner similar to that in which it alters the attraction or repulsion between electrically charged particles.  As then the form and the dimensions of a solid body are determined in the last resort by the intensity of the molecular forces, an alteration of the dimensions cannot well be left out of consideration.

The solution to the mystery was provided at last by Albert Einstein in 1905, in an amazing and genuinely paradigm-shifting paper.  Whereas FitzGerald and others focused on the aether as an “absolute” property of nature around which to construct the laws of nature, Einstein instead declared that the speed of light is the “absolute”: every observer, regardless of what their state of (constant) motion, measures exactly the same speed of light.  Einstein coupled this postulate with the postulate of relativity: the laws of physics are the same for every observer moving at constant velocity.

Einstein’s interpretation of the Michelson-Morley experiment is quite simple: since the speed of light is the same for every observer, the light traveling in each arm of the interferometer is simply traveling at the same speed, meters/second.  No motion of the Earth will change these values.  The laws of physics for an observer moving at a constant velocity are the same as those for an observer standing still.  This also results in the observation that there is no such thing as “standing still”, in an absolute sense — there is no experiment that can be done to measure one’s “absolute motion”.

As a side effect, Einstein’s theory actually includes a form of length contraction, but of a fundamentally different nature than FitzGerald’s.  Let us imagine a modified version of the Michelson interferometer, which fires simultaneously two ultrashort pulses each of which propagates a distance  to a mirror and returns to a detector at the point of origin. If both pulses arrive at the same time, a siren goes off to unambiguously announce their simultaneous arrival. From the point of view of the laboratory, each pulse travels the same distance at the speed of light  and returns, firing off the siren.

Now let us consider this from the point of view of an observer who sees the experiment moving past him at a velocity :

The new observer sees the light heading towards mirror 1 on a diagonal trajectory, and the light heading towards mirror 2 must chase the moving mirror, taking longer to reach it.  According to Einstein’s relativity, however, and distinct from the aether/boat case discussed earlier, the new observer measures the velocity of light to be  for the light moving along both paths!

Because the siren goes off, we know that the light pulses arrive back at the detector at the same time.  As we have seen from the boat example, if the distances to the two mirrors are equal, it should in fact take a longer time for the light to return from mirror 2.  The only conclusion that the second observer can draw is that the distance between the source and mirror 2 has contracted, i.e. length contraction has occurred!

Herein lies the difference between the FitzGerald form of length contraction and the Einstein form.  In the FitzGerald version, both the first and second observer can agree that the experiment is moving.  The observer moving with the experiment would in principle be able to measure a change in the electrical forces between molecules, demonstrating a motion in the aether.  In the Einstein version, the two observers agree that the speed of light is c, but disagree on the distances that the light has to travel.  The ideas of absolute motion and absolute distance for every observer have been replaced by the idea that the laws of physics are absolute for every observer, and the details such as space (and time) are arguable!

Einstein’s theory represented a huge change in mankind’s perception of nature and the universe.  Space and time were relegated from absolute quantities to relative ones depending on an observer’s state of relative motion.  Nevertheless, others were already anticipating Einstein’s discovery before him.  Lorentz, who was a proponent of the FitzGerald contraction hypothesis, also derived a mathematical transformation between observers — the Lorentz transformation — that left the speed of light constant for all observers.  This transformation is now a part of Einstein’s relativity.  Other scientists such as Ernst Mach and Henri Poincaré had already in the late 1800s begun to dismiss the notion of absolute space and time, and the aether along with it, and propose a new theory of relativity.  In fact, one distinguished scientist stubbornly refused to accept any real significance in Einstein’s relativity work.

Einstein, however, was the one to bring all the scattered musings of the different scientists together into a new principle of physics.  Where the others were inching towards a new theory of relativity, Einstein leapt towards it wholeheartedly.  It was a great discovery, but the earlier efforts of others such as FitzGerald show that Einstein’s work was part of a greater effort to understand the nature of space and time.

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* I’ve adopted the boat discussion from an earlier post on measuring the speed of light, with some updates.

** G. FitzGerald, “The Ether and the Earth’s Atmosphere,” Science 13 (1889), 390.

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The math of the time-delayed boat problem:

Each boat has a top speed of c, and the river flows to the right with velocity v. Boat 1 travels a distance d from point A to point B, and then returns, while boat 2 travels a distance d from point A to point C, and then returns. How long does it take each boat to return to the starting point? We can use a little geometry and velocity addition to determine this. Boat 2 will have a total velocity of c + v on the way out to point C, and will have a total velocity c – v on the way back. Boat 1′s velocity is slightly more difficult to calculate; in order to travel straight across the river, it must have a velocity v against the river flow, and a total velocity of c. Using some simple geometry,

Boat 1 must have a speed moving from A to B, and an identical speed for the return trip. Since the amount of time it takes for a trip is the distance divided by the speed, we therefore find that the transit times for the boats are as follows:

,

.

The net result is that boat 2 takes a longer time to make its round trip than boat 1.

FitzGerald’s length contraction “fixes” the time discrepancy if the distance from A to C is shrunk to a value

.

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G. FitzGerald (1889). The Ether and the Earth’s Atmosphere. Science (New York, N.Y.), 13 (328) PMID: 17819387

Enter Conservapedia, the right-wing version of Wikipedia intended to combat the liberal bias in reality!  Over the past day, Twitter has been abuzz with tweets¹ on the Conservapedia page on “Counterexamples to relativity“, provides a list of 24 “points” that attempt to show the weakness of Einstein’s crazy ideas!

In my mind, perhaps the most despicable sort of denialism or crankery, however, is that which is based on some sort of political or religious ideology.  This is clearly what is going on here, and the author relies on a familiar form of rhetorical trickery known as the “Gish Gallop“: throw as many claims out there as possible, regardless of their validity, with the realization that most people will be swayed by the amount of “evidence”, and not look too closely at the details.

Looking at the “evidence”, it is clear that there isn’t a single point made that isn’t misleading, incoherent, or simply dishonest.  A person reading the Conservapedia post will be measurably more ignorant afterwards, and I get the distinct impression that this is what the author intended.

But never fear, dear reader!  I’m here to go through the list of some of the most entertaining assertions, and explain why they’re nonsense. Why bother, you ask?  For one thing, entertainment.  For another, there’s always a chance that someone may come across the Conservapedia entry and look for some sort of counterbalance… someone should write one!

One caveat: I can’t guarantee that the list I present will match the list on the Conservapedia page.  I saved the tweeted list, but after all the internet attention, it was reduced to four points.  Soon afterwards,  it reverted to the original list again.  There’s no guarantee that it will remain in its current form, though…

Let’s start with a few observations about Einstein’s relativity, which may be broken into the special theory of relativity, published in 1905, and the general theory, published in 1915.

The special theory of relativity is based on two postulates: 1) the laws of physics are the same for all observers moving at constant speed, and 2) the speed of light is the same for all observers.  The first point is the crux of “relativity”: there is no such thing as “absolute” motion of an object, and the laws of physics have the same form for any observer moving at constant velocity.  This statement goes back as far as Galileo, who realized that a person sitting within the depths of a moving ship has no local means of telling that they are in fact moving.  The second point is the one that results in all the crazy, counterintuitive notions.  In order for all observers, regardless of their motion, to agree on the value of the speed of light, traditional notions of space and time must be modified, and in fact we must consider space and time as interrelated entities.  From my point of view, a moving observers has clocks which run slow, while that observer will say that my clocks are running slow.  Built into the constancy of the speed of light is the fact that the speed of light is the “speed limit” of the universe, and nothing can be moved faster than that speed… with certain cosmological caveats.

The general theory of relativity adds gravity to the mixture.  The fundamental idea is that, locally, a gravitational force is indistinguishable from accelerated motion in the opposite direction.  The simple illustration of this is being in an elevator — when the elevator is accelerating up, you feel heavier.  When the elevator is accelerating down, you feel lighter.  From the theory of general relativity follows all sorts of weird stuff such as black holes and the possibility of wormholes.   In general relativity, matter is viewed as fundamentally distorting the shape of space and time.

With that in mind, let’s get to the list, but start with the introduction to it (I’m leaving out the hyperlinks and citations, but will refer to them when appropriate):

The theory of relativity is a mathematical system that allows no exceptions. It is heavily promoted by liberals who like its encouragement of relativism and its tendency to mislead people in how they view the world. Here is a list of 24 counterexamples: any one of them shows that the theory is incorrect.

The first sentence makes no sense!  It is not just a “mathematical system”, it is a physical theory that has been tested countless times and is used by experimental physicists on a daily basis.  I have a book in my office that is all about the analysis of the different experimental tests specifically of special relativity.  It is not clear what “allows no exceptions” means; physics is all about looking for “exceptions” to existing physical law, and relativity is not necessarily immune to this.  The basic tenets of both the special and general theories have been well established, however, and any new discoveries are expected to build upon them, not refute them.

I hate to go all Godwin early in this post, but replace “liberals” by “Jews” in the second sentence, and the sentence might as well have been written by a Nazi circa 1930s-era Germany.  Nazi scientists specifically rejected Einstein’s theory of relativity as “Jewish science”, and they founded their own theory of physics referred to as Deutsche Physik.  The result of their ideological and racist hubris was to cripple Germany as a scientific giant for decades.  For a good discussion of the poisonous effects of ideology on science, read John Grant’s awesome book, Corrupted Science.

The third sentence is also absurd: as I’ve said, the theories of relativity have been experimentally tested for decades, and have survived all tests.  Even if a discrepancy between theory and experiment is found, it does not invalidate the entire theoretical framework, but builds upon it.  Einstein’s theory of relativity built upon Newton’s earlier theory of relativity — Newton’s theory was not shown to be wrong, but rather incomplete.  Assuming that a new piece of evidence somehow invalidates a century of observations shows a complete lack of understanding of science and how it works.

Let’s get to the list:

1.  The Pioneer anomaly.

Humankind has launched a number of unmanned spacecraft that are on course to leave the solar system.  These craft, such as Pioneer 10 and 11, are continually slowing down under the influence of the Sun’s gravity, but observations in recent years suggest that they are slowing down slightly more than expected.  A number of explanations have been proposed to explain this effect, including observational errors and previously unobserved gravitational effects.  There is a small possibility of new physics — we have relatively few direct measurements of the effects of gravity over long distances and at slow speeds, and it has been suggested that the familiar Newtonian laws of gravity may be slightly different at large scales.

Notice that I didn’t mention relativity in that paragraph?  Though any new physics could potentially involve changes to relativity, the Pioneer anomaly doesn’t directly relate to relativity, and there’s no reason to say that this small effect in any way invalidates the long established theory.

2.  Anomalies in the locations of spacecraft that have flown by Earth (“flybys”).

This is really in essence a repeat of point #1!  In 1990, it was observed that the Galileo spacecraft, passing close to the Earth, experienced an unexplained change in speed as it went by.  Similar observations have been made of other spacecraft.  These anomalies are quite exciting for science, as they do offer the potential for new physics.  However, they don’t threaten to do anything to relativity other than perhaps modify it.

3.  Increasingly precise measurements of the advance of the perihelion of Mercury show a shift greater than predicted by relativity, well beyond the margin of error.

The first success of Einstein’s general relativity was providing an explanation for a previously-unexplained slow evolution in the motion of planet Mercury.  No reference is provided on Conservapedia for these so-called shifts “well beyond the margin of error”, but even if they do exist they will likely involve a correction of existing theory, and do not invalidate relativity.

4.  The discontinuity in momentum as velocity approaches “c” for infinitesimal mass, compared to the momentum of light.

Completely nonsensical.  Special relativity results in expressions for the momentum of massive particles, which approaches infinity as the speed of the particle approaches the speed of light.  The energy/momentum relation for a massless particle is perfectly well-defined, and the expression for the momentum of a massive particle as a function of speed contains a “loophole” that suggests that massless particles with momentum are possible.  The specific expression for the momentum of light comes from Maxwell’s equations, and the momentum of photons (light particles) is associated with quantum mechanics.

5.  The logical problem of a force which is applied at a right angle to the velocity of a relativistic mass – does this act on the rest mass or the relativistic mass?

I’ve never heard of this being a problem for anyone studying relativity.  In classical, pre-relativity Newtonian physics, momentum is directly proportional to velocity, while in Einstein’s relativity, the momentum of a particle approaches infinity as the speed of the particle approaches the speed of light.  There are two ways to interpret the new momentum formula: you can keep Newton’s formula and interpret mass as increasing, or you can just accept that Newton’s formula doesn’t hold for really fast objects.  The idea of relativistic mass is hardly ever taught these days, because it leads to pointless confusion as given in point 5 above.

6.  The observed lack of curvature in overall space.

Another point without external reference!  As I noted, general relativity supposes that mass creates a curvature of the fabric of space and time itself, with a greater distortion associated with greater mass.  In a footnote, the author argues that space is “almost flat”, but doesn’t say how they reach that conclusion, or how that point disproves relativity.

7.  The universe shortly after its creation, when quantum effects dominated and contradicted Relativity.

Why is this a counterexample to relativity, and why do quantum effects “contradict” relativity?  The author doesn’t say, and clearly doesn’t know.

8.  The action-at-a-distance of quantum entanglement.

“Quantum entanglement” refers to the idea that two quantum mechanical particles can in principle have a “connection” even after being separated by great distances.  A measurement of one entangled particle supposedly must “instantaneously” influence the behavior of the second particle, seemingly at odds with special relativity.  In fact, Einstein himself used this to argue against the idea of quantum mechanics, referring to this effect as “spooky action at a distance“.  A more sophisticated analysis, and experimental tests, however, have shown that it is impossible to use this “spooky action” to convey information at a speed faster than that of light.  Entanglement has in fact broadened our understanding of quantum mechanics and relativity, and not discounted either.  There is still some subtlety to the story of entanglement, and investigations are still underway, but the author of this post clearly understands none of it.

9.  The action-at-a-distance by Jesus, described in John 4:46-54.

Seriously?  You’re kidding, right?  Okay, let me explain something about science to this sad Conservapedia author: Bible quotes are not scientific evidence.  Now let me explain some theology to this author: pretty much by definition, a miracle is an act that goes against the laws of nature.  If you think that Jesus’ acts disprove relativity, you’re saying that Jesus wasn’t performing miracles at all.  Idiot.

10.  The failure to discover gravitons, despite wasting hundreds of millions in taxpayer money in searching.

Luv the conservative whine about taxes, dude!  The problem is: the graviton isn’t a part of relativity.  A graviton is a hypothetical elementary particle that is the origin of the gravitational force, just as the photon is the origin of the electromagnetic forces.  It is not a part of special or general relativity, however, and was introduced as a way to try and explain gravity in a similar manner to other fundamental forces such as the strong and weak nuclear forces.  The validity of general relativity does not depend on the graviton’s existence.

11.  The inability of the theory to lead to other insights, contrary to every verified theory of physics.

If you neglect all of cosmology and astronomy, I guess you could say that relativity provides no insights.  That would be a pretty big neglect, however.

12.  The change in mass over time of standard kilograms preserved under ideal conditions.

Finally — an external link!  Unfortunately, said link demonstrates that the point in question is irrelevant.   Unlike all other fundamental metric units, the kilogram is not defined by some sort of physical phenomena but by a block of platinum and iridium kept in Paris.  In recent years, it has been found that this lump of metal is losing mass, or that copies are gaining mass.  What does this have to do with relativity?  Nothing.

14.  “The snag is that in quantum mechanics, time retains its Newtonian aloofness, providing the stage against which matter dances but never being affected by its presence. These two [QM and Relativity] conceptions of time don’t gel.”

Ah, now we have an out of context quote from Scientific American!  It has been a long-standing problem in physics to try and combine quantum mechanics and relativity.  The Scientific American article describes one hypothesis for modifying gravity to incorporate quantum effects.  The quotation in question describes a problem that physicists are trying to overcome, not an experimental problem with relativity.  Quote fail.

15. The theory predicts wormholes just as it predicts black holes, but wormholes violate causality and permit absurd time travel.

Scott Adams of Dilbert fame once compiled a really killer list of logical fallacies relating to science, including the fallacy, “Incompleteness as proof of defect.”  That is, pretend a theory is wrong because it can’t explain every problem anyone has ever proposed!  Nobody pretends that general relativity is complete; it does, however, explain cosmological observations really well.

18. The lack of a single useful device developed based on any insights provided by the theory; no lives have been saved or helped, and the theory has not led to other useful theories and may have interfered with scientific progress. This stands in stark contrast with every verified theory of science. The only device based on relativity is the atom bomb, but that has destroyed far more lives than it’s saved so it can hardly be considered useful.

The second part of this point is just a repeat of point 11!  The first part is somehow an argument conflating the “accuracy” of a scientific theory with its “usefulness”, though the two are not equivalent.  A theory must be accurate to be useful, but an accurate theory does not necessarily have direct uses.  Reality is reality, regardless of its usefulness.  However, it should be noted that all of high-energy physics, including the operation of particle accelerators, depends on the results of special relativity to function — particles are accelerated to speeds that are within a fraction of a percent of the speed of light.  The third part of this statement is again a statement of usefulness, which is utterly irrelevant to the accuracy of the theory.

19.  Relativity requires different values for the inertia of a moving object: in its direction of motion, and perpendicular to that direction. This contradicts the logical principle that the laws of physics are the same in all directions.

Nonsense.  The term “inertia” itself is usually described as an object’s “resistance to a change in motion”; in Newtonian physics, this is typically equated with mass.  In special relativity, however, the effect of forces on an object are typically described in terms of the object’s momentum, and there is no problem of “different values”.  This is, in fact, an undergraduate-level calculation.

20.  Relativity requires that anything traveling at the speed of light must have mass zero, so it must have momentum zero. But the laws of electrodynamics require that light have nonzero momentum.

*BUZZZZ!!!!*  Wrong answer!  The problem is the statement, “so it must have momentum zero.”  As noted in point 4, special relativity actually suggests the opposite. If one looks at the equation for the momentum of a massive particle in special relativity as a function of speed, it turns out that the expression has an undefined form in the limit of zero mass and light speed. This provides a “loophole” that allows a massless particle to have nonzero momentum. (Update: As noted by a commenter below, the relation between energy and momentum in relativity is perfectly well-defined for a massless particle.)

23.  The Twin Paradox: Consider twins who are separated with one traveling at a very high speed such that his “clock” (age) slows down, so that when he returns he has a younger age than the twin; this violates Relativity because both twins should expect the other to be younger, if motion is relative. Einstein himself admitted that this contradicts Relativity.

*BUZZZZ!!!!* Wrong answer, again!  The “twin paradox” hasn’t been a paradox of relativity theory for pretty much 100 years.  The statement of the problem is roughly correct, if oversimplified — as noted in the introduction, observers in uniform motion relative to one another both, correctly, observe each other’s clocks as running slow.  The key, though, is the word “uniform”: in the twin paradox, in order for the two twins to end at the same place, one of them must have accelerated — undergone non-uniform motion — in order to return home.  Counter to footnote 13, this acceleration can never be “neglected” — a weaker acceleration must be applied over a longer period of time in order to send the twin home.  The calculation and resolution of the twin paradox is not that difficult to do, actually, and I will return to it in my future relativity posts.

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So, where do we stand?  Other than a few points I got too tired to explain (the power of the Gish Gallop), everything we’ve seen has been deceptive, incorrect — or just plain crazy.

Really, to assume that relativity is a liberal conspiracy requires one to believe that physicists for over one hundred years have been conspiring to hide the “truth” from the people, and that none of them have ever stepped forward to reveal said conspiracy!  If you believe that, you’re probably hiding in your mother’s basement, wearing a tinfoil hat and living on a diet of grade school paste.

Update: Tom at Swans on Tea tackles the Conservapedia claim that GPS doesn’t use relativity, and convincingly shows the stupidity of that claim.

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¹ I have to credit Sarah at The Language of Bad Physics for pointing me to the link.

One of the most fascinating aspects of 19th century physics is that many remarkable ideas and ingenious experiments were motivated by a physical hypothesis which we now know to be incorrect: namely, the aether.   When light was demonstrated to have wavelike properties in the early 1800s, it was natural to reason that, like other types of waves, light must result from the excitation of some medium:  after all, water waves arise from the oscillations of water, sound waves arise from the oscillation of air, and string vibrations are of course the oscillations of string.  The hypothetical medium which carries light vibrations was dubbed the “aether”, due to its unknown, “aetherial” nature.

A lot of scientists speculated on the physical properties of the aether, and sometimes this speculation produced lasting results in other fields; for instance, Earnshaw’s theorem was originally conceived to try and describe the forces involved in the aether’s oscillation.

By the late 1800s, however, more and more research cast doubt on the very existence of the aether, notably the Michelson-Morley experiment (to be discussed below).  In response, theoreticians produced more and more “patches” to the aether theory, until at last Einstein published his special theory of relativity, which eliminated the need for an aether and in fact suggested that the idea of an aether was incompatible with the experimental evidence.

Before this happened, however, at least one brilliant researcher took up the challenge of testing one of the “patches” to the aether.  Lord Rayleigh (1842-1919), distinguished physicist and eventual Nobel Prize winner, conceived of and carried out a very clever optical experiment to see whether objects shrink in the direction of motion, a phenomenon known as length contraction.

As is often the case, even though the experiment was unsuccessful, we can still learn many useful lessons about the workings of science from it!

To understand the experiment, we need to discuss first the 1887 Michelson-Morley experiment and the questions that it raised about the nature of the aether.  I’ve discussed this in some detail in a previous post; here we review the essentials.

By the 1880s, the aether was pretty much as mysterious as it had been when first postulated, as no experiments had directly detected its existence.  By analogy with sound waves, however, it was assumed that the speed of light m/s was fixed with respect to the aether.  That is, an observer at rest in the aether would see all light waves traveling at , but observers in motion would measure different speeds for light, depending on their motion relative to the light. For instance, according to Newtonian relativity, an observer moving at velocity v parallel to a beam of light would measure the speed of the light beam as , while an observer moving at velocity v antiparallel to the beam would measure its speed as .

The Earth was known to be traveling at 30 km/s in its path around the Sun, which meant that at some point in its motion it had to be traveling at least 30 km/s with respect to the underlying aether.   This suggested that the speed of light along the direction of Earth’s motion would be different than the speed of light perpendicular to the direction of Earth’s motion.

In 1887, Albert Michelson and Edward Morley refined an ingenious optical experiment to measure this difference.  They used a device known now as a Michelson interferometer, shown in simplified form below:

A light beam is incident from the left. It hits a half-silvered mirror which is inclined at a 45 degree angle, which splits the beam into two parts: one which is reflected, one of which is transmitted. Each fraction of the beam travels to a mirror and back, and at the half-silvered mirror they are recombined and their sum is projected onto a screen. Because the light has traveled a different distance in each arm of the interferometer, we get an interference pattern on the screen.

To understand how such an interferometer allows, in theory, one to detect relative motion with respect to the aether, we make an analogy between light traveling in the two arms of the interferometer and boats traveling on different paths on a river:

Each boat has a top speed of , and the river flows to the right with velocity . Boat 1 travels a distance from point A to point B, and then returns, while boat 2 travels a distance from point A to point C, and then returns. How long does it take each boat to return to the starting point? We can use a little geometry and velocity addition to determine this. Boat 2 will have a total velocity of on the way out to point C, and will have a total velocity on the way back. Boat 1’s velocity is slightly more difficult to calculate; in order to travel straight across the river, it must have a velocity against the river flow, and a total velocity of . Using some simple geometry,

boat 1 must have a speed   moving from A to B, and an identical speed for the return trip. We therefore find that the transit times for the boats are as follows:

,

.

The two times are different!  In principle, this difference manifests itself in the position of interference fringes in the output of the interferometer.  If the interferometer is rotated 90°, the roles of the two interferometer arms are reversed, and the fringes should visibly shift in position.

Surprisingly, no shift in the fringes was observed by Michelson and Morley.  Their experimental apparatus was sensitive enough to detect the known motion of the Earth within the aether, and utterly failed to do so.  None of the theories of the aether at the time could explain this absence of relative motion without contradiction.

A possible solution to the conundrum was suggested independently in a short letter¹ by George Fitzgerald in 1889 and in more detail² in 1895 by Hendrik Lorentz. It is worth quoting Fitzgerald’s letter in its entirety to explain the effect:

I HAVE read with much interest Messrs. Michelson and Morley’s wonderfully delicate experiment attempting to decide the important question as to how far the ether is carried along by the earth. Their result seems opposed to other experiments showing that the ether in the air can be carried along only to an inappreciable extent. I would suggest that almost the only hypothesis that can reconcile this opposition is that the length of material bodies changes, according as they are moving through the ether or across it, by an amount depending on the square of the ratio of their velocity to that of light. We know that electric forces are affected by the motion of the electrified bodies relative to the ether, and it seems a not improbable supposition that the molecular forces are affected by the motion, and that the size of a body alters consequently. It would be very important if secular experiments on electrical attractions between permanently electrified bodies, such as in a very delicate quadrant electrometer, were instituted in some of the equatorial parts of the earth to observe whether there is any diurnal and annual variation of attraction, — diurnal due to the rotation of the earth being added and subtracted from its orbital velocity; and annual similarly for its orbital velocity and the motion of the solar system.

In short: suppose that an object moving in the aether shrinks along the direction of motion, as illustrated below:

One can readily determine the amount of contraction necessary to give a null result for the Michelson Morley experiment; suppose the distance that boat 2 moves, with/against the river flow, is contracted to a new length

.

Then the time delay that would be experienced by boat 2 in its round trip would be

.

This would be exactly the time delay experienced by boat 1, as well,  and no difference between the boats’ travel times would be observed.

In the case of the Michelson Morley experiment, the “shrinkage” in the length of the interferometer in the direction of motion would be ridiculously small, assuming km/s and m/s:

.

In other words, if the arm of the interferometer was a meter long, the overall length of the interferometer would change by 10 nanometers, an almost negligible amount.

It is hard not to view the contraction hypothesis as a bit of an ad hoc “patch” to the theory of the aether, though Lorentz himself apparently viewed it as the only possible conclusion from the experimental results.  He even gave a plausibility argument concerning it (translation from Larmor’s Aether and Matter, 1900):

However extraordinary this hypothesis may appear at first sight, it must be admitted that it is by no means gratuitous, if we assume that the intermolecular forces act through the mediation of the aether in a manner similar to that which we know to be the case in regard to electric and magnetic forces.  If that is so, the translation of the matter will most likely alter the action between two molecules or atoms in a manner similar to that in which it alters the attraction or repulsion between electrically charged particles.  As then the form and the dimensions of a solid body are determined in the last resort by the intensity of the molecular forces, an alteration of the dimensions cannot well be left out of consideration.

I had always assumed that the contraction effect was much too small for anyone to consider measuring in Lorentz’s time.  In 1902, however, Lord Rayleigh suggested that the contraction effect might actually manifest itself through the phenomenon of double refraction — and he proceeded to test his hypothesis³!

Double refraction, also known as birefringence, occurs when light passes through a material such as a crystal which has an anisotropic structure, i.e. light of one polarization ‘sees’ a different structure than light of a perpendicular polarization.  More quantitatively, light of one polarization passes through a material faster than light of another polarization.  In materials such as calcite this results in the appearance of ‘twin’ images (via Wikipedia):

If polarized light is passed through a birefringent material, the direction of polarization is rotated

Rayleigh realized that length contraction would by its nature make a material birefringent, as illustrated below:

Light traveling perpendicular to the direction of motion will ‘see’ two different spacings of the atoms in the material; light x-polarized will see a smaller atomic spacing than light which is y-polarized.

A schematic of Rayleigh’s experiment is shown below.  Because it is difficult to present polarization changes in a figure with perspective, I also illustrate the ‘head on’ view of the important elements:

Unpolarized light is produced and focused onto a Nicol polarizer, which produces a polarized beam of light oriented 45º to the vertical.  Ignoring the strained glass for a moment, the light enters the liquid sample.  Provided the motion of the Earth is perpendicular to the direction of light propagation, one would expect length contraction-induced birefringence to rotate the direction of polarization.  Light then passes through an “analyzing” Nicol prism.

In the ideal situation, one would orient the analyzing prism until it completely extinguishes the polarized light.  By comparing the orientation of the two Nicol prisms, one could see if the state of polarization rotates.  This is a dubious comparison, however, because other factors, such as a natural birefringence of the liquid chamber, might perform an indistinguishable rotation.  To distinguish between the two cases, Rayleigh fixed his experimental apparatus to a turnable platform. Because length contraction-induced birefringence would be expected to depend significantly upon the orientation of the experiment, any changes in birefringence upon changing the orientation would be a sign of the effect.

To make any changes easier to observe, Rayleigh placed a piece of strained glass between the first polarizer and the liquid chamber.  The strained glass has its own natural birefringence, which results in an interference pattern at the output of the experiment.  Changes in birefringence (presumably due to length contraction) could be easily seen as shifts in the position of the interference pattern.

Rayleigh searched for length contraction-induced birefringence in bisulphide of carbon and in water.   As we would expect, he found no such contraction, even though his device was in principle sensitive to length variations of a tenth of a nanometer, a factor of a hundred more sensitive than needed to detect the effect.  Being a very good scientist, Rayleigh did not shy away from pointing out potential weaknesses in his own analysis,

So far as liquids are concerned, the experiment is of no great difficulty, and the conclusion may be stated that there is no double refraction of the order to be expected, that is comparable with of the single refraction.  But the question arises whether experiments upon liquids really settle the matter.  Probably no complete answer can be given, unless in the light of some particular theory of these relations.  But it may be remarked that the liquid condition is no obstacle to the development of double refraction under electric stress, as is shown in Dr. Kerr’s experiments.

In other words, it is natural to wonder whether liquids might slosh about in such a way that the double refraction effects do not appear.  This is a reasonable concern, but Rayleigh notes that double refraction is achievable in liquids via electrical forces, namely through the Kerr effect.  Rayleigh furthermore noted attempts to repeat the experiments in solids, and he again found negative results, though less accurate ones.

Einstein’s relativity was published in 1905, only a few years after Rayleigh’s experiments, and the new theory could account for them quite succinctly.  In special relativity, the laws of physics, including the speed of light, are the same in every inertial reference frame (i.e. for every observer moving at constant velocity).  According to an observer in the laboratory, the speed of light in each arm of the Michelson interferometer is simply , regardless of how the Earth moves relative to its neighbors; the notion of “length contraction induced by aether” is unnecessary.  There is no “aether” which light propagates in, so there is no “relative velocity with respect to the aether” to be measured.

Curiously, though, length contraction does make an appearance in special relativity, but it is a much more subtle effect than that which appears in Lorentz’s initial theory. Let us imagine a modified version of the Michelson interferometer, which fires simultaneously two ultrashort pulses each of which propagates a distance to a mirror and returns to a detector at the point of origin⁵. If both pulses arrive at the same time, a siren goes off to unambiguously announce their simultaneous arrival. From the point of view (“reference frame”) of the laboratory, each pulse travels the same distance at the speed of light and returns, firing off the siren.

Now let us consider this from the point of view of an observer who is traveling at velocity relative to the experiment along a line antiparallel to the path towards mirror 2.  From the observer’s point of view, the experiment is moving to the right at velocity :

The moving observer sees the light heading towards mirror 1 on a diagonal trajectory, and the light heading towards mirror 2 must chase the moving mirror, taking longer to reach it.  On the way back, the light returning from mirror 1 also follows a diagonal trajectory, and the light returning from mirror 2 has the detector moving towards it at velocity .  According to Einstein’s relativity, however, and distinct from the aether/boat case above, the moving observer measures the velocity of light to be for the light moving along both paths!

Let us assume that the physical separation, according to the observer, between detector and mirror 1 is , and the separation between detector and mirror 2 is .  With geometry, we can show that the total distance the light travels on a round trip from mirror 1 is:

,

while the total distance the light travels on a round trip from mirror 2 is:

.

Because the siren goes off, we know that the light from the two paths arrives back at the detector simultaneously, which means they must have traveled the same distance.  Because the speed of light is the same for both cases, this implies that

,

which is exactly the formula of Lorentz contraction!

Though the formulas are superficially the same, the interpretation of the physics in the Einstein case and the Lorentz case are dramatically different.  Both interpretations result in the observation that changes in the speed of light cannot be measured with the Michelson interferometer; however, the means by which this invariance is achieved are quite different.  The Lorentz interpretation says that a laboratory observer could in principle physically measure the shrinking of his apparatus due to relative motion with the aether; this is what Rayleigh tried, and failed, to measure.  The Einstein interpretation says that the laboratory observer sees no measurable shrinkage of his device, but an observer moving past the experimental setup will observe the shrinkage.

Lots of anti-relativity “crackpots” like to trumpet the idea that Lorentz’s relativistic formulation is equally valid to Einstein’s.  Experiments like Rayleigh’s, however, highlight that, at least in its earliest form, Lorentz’s relativistic aether theory had measurable experimental differences with the Einstein theory, and those tests which were performed in fact failed.  If you ever have to argue with a relativity denialist about the existence of an aether and “Lorentzian relativity”, keep this post in mind!

There are a lot of subtle points to digest in the above discussion, and a lot of Einstein’s relativity which has been more or less left out for the purposes of brevity⁶.  I’ll come back to a detailed analysis of Einstein’s relativity theory in future posts.

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¹ G. Fitzgerald, “The Ether and the Earth’s atmosphere,” Science 13 (1889), 390.

² H. Lorentz, “Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern” (Leiden, 1895).

³ Lord Rayleigh, “Does motion through the Æther cause double refraction?” Phil. Mag. 4, sixth series (1902), 678-683.

⁴ We give a somewhat unconventional explanation here, which allows us to compare the results with the already-discussed expectations of the aether. A more concise explanation will be given in a future post, when I formulate Einstein’s relativity and its consequences rigorously.

⁵ This point is important. In Einstein’s relativity, events which are simultaneous but spatially separated in one reference frame are not necessarily simultaneous in another reference frame. If the two pulses arrive at the same point at the same time in one frame, however, they do so in all frames.

⁶ And the post is still too darn long!

This gives me an opportunity/excuse to discuss one of my favorite exchanges between the pair, concerning Sir Edmund Whittaker’s book A History of the Theories of Aether and Electricity.

Sir Edmund Whittaker (1873-1956) was a very successful and influential mathematician and researcher. In 1910, he published a book entitled A History of the Theories of Aether and Electricity, which describes the speculation and search for the medium of light propagation (the ‘aether‘, a topic we’ve discussed many times) from the time of Rene Descartes through the end of the nineteenth century. (The complete 1910 edition can be downloaded here.)

Whittaker likely expected his book to be the fundamental reference on the physics of the aether. However, in 1905, before the book was even published, Einstein postulated his special theory of relativity which made the aether a moot concept. Whittaker’s book was relegated from fundamental scientific work to a historical footnote, of sorts, which no doubt caused him no small amount of irritation.

The 1910 version of the book included no description of the special theory of relativity. When Whittaker published a revised version of the book in the early 1950s, he included a chapter on, “The Relativity Theory of Poincaré and Lorentz”, and basically ranked Einstein’s contributions to the theory as of little importance.

Max Born was good friends with both Whittaker and Einstein, and tried to convince Whittaker that Einstein’s contributions to the theory were much more fundamental (we’ll come back to this point when I pick up my posts on relativity theory). Whittaker remained unconvinced, so Born wrote the following in a letter to Einstein in October of 1953:

Very often I feel the need to write to you, but I usually suppress it to spare you the trouble of replying. Today, though, I have a definite reason — that Whittaker, the old mathematician, who lives here as Professor Emeritus and is a good friend of mine, has written a new edition of his old book History of the Theory of the Ether, of which the second volume has already been published. Among other things it contains a history of the theory of relativity which is peculiar in that Lorentz and Poincaré are credited with its discovery while your papers are treated as less important. Although the book originated in Edinburgh, I am not really afraid you will think that I could be behind it. As a matter of fact I have done everything I could during the last three years to dissuade Whittaker from carrying out his plan, which he had already cherished for a long time and loved to talk about. I re-read the originals of some of the old papers, particularly some rather off-beat ones by Poincaré, and have given Whittaker translations of German papers (for example, I translated many pages of Pauli’s Encyclopaedia article into English with the help of my lecturer, Dr. Schlapp, in order to make it easier for Whittaker to form an opinion). But all in vain. He insisted that everything of importance had already been said by Poincaré, and that Lorentz quite plainly had the physical interpretation. As it happens, I know quite well how sceptical Lorentz was and how long it took him to become a relativist. I have told Whittaker all this, but without success. I am annoyed about this, for he is considered a great authority in the English speaking countries and many people are going to believe him. It is particularly unpleasant in my opinion that he has woven all sorts of personal information into his account of quantum mechanics and that my part in it is extolled. Many people may now think (even if you do not) that I played rather an ugly role in this business. After all, it is common knowledge that you and I do not see eye to eye over the question of determinism. What is more, I have written a small article which is shortly to appear in which I give a theoretical interpretation of an idea of Freundlich’s about stellar red shift, which could, if correct, cause difficulties for the relativistic interpretation. Therefore my feeling towards you is that of a cheeky urchin who can get away with certain liberties without offending you. But it may well seem less harmless to other people. Well, I had to write this and get it off my chest.

Einstein’s response is short and priceless:

Don’t lose any sleep over your friend’s book. Everybody does what he considers right or, in deterministic terms, what he has to do. If he manages to convince others, that is their own affair. I myself have certainly found satisfaction in my efforts, but I would not consider it sensible to defend the results of my work as being my own ‘property’, as some old miser might defend the few coppers he had laboriously scraped together. I do not hold anything against him, nor of course against you. After all, I do not need to read the thing.

What was Whittaker’s problem? I have previously referred to G.A. Schott as the last of the respectable ‘anti-quantumists’; Whittaker may have very well been the last of the respectable ‘anti-relativists’, though he never admitted this to be the case. I wouldn’t be surprised if he also wasn’t just a little irked, as I mentioned above, that his magnum opus on the aether was made a historical curiosity by Einstein even before it was published.

I’ve been meaning to get back to my series of posts on relativity, but things have gone slower than I expected because of my obsessive desire to truly understand the historical scientific issues that were prevalent at the time.

In the meantime, I’ve been thinking about an interesting, infrequently-discussed topic in special relativity: the behavior of light on propagation through moving matter. This question was inspired by a comment on Uncertain Principles some time ago. In fact, one of the earliest hints of special relativity came from an experiment performed by François Arago in 1810 on ‘stellar aberration’, nearly 100 years before Einstein’s landmark 1905 paper! In this post I’ll discuss Arago’s experiment, its historical context, and the conclusions that were drawn from it.

From a historical point of view, Arago’s experiment* is absolutely fascinating: as we will see, it was a failed experiment, based on incorrect theories of light propagation, which was interpreted incorrectly by Fresnel, but this incorrect interpretation helped lead to the (correct) view that light has wavelike properties! The incorrect interpretation, however, also led physics into a hundred-year ‘red herring’ search that only ended with the advent of Einstein’s relativity. These are a lot of twists and turns to untangle, so let’s take them one step at a time.

Before 1800, most scientists were proponents of the so-called corpuscular theory of light propagation. In this view, which was championed and solidified by Isaac Newton in his 1704 book Opticks, held that light consisted of a stream of particles. Newton explicitly argued against the wave theory of light and (seemingly) refuted arguments by early wave theory proponents such as Christiaan Huygens. Newton’s arguments, and his personal gravitas, left his particle theory mostly unchallenged until the early 1800s.

There is one aspect of the particle theory of light which will be important later in the post: the explanation of refraction. When a ray of light is incident upon the flat surface of a medium, part of the ray is reflected and part of the ray is transmitted into the medium. The transmitted ray, however, is ‘bent’: it travels in a different direction than the incident ray. According to Snell’s law (experimentally determined, originally), the relationship between the angle of incidence of the ray and the angle of refraction is given by

,

where and are the (experimentally determined) refractive indices of the two mediums, and are the angles which the incident and transmitted ray make with respect to the normal to the surface, and represents the trigonometric sine function. These symbols are illustrated below:

What happens to the speed of light when it enters a medium? According to the wave theory, refraction occurs because light slows down as it enters the medium, so that

,

where is the refractive index of the medium, usually greater than 1, and is the speed of light in vacuum. According to Newton’s particle theory, however, refraction occurs because light speeds up as it enters the medium, so that

.

(This difference is curiously similar in form to a more modern controversy: what is the momentum of a photon as it enters a medium?)

The relationship of this speed to the phenomenon of refraction differs depending on whether one applies a particle theory of light or a wave theory. Let’s look at both views, and do a little math to explain them:

1. Newton’s corpuscular (particle) theory of light refraction. Newton argued that, upon passing the boundary between materials, the particle experienced a force in a direction perpendicular to the surface. This force results in a change of velocity in that direction, with no change of the velocity in a direction parallel to the surface. A light particle is ‘sped-up’ when it passes from a rarer medium (low index) to a denser medium (high index), and a light particle is ‘slowed down’ when it passes from a denser medium to a rarer medium. One can visualize this by picturing the surface of the medium to represent a very steep hill: a particle passing from rarer to denser rolls down the hill and gains speed at the bottom:

Let’s do a little math to see how this would work. Suppose a ‘corpuscle’ is incident from vacuum at speed with horizontal component and vertical component . This is illustrated in the figure below:

Because the total speed is , we have by simple geometry:

.

We may rewrite this equation with some simple algebra to the form:

.

Upon entering the medium, the new (net) speed of the corpuscle is , the horizontal component of this is still , and by simple geometry the vertical component has become:

.

Here is how Newton himself described the same situation (Opticks, Book one, Part I, Experiment 15):

If an Motion or moving thing whatsoever be incident with any Velocity on any broad and thin space terminated on both sides by two parallel Planes, and in its Passage through that space be urged perpendicularly towards the farther Plane by any force which at given distances from the Plane is of given Quantities; the perpendicular velocity of that Motion or Thing, at its emerging out of that space, shall be always equal to the square Root of the sum of the square of the perpendicular velocity of that Motion or Thing at its Incidence on that space; and of the square of the perpendicular velocity which that Motion or Thing would have at its Emergence, if at its Incidence its perpendicular velocity was infinitely little.

We can check that we reproduce Snell’s law by noting that

and

.

Since is the same in both equations, we can solve one for the other to find that

,

which is simply Snell’s law.

2. Wave theory of refraction. In the (correct) wave theory of refraction, the speed of light is reduced by the refractive index, so that . This in turn suggests that the wavenumber of the light, , where is the frequency of the light oscillation, is increased on entering the medium. It is assumed that the horizontal component of the wavenumber is unchanged, i.e. is the same both inside and outside the medium, which means that the vertical component of the wavenumber, , is increased. This is illustrated below:

It is to be noted that this picture is essentially the same as the Newtonian picture, with wavenumber replacing velocity! The formulas for the z-components of the wavenumber are given by

,

,

which are structurally similar to Newton’s equations for light velocity. They also produce Snell’s law, by the same geometrical arguments.

So we have two theories for the nature of light, both of which can reasonably produce Snell’s law. In Newton’s time, additional evidence leaned towards the particle theory of light, but in the early 1800s a number of experiments were performed which eventually led to the wave theory winning out**. One of the most significant was Young’s double slit experiment (discussed here and here), the results of which were reported by Thomas Young in 1807. Young demonstrated that light passing through a pair of small holes in an opaque screen will produce interference fringes on a secondary screen beyond; this interference could only be explained by a wave theory of light:

As with most revolutionary results in science, however, the full import of Young’s work would take some time to resonate with the broader community. By 1810, plenty of scientists, including Arago, were still operating under the assumption of light as a particle.

One thing that everyone agreed upon in Arago’s time, however, was the speed of light. As I’ve discussed previously, as early as the 1670s Ole Christensen Römer determined a quite good estimate of the speed of light by timing the eclipses of one of Jupiter’s moons. In short, when Jupiter is moving towards us, the eclipses seem to occur more often (the light reaches us quicker), and when Jupiter is moving away from us, the eclipses seem to occur less often (the light takes longer to reach us). From these variations, one can estimate the speed of light. Römer’s estimate was remarkably close to the modern working value of the speed of light, m/s.

The finite speed of light results in a number of interesting astronomical consequences, one of which is referred to as “stellar aberration”. If the Earth is moving transversely relative to a distant star, we will see the starlight arrive at an angle which depends on the speed of relative motion. If the Earth changes its direction of motion (as it does during its orbit around the Sun), we will see the starlight arrive at a different angle. In 1725, British astronomer James Bradley first observed this effect, noting that the apparent angular position of stars in the sky depends on the time of year.

This effect is actually not difficult to understand. Expanding upon an analogy from Wikipedia, suppose you are in a rainstorm, and the rain is falling directly from above. When you start to walk, you will need to tilt your umbrella slightly forward, as you are now ‘running into’ the rain. From your (moving) point of view, the rain is falling at a slight angle:

If you were to change your direction of motion (to avoid a big puddle, say), from your perspective the rain would approach from a different angle, and you would need to tilt your umbrella in a different direction.

Stellar aberration is a similar effect, but with rays of light from the Sun replacing rain. Since the Earth is moving relative to the stars in the sky, it ‘runs into’ the starlight. As the Earth changes direction in its yearly orbit, the angle at which the starlight approaches changes as well. This is roughly illustrated below for the Earth and the Sun:

We can quantify the angle of the starlight in a rough mathematical sense as follows: suppose a star is directly above the Earth, and the Earth is moving horizontally below at velocity . To an observer on the Earth, the starlight does not come directly from above, but instead at an angle defined by

,

where represents the tangent function. In order to observe the star, the telescope must be oriented at an angle θ from the vertical.

It is to be noted that the formula above suggests that one might be able to determine the speed of light from the stellar aberration, since the velocity of the Earth is known and the angle of aberration is measurable. There was a great interest in doing so: though all measurements of the speed of light up to that point had returned the same value (within experimental error), it was assumed that variations in that speed had to exist***. Light escaping from very massive stars would have to escape a greater gravitational field, and would presumably be slowed by that field. Arago, and others of the time, assumed that light from larger stars would be traveling slower when it reached the Earth, though we know now by general relativity that this is not the case****. According to the aberration formula above, heavier stars should produce a larger aberration angle than lighter stars.

Unfortunately, telescopes of Arago’s time were not precise enough to detect this small variation in aberration angle. Arago, however, came up with a clever idea: according to Newton’s theory of refraction, the angle of refraction will be different for light particles moving at different speeds. Let’s see how this works with a little bit of math. Suppose two rays of light, one with components and and total speed , and one with components and and total speed , are incident on a material surface at the same angle. Let us further suppose that the first ray is moving faster than the second, i.e. . Because the angles are the same, the ratios of the components are equal, i.e.

.

What do these ratios look like for the refracted rays? Using our formula based on Newton’s theory of refraction, we have:

,

.

But since , we find that

,

which directly tells us that the angle of refraction depends on the speed of the incoming light particles!

Arago’s experimental arrangement was exceedingly simple. He glued a prism to the objective lens of a telescope and looked for the deviation in the light rays on passing through the prism. The prism for his first experiments was a piece of crown-glass and a piece of flint glass fixed together, with a total angle of roughly 24 degrees. He later modified the setup so that the prism only covered half of the objective lens; in this manner he could observe the position of the star directly, as well as the position as deviated by refraction, and deduce the angle of refraction of the star. This is illustrated schematically below (note: this is a little speculative, as Arago did not include figures in his paper):

With his prism experiment, Arago could in principle directly measure the speed of light arriving from distant stars.

We need not give any more experimental detail because Arago’s experiment failed to detect any variations in the speed of light. In his own words, translated from the French,

However, by examining the preceding tables attentively, one finds that the rays of all stars are prone to the same deviations…

Light from every star is refracted the same amount! This was extremely difficult to justify using Newton’s theory of refraction, but Arago made a first faltering attempt to do so:

This result seems to be, with the first aspect, in manifest contradiction with the Newtonian theory of the refraction, since a real inequality in the speed of the rays however does not cause any inequality in the deviations which they test. It even seems that one can return of it reason only by supposing that the luminous elements emit rays with all kinds speeds, provided that it is also admitted that these rays are visible only when their speeds lie between given limits. On this assumption, indeed, the visibility of the rays will depend their relative speeds, and, as these same speeds determine the quantity of the refraction, the visible rays will be always also refracted.

In short, Arago speculated that stars radiate light over an infinite variety of speeds, and that we can only observe light whose speeds lie within a limited range of values. This was not so completely off-the-wall as one might first expect. Infrared light had been recently discovered in 1800 by William Herschel, and in 1801 Johann Ritter discovered ultraviolet light; both types of radiation are invisible to the naked eye. These discoveries are used by Arago as evidence for his supposition.

Arago’s theory is, however, wrong: the colors of light are not dictated by the speed of light, but by the frequency of light. This frequency can be changed by relative motion of source and observer, in what is known as a Doppler shift, but the speed of light is always the same, regardless of the motion of source and observer: this is in essence one of the postulates of Einstein’s relativity.

We’ll get to how Einstein’s relativity explains Arago’s results at the end of the post; before we get there, though, we take a moment to explain how researchers at the time explained them.

Newton’s particle theory of light propagation couldn’t explain Arago’s results, but the wave theory of light, at first glance, fared little better. In the wave theory of light, as understood at the time, the speed of light would be constant with respect to an all-pervasive ‘aether’. This by itself cannot be used to explain Arago’s experiment, however, because the speed of light would now depend on the relative motion of the Earth with respect to the aether: if the Earth was approaching a source of light, the speed should be increased, and if the Earth was receding from a source of light, the speed should be decreased. Such variations in speed would result in differences in the angle of refraction of light, like the Newtonian theory predicts, but in disagreement with experiment.

Another possibility is to assume that the Earth ‘drags’ the aether along with it, so that light which enters the Earth’s ‘aether field’ is not affected by the relative motion of the Earth. This fixes the problem of the refraction of light, but now predicts that stellar aberration should not occur, in disagreement with experiment!

In 1818 Augustin Jean Fresnel suggested another possibility to Arago*****: that the aether is partially dragged along with a material object. When light enters a moving medium of vector velocity and refractive index , it has a velocity:

,

where

is the so-called Fresnel drag coefficient. Fresnel’s approach represents a compromise between the ‘complete drag’ theories and ‘no drag’ theories. First, it suggests that objects with a low refractive index (for instance, the Earth’s atmosphere) produce almost no drag at all: therefore stellar aberration can occur. Second, it suggests that objects with a higher refractive index produce higher drag: this produces an agreement with Arago’s experiment.

Fresnel’s approach was convincing to Arago, who was led to abandon the particle theory of light and embrace the wave theory of light. With the confirmation by other means of the wave theory of light, Fresnel’s aether drag became an accepted part of the (incorrect) theory of the aether. In fact, an experiment was performed by Fizeau in the 1850s to explicitly measure the ‘drag coefficient’, and the results were in agreement with Fresnel’s theory.

But although Fresnel’s formula was correct, his interpretation of it was wrong. Einstein’s special theory of relativity produces Fresnel’s formula as a low-velocity special case of the relativistic velocity addition formula, as we briefly show.

*** warning: gratuitous math content! ***

In Newtonian relativity, velocities add in a straightforward way. For instance, a person on a bus moving at 50 mph who walks towards the front of the bus at 2 mph will have a speed of 52 mph relative to the street. This formula can be expressed as:

,

where is the speed relative to the ground, is the speed of the bus relative to the ground, and is the speed of the man relative to the bus. In Einstein’s relativity, this velocity-addition formula takes on a more complicated form:

.

At low speeds (when ), this formula becomes the classic Newtonian formula.

Light traveling in a moving medium is analogous to a man walking in a moving bus (light = man, medium = bus). The speed of light in the medium is , so our formula for the speed of light relative to the ground is:

.

If is significantly smaller than the speed of light, which it is for almost all terrestrial applications, we may make the following Taylor series approximation:

If we substitute back into the equation for , and keep only those terms which are linear in velocity, we arrive at:

,

which is exactly the Fresnel drag formula.

*** end gratuitous math content! ***

The Fresnel drag theory helped convince scientists for six decades that the aether existed. It was only when Michelson and Morely failed in 1887 to detect any motion of the Earth with respect to this aether that the concept began to falter. Even then, it wasn’t until Einstein unveiled his special theory of relativity that scientists began to realize that the concept of an aether was unneccessary.

So Arago’s work involved many twists and turns: it was a failed experiment (no variations in the speed of light were detected), based on incorrect theories of light propagation (Newton’s corpuscular theory of light), which was interpreted incorrectly by Fresnel (aether drag), but this incorrect interpretation helped lead to the (correct) view that light has wavelike properties!

Often lost in the hubbub is the realization that Arago produced what we now know to be the first experimental evidence for the special theory of relativity, but it took one hundred years for the theory to catch up!

*********************************************

* Arago’s original paper was reprinted in his Œuvres Complètes, Tome 7, Volume 4 (1858), p. 548-568.

** Of course, when quantum mechanics arrived, light ‘regained’ at least part of its particle-like properties. We now know that light has properties of both particles and waves.

*** Though not necessarily for the reasons one might think. The Big Bang hypothesis and the idea of an expanding universe would not appear until the first half of the 20th century.

**** Gravity does have an effect on light: the frequency of light which escapes a gravitational well is shifted towards lower energies/frequencies. This is known as a gravitational redshift.

***** A translation of Fresnel’s letter to Arago can be found here.

(If you’ve read this far, I hope you’ve enjoyed the post: let me tell you, this was one of the hardest bits of research I’ve ever done. It was a drag in more ways than one!)

A court case was being tried in New Mexico. A group of pornographers were charged with smuggling pornography from Mexico by projecting it across the border to a camera. The defense argued that nothing physical was transported, and in the end the argument boiled down to this: if light moved at a finite speed, the films were being transported; if it moved at infinite speed, the defense was correct. A physicist was brought in to discuss the speed of light but, after a number of figures were presented, the judge interrupted. “When I put my hands over my eyes, the light stops coming immediately, and when I move my hands, it reappears instantly. The speed of light is infinite – the defendants are not guilty!”

The reason I suspect this story is apocryphal is that science has accepted that the speed of light is finite – albeit very large – for centuries. The value, usually denoted c, is approximately meters/second, or 186,282 miles/second. In fact, as we will see in later posts, light is the fastest thing in the universe. The topics we address in this post: a brief history of measuring the speed of light, and how these measurements led inexorably to Einstein’s special theory of relativity.

We will, in fact, be concerned with two distinct classes of measurements. The first class are techniques for measuring the value of c itself. The second class are attempts to measure the change in the speed of light due to relative motion of the source or observer. We start with the first of these, and then discuss the reasons for the second class of measurements.

Many early ‘scientists’ or ‘natural philosophers’, though not all, imagined light to be of infinite speed. Aristotle was a proponent of this view, in large part because of his flawed idea that sight was a process of emission from the eye. Though some scientists challenged this idea (notably Ibn al-Haytham), the infinite speed seemed to be the prevailing view for most natural philosophers for quite some time.

One of the first attempts to experimentally measure the speed of light was undertaken by Galileo. The technique was simplicity itself: an experimenter would unveil a lantern on a mountaintop, an observer at a distant point would unveil his own lantern at the sight of the first lantern, and the first observer measures the amount of time before the second signal arrives. This is illustrated below:

This experiment produced a result indistinguishable from infinity, though it is not difficult in hindsight to understand why. The round-trip time for light traveling a distance d between mountaintops is

.

If we conservatively assume that Galileo had observation points a mile apart, and that the second observer instantaneously unveils his own lantern, the signal will return to Galileo in seconds! Considering human response time is on the order of 0.1 seconds, it’s clear that Galileo could not detect this time delay. One of the longest straight-line paths on the Earth is from Uncompaghre Peak, Colorado to Mount Ellen in Utah, a 183-mile separation. Galileo would fare no better with this distance; seconds. Even if we get unrealistically optimistic and imagine using a system of mirrors to create a path around the circumference of the Earth (24,900 miles),

we still find a time delay seconds, still not within measuring capabilities of an unaided human.

In modern terms, I would say that the speed of light pwned Galileo! He realized, though, as others did, that his negative result simply meant that the speed of light could be much faster than he could detect.

The first successful measurement of the speed of light was apparently made by Ole Christensen Römer (1644-1710), a Danish astronomer. In the 1670s, Römer was studying eclipses of Jupiter’s moon Io, which should presumably appear at near regular intervals because Io is moving in a circular orbit around Jupiter. Römer observed, however, that the time between eclipses varied, and in fact came slightly less frequently when Earth is moving away from Jupiter, and slightly more frequently when Earth is moving towards Jupiter. He concluded (as did his supervisor Cassini before him), that this variation arose from the finite speed of light.

How does this work? Let us quote from the English translation of his 1677 paper (figure from Wikipedia, from Römer’s original paper: note the ‘smiley face’ in the Sun!):

Now, suppose the Earth, being in L towards the second quadrature of Jupiter, hath seen the first satellit at the time of its emersion or issuing out of the shadow in D; and that about 42 1/2 hours after (vid. after one revolution of this satellit,) the Earth being in K, do see it returned in D; it is manifest, that if the Light require time to traverse the interval LK, the Satellit will be seen returned later in D, than it would have been if the Earth had remained in L, so that the revolution of this Satellit being thus observed by the Emersions, will be retarded by so much time, as the Light shall have taken in passing from L to K…

Let us try and illustrate this a bit more simply below.

Suppose Jupiter starts out a distance from Earth at the beginning of one of Io’s eclipses (eclipse 1), that Io makes a full revolution of Jupiter in time T, and that Jupiter is moving away from Earth at velocity v. By the time the next eclipse begins (eclipse 2), Jupiter is now a distance d = vT further away from Earth than it was during eclipse 1. The light from eclipse 2 must travel a distance d = vT further than the light from eclipse 1 needed to travel, and will therefore arrive ‘late’ by a time

.

The apparent duration of the orbit of Io is increased by ! Similarly, when Jupiter and Earth are moving closer together, the apparent duration is decreased by . If we know the period of Io (or its average value), and we know the relative speed difference between the Earth and Jupiter, we can experimentally calculate the value of c! The change in a single orbital period is small, but Römer looked at the cumulative change over a large number of orbits (40, according to his paper), and found a measurable deviation of 22 minutes. Römer himself did not explicitly calculate the speed of light in his work, but others (including Dutch smarty-pants Christiaan Huygens) did, and estimated the speed within 20% of the current accepted value.

The astute observer may notice a similarity between Römer’s calculation and the Doppler shift, in which the frequency of a wave signal is increased/decreased as the source moves towards/away from the observer, respectively. In fact, Römer’s technique is a simple application of a Doppler shift; the observed angular frequency of Io as measured on Earth is, according to the equation above,

,

which is exactly the classical Doppler formula for a slowly moving source.

We skip ahead now to the work of Hippolyte Fizeau (1819-1896), who in 1849 designed a new technique to measure the speed of light. In essence, it is a reimagining of Galileo’s unsuccessful experiment, and is illustrated schematically below:

Light leaves a source, is reflected off of a partially reflecting mirror, and illuminates the spokes of a rapidly rotating toothed wheel. The teeth of the wheel break the light up into ‘pulses’, each of which travel to a distant mountaintop and are reflected by a mirror there. If the pulse returns while a gap in the teeth is present, the pulse passes through and reaches the eye of the observer. Knowing the distance to the mountain, the number of teeth on the wheel and the frequency of rotation at which the returning pulse reaches the eye, one can accurately measure the speed of light.

Fizeau’s results were published in Comp. Rend. Acad. Sci. (Paris) 29 (1849), 90-92. (This reference was extremely hard to pin down, and I’m going to reprint it and a translation in its entirety in another blog post.) How is this an improvement over Galileo’s mountain to mountain experiment? To quote from (a Babelfish-translation of) Fizeau’s own words,

When a disc turns in its plan around the centre of face with a great speed, one can consider the time employed by a point of the circumference to traverse a very small angular space, 1/1000 of the circumference, for example.

When the number of revolutions is rather large, this time is generally very short; for ten and hundred turns a second, it is only 1/10000 and 1/100000 of second. If the disc is divided has its circumference, with the manner of the toothed wheels, in equal intervals alternatively empty and full, one will have, for the duration of the passage of each interval by the same point of space, the same very-small fractions.

During such short times the light traverses limited enough spaces, 31 kilometers for the first fraction, 3 kilometers for the second.

The fast moving disc takes the place of the slow human reaction time. Supposing the wheel has N teeth, the gaps in the teeth are separated by an angle

.

(Using radians =360 degrees.) The time for a pulse to traverse the distance to the mountain and back again is

.

A bright spot will appear on the observation screen if the disc has rotated to the adjacent gap at exactly the time the pulse returns; the ‘magic’ angular frequency is given by

;

in terms of rotations/second, this becomes:

.

Solving this equation for c gives us

.

What estimate can we get with Fizeau’s results? Fizeau used a disc with 720 teeth. The experiment was done with two posts, one at “a house located at Suresnes, the second on the height of Montmartre, at an approximate distance of 8633 meters.” The first appearance of a luminous point occurred at turns/second. Plugging these numbers into our equation, we get a result of

m/s,

which is roughly 4% off the actual value!

Let us now turn our attention to the second class of measurements: measuring changes in the speed of light due to the Earth’s motion. As we discussed in our previous relativity posts (pre-history, Newtonian relativity), by the late 1800s scientists had realized that there was something problematic when combining Newtonian relativity (the laws of motion are the same in all inertial reference frames) with Maxwell’s equations (the laws which describe light and electromagnetic fields). In particular, one can easily use Maxwell’s equations and Newtonian relativity to show that observers in different reference frames will predict different forces between electric charges! As we’ve stated in previous posts, this would be comparable to saying that a stationary witness to a car accident sees both cars totaled while a moving witness claims it was only a fender-bender!

This was a clearly at odds with reality, and so physicists were forced to rethink their views. Clearly only one of the (potentially infinite) number of observers can be correct about the forces, but which one?

The (incorrect) answer came by analogy. At that point, all waves known to science required a medium of some sort to travel through: sound waves propagate through air, water waves travel through water, string vibrations travel along strings. It was perfectly reasonable at the time to assume that light, an electromagnetic wave, must also be traveling through some as yet unobserved medium. This medium was dubbed the aether or luminiferous aether. An observer at rest with respect to this aether would see all light waves traveling at c, but observers in motion with respect to the aether would measure different speeds for light, depending on their motion relative to the light. For instance, according to Newtonian relativity, an observer moving at velocity v parallel to a beam of light would measure the speed of the light beam as , while an observer moving at velocity v antiparallel to the beam would measure its speed as .

The biggest problem with the idea of an aether is that nobody had ever observed it. It had to be robust enough to carry high-frequency light waves but tenuous enough that its presence was otherwise unobservable. The Earth, however, was already known to move at 30 kilometers/second in its path around the Sun. It would have to be moving, at some point in its orbit, at least at this speed relative to the hypothetical aether. Even though nobody knew how to make a direct measurement of aether, one could presumably indirectly detect it by measuring changes in the speed of light as the Earth moves.

Therein lies another problem, though: the speed of light is km/s, while the speed of the Earth is only 0.01% of that value. The expected change in light speed due to the Earth’s motion would be incredibly small.

To imagine how difficult this would be to detect, let us look at the Fizeau experiment again. A simplified schematic is illustrated below*:

In the ideal case, our entire experimental apparatus is moving parallel to the light beam on the way out, and the light is traveling at c – v. On the return trip, the apparatus is moving antiparallel to the light beam and the light is traveling at c + v. Because of these velocity differences, the round trip time the light takes will be different than that of an unmoving Fizeau experiment, by an amount:

.

Using the Fizeau experiment numbers, we find that the discrepancy would be on the order of

seconds! In order for a Fizeau-type experiment to be sensitive to that variation in time, it would have to be turning at a speed such that this time scale resulted in one ‘tooth’ rotation, or

rotations/s!

To appreciate how hopeless this is, it is to be noted that a commercial jet engine turbine has a rotation speed on the order of 36,500 rpm, or 608 rotations/second.

Is there anything that can oscillate or rotate that fast? In fact, as we have noted in an optics basics post, visible light oscillates with an angular frequency on the order of cycles/second. A light beam can therefore serve as its own ‘timer’ in light velocity measurements!

This was, in essence, the strategy employed by Albert Michelson and Edward Morley in 1887 in what is now known as the Michelson-Morley experiment. The experimental setup, of a device known as a Michelson interferometer, is illustrated below:

A light beam is incident from the left. It hits a half-silvered mirror which is inclined at a 45 degree angle, which splits the beam into two parts: one which is reflected, one of which is transmitted. Each fraction of the beam travels to a mirror and back, and at the half-silvered mirror they are recombined and their sum is projected onto a screen. Because the light has traveled a different distance in each arm of the interferometer, we get an interference pattern on the screen.

It’s worth noting that a real Michelson interferometer is a bit more complicated than the cartoon picture above, and includes a compensator plate and a lens:

These components are only present so that the behavior of the interferometer closely corresponds with the ‘ideal’, so we can neglect them in the discussion which follows.

How does this interferometer allow us, in principle, to detect the motion of the Earth in aether? (IMPORTANT: Remember, there is no aether! We’re only speaking from the point of view of what scientists of the late 1800′s believed.) We can make an analogy between light traveling in the two arms of the interferometer and boats traveling on different paths on a river*:

Each boat has a top speed of c, and the river flows to the right with velocity v. Boat 1 travels a distance d from point A to point B, and then returns, while boat 2 travels a distance d from point A to point C, and then returns. How long does it take each boat to return to the starting point? We can use a little geometry and velocity addition to determine this. Boat 2 will have a total velocity of c + v on the way out to point C, and will have a total velocity c – v on the way back. Boat 1′s velocity is slightly more difficult to calculate; in order to travel straight across the river, it must have a velocity v against the river flow, and a total velocity of c. Using some simple geometry,

boat 1 must have a speed moving from A to B, and an identical speed for the return trip. We therefore find that the transit times for the boats are as follows:

,

.

With the help of a little calculus approximation, we find that boat 1 will beat boat 2 by a time:

.

This is exactly the time discrepancy that we expected with the Fizeau experiment, as well; the difference is that with the Michelson interferometer, we are using light interference to detect this difference.

For light of angular frequency ω, the light from arm 1 and arm 2 of the interferometer will be out of phase by an amount:

.

Michelson and Morley made this shift more noticeable by rotating their entire apparatus 90 degrees; this reverses the roles of arm 1 and arm 2 of the interferometer, and results in a total phase change of

.

What did Michelson and Morley actually see? They used light from a sodium lamp (wavelength nm, angular frequency cycles/second), a path length d = 11 meters, and with the Earth’s speed known to be km/s, we find that we expect to see a phase change of

cycles/second.

In an actual Michelson interferometer, one actually sees a pattern of bright and dark circles of light, as shown below:

A bright spot corresponds to a phase difference of 2π, while a dark spot corresponds to a phase difference of π. When the interferometer is turned, these bright and dark circles spread outward, corresponding to a change in phase. We define the ‘fringe shift’ ΔN as the amount one dark circle moves towards the position of the next outward circle, and this shift is simply given by .

From our numbers above, Michelson and Morley expected that the motion of the Earth would result in a fringe shift of . What they in fact found, was , which was essentially within the experimental uncertainty of their device.

In other words, Michelson and Morley could not detect any motion of the Earth with respect to the aether! This was an astonishing result. If the aether in fact existed, one could not detect one’s own motion through it, which inevitably made scientists doubt the existence of this hypothetical material.

A number of scientists tried to explain these results using arguments which can now be seen to be ad hoc: Lorentz suggested that objects moving through the aether are shrunk along their direction of motion in such a way as to equalize the time delay of light between the two arms. Others suggested that the Earth was ‘dragging’ the aether along with it, meaning that, at least locally, the aether was stationary with respect to the Earth. These suggestions, however, have an air of desperation about them, in that they raise more questions than they answer (why do objects shrink? how does the Earth drag the aether?).

It was Albert Einstein who, in 1905, proposed a new theory of relativity of such beauty and simplicity that it made the aether unnecessary. In the next relativity post, we move away from our long discussion of the historical origins of relativity and into Einstein’s amazing theory itself.

* These discussions are adapted from Tipler and Llewellyn’s Modern Physics.

So what is ‘Newtonian relativity’, exactly? Newton, in his own words, introduced it in a rather roundabout way; in modern parlance, we may summarize it as:

The laws of physics (mechanics) are the same for any observer moving at constant speed.

In other words, we all play by the same physical rules, whether we are ‘moving’ or ‘standing still’. Indirectly, this tells us that there is no such thing as absolute motion: if there is no physical experiment which can distinguish between an experimenter moving at (absolute) constant speed or sitting at (absolute) rest, the term ‘absolute motion’ has no meaning. The only motion that matters is relative motion between objects.

There are a few observations we can immediately make about ‘Newtonian relativity’. First of all, we emphasize that Newton himself was only concerned with the physics of moving objects (i.e. mechanics), and said nothing about other physical laws, such as electromagnetism. A consistent physical theory of electromagnetism had not been formulated at Newton’s time. Second, it is to be noted that the definition above begs the question somewhat: What does ‘constant speed’ mean, if there is no such thing as absolute motion? Constant speed with respect to what? We will address this somewhat as we go on, and in fact it will play a big role in later posts. For now, we simply say that we ‘instinctively’ can tell the difference between, for instance, driving down the road at a constant speed and slamming on the brakes of the car.

To further understand what Newtonian relativity implies about the world, it is helpful to imagine an ‘alternate reality’ in which the laws of physics are not the same for every observer in constant motion. What might such a world look like? As mentioned in the previous post, Aristotle imagined that the natural state of an object is to return to a state of ‘absolute rest’. If the laws of physics worked in this way, an object in (absolute) motion would naturally experience a braking force which would attempt to make the object stop moving. Though Aristotle never formulated his theory mathematically, it is natural to expect that the force experienced would get stronger the faster one moved. We would evidently have a way of distinguishing, by experiment, whether we are moving or not: we would simply measure the ‘braking force’ exerted on us as a measure of our current motion. No such braking force exists, however, and Newton surmised that no experiment would be otherwise able to detect absolute motion.

Up to this point, we’ve been using terms like ‘observer’ rather loosely. In anticipation of adding a little mathematics to the discussion, let’s define a few terms that will be applied consistently from now on…

Observer: An ‘observer’ is someone making measurements of physical phenomena. Typically we will pretend the observer is a person, though it could of course be a scientific apparatus as well. An observer carries with him two sets of tools for measuring motion: a clock, for measuring time, and a set of rulers, for measuring the position of objects. Using his rulers and clock, for instance, an observer could measure the distance Δx an object moves relative to him and divide by the amount of time Δt it takes for the object to travel that distance: the ratio of the two is the speed of the object:

Frame of reference: As all motion is taken to be relative, an observer only measures the positions of objects and their motions relative to himself. This ‘point of view’ of the observer is referred to as a ‘frame of reference’. Quantitatively, this frame of reference consists of the clock and set of rulers that a specific observer is using. For instance, I measure time using my own wristwatch, and I measure distances relative to where I am now.

Inertial frame of reference: When an observer is moving at a constant speed, they are said to be in an ‘inertial frame of reference’, i.e. a frame in which Newton’s law of inertia holds.

Event: Something that happens. In an idealized case, an ‘event’ happens at a single instant in time at a single point in space. A simple visual example of an event is the detonation of a firecracker. Different observers can measure the same event; each observer will use a different set of rulers and clocks to measure ‘when’ and ‘where’ the event occurs, but they will all be measuring the same thing (“Where were you when Kennedy was assassinated?”).

Coordinate system: We quantify measurements in a particular frame of reference using a coordinate system. Each observer measures the location of events in terms of four algebraic variables: x, y, z for spatial position (left/right, forward/backward, up/down) and t for time.

We can easily relate the behavior of objects, and events, as portrayed in different coordinate systems. In fact, most of us do this all the time without thinking about it formally. Consider a pair of cars driving down the expressway, and a third car stalled on the side of the road, as shown below*:

We have two observers in this picture: the ‘stationary’ observer O in the red car, and the moving observer O’ in the yellow car. The yellow car is moving at speed mph and the blue car is moving at speed mph, according to the person at the side of the road. (These are, of course, the values the speedometers of the two cars will read.) How fast is the the blue car moving relative to the observer O’? Clearly, the velocity of the blue car in the ‘primed’ reference frame is simply:

mph

In words, the person in the yellow car sees the blue car moving to the left at 10 mph relative to him. This sort of transformation is referred to as a Galilean velocity transformation. Such a transformation can be used to relate the velocities of objects as observed in different reference frames. More generally, suppose observer O sees observer O’ moving at velocity in the x-direction, in the y-direction, and in the z-direction. Similarly, observer O sees an airplane moving at velocity in the x-direction, in the y-direction, and in the z-direction. How fast does observer O’ see the airplane moving? The general Galilean velocity transformations are:

,

,

.

To change from the ‘point of view’ of observer O to the point of view of observer O’, we simply subtract the velocity of observer O’ from every velocity of interest.

We may also write a Galilean transform for position. If x, y and z are the positions of an event according to observer O, and x’, y’ and z’ are the positions of an event according to an observer O’, then the positions are related by

,

,

.

There are several important things to note about these equations:

1. The measurement of time is the same for observer O and observer O’: Newton assumed that there is an absolute measure of time. This means that every observer’s (perfect) clock runs at the same rate, no matter how they move. We will see that this is not true in reality.
2. All observers agree on the spatial distance between two events. If one event happens at position and another event happens at position , observer O will say the two events were separated by a distance . Using the Galilean transforms, we can readily show (assuming that the two events happen at the same time) that the observer O’ measures the same distance between events, i.e. .
3. We can derive the Galilean velocity transformations from these equations. Looking only at x, let us suppose two events which happen are separated by a distance and a time interval . We may then write, using the Galilean transform, that the two events are related by . If we divide both sides of this equation by , we get . Finally, we note that velocity = distance/time, so the above equation results in the Galilean velocity transform for x.

With the Galilean velocity transforms, we’re in a position to explain, semi-mathematically, why Newton’s laws are the same for any observer moving at a constant velocity. The crux of Newton’s laws of motion is Newton’s famous second law,

,

Which says that a force F acting on an object produces an acceleration (change in velocity per unit time) equal to . The right-hand side of Newton’s second law does not depend upon velocity at all, only on change in velocity; every observer moving at constant velocity (i.e. moving in an inertial reference frame) will measure exactly the same forces on an object, because they will agree on the acceleration of that object.

The best way to illustrate this is to look at a situation when observers disagree on the forces involved, because one of them is accelerating. Suppose we have a situation as illustrated below**. One observer is standing by the side of a set of railroad tracks, while another observer is sitting inside a closed railroad car. A pendulum is hanging from the ceiling of the railroad car.

If the train is moving at constant speed, the observer inside the train car is in an inertial reference frame and feels nothing out of the ordinary. Both the observer by the side of the train and the observer inside agree that there are no forces acting on the train car (except, of course, gravity).

Now suppose the train is accelerating, i.e. it is increasing its speed rapidly.

The person outside of the train will say that there is a force acting on the train which is pulling it to the right. The person inside the train, however, thanks to Newton’s first law (things at rest remain at rest, things in motion tend to remain in motion), will feel himself sliding towards the rear of the train car – the train is being pulled out from under him! The pendulum will be hanging at an angle, because the support of the pendulum is being pulled away from the pendulum as well. The person in the train, without any window to look out, can only conclude that a force is pulling him and everything inside the train car to the left.

The observer by the side of the tracks says a force is pulling the train to the right, while the person inside says that a force is pulling him to the left. They have a fundamental disagreement as to the actual physics occurring!

Such discrepancies are actually very familiar to most people. When you take a sharp turn around a corner in your car, you are pressed up against the side of the car by what seems to be an outward force, the ‘centrifugal force.’ In fact, you are a ‘victim’ of the law of inertia: your body wants to continue in a straight line, while your car is turning. An observer standing nearby will tell you that the actual force taking you around the turn is centripetal (inward). Centrifugal force is typically referred to as a pseudo-force: the turning of the car gives you the illusion that you are in fact being pushed against the side of the car. In reality, the car is pulling you around the turn!

Such discrepancies seem on the surface very troubling. Shouldn’t the laws of physics be the same no matter how an observer is moving? We will see, much later, that Einstein solved this problem as well, with his general theory of relativity. For now, we’ll stick to physical situations where the observers are in inertial reference frames.

So Newtonian relativity seems to work just fine; why does it in fact break down when electricity and magnetism get involved? Returning to Newton’s second law,

,

We have explained that the right-hand side of the equation is necessarily invariant under Galilean transformations. For forces of the form that Newton studied, such as gravity, the left-hand side of the equation is also invariant. The force of gravity only depends on the distance between two objects, and we have shown that the distance between two objects is the same for every observer.

However, the force a magnetic field exerts on a charged particle depends on the velocity of the particle! As we have seen, the Galilean transforms tell us that different observers will measure different velocities for an object. When studying magnetic fields, then, one finds that the left-hand side of Newton’s law depends upon what frame of reference one is traveling in. Different observers cannot measure different forces – as mentioned in the previous post, this would be akin to one person saying a car crash totaled both cars and another person saying that the crash was only a fender bender – so evidently we have reached a paradoxical situation. This paradox made scientists in the late 1800s/early 1900s suspect that, for magnetic forces, there was a ‘correct’ frame of reference (akin to Aristotle’s system of absolute rest). These scientists began to make measurements of the speed of light in an attempt to measure the Earth’s motion relative to this absolute rest frame. In the next post, we’ll give a brief historical background of measurements of the speed of light.

Whew! It’s tough to write physics posts without using calculus or vectors! If any readers have any questions, or anything seems unclear, feel free to leave me a comment!

Update:  I should really note that, in spite of fact that Einstein’s theory supplanted Newton’s, Newton’s relativity is in essence correct: it accurately describes most terrestrial phenomena, as long as the relative speeds between objects are small.  What Einstein showed is that Newton’s theory is a subset of a much broader relativistic formalism.  We’ll be discussing this in more detail as these posts progress.

* As you can see, this picture clearly demonstrates that I majored in physics, not art…

** This example is inspired by a similar example in Tipler and Llewellyn’s textbook, Modern Physics.

Einstein’s theories of relativity are certainly the among most elegant of all of physics. Incredibly deep and counterintuitive consequences can be derived from the statement of a small number of simple postulates, and general features of the special theory of relativity are accessible to anyone who has some familiarity with algebraic manipulation.

But no theory is created ‘in a vacuum’ (pun intended), and Einstein’s is no exception. Relativity has its roots in the very beginnings of what we now call physics, so we begin our discussion with a short introduction to the events and observations that led up to Einstein’s magnificent theories. This post will be pretty much bereft of math; later posts will include algebraic operations as needed.

The foundations of relativity were set by Galileo Galilei (1564-1642) in the 16th century. Galileo, pictured below, was an Italian scientist/philosopher who is generally credited with giving birth to modern physics.

Among his many contributions, Galileo was a master astronomer, and produced some of the best telescopes of his age. With these telescopes he made many important discoveries, including the first observations of Jupiter’s four largest moons, the rings of Saturn, and lunar mountains and craters. A number of his discoveries lent support to a heliocentric (Sun-centered) model of the solar system, and Galileo became an ardent supporter of this model.

At the time, however, scientific consensus and Church doctrine proclaimed that the Earth was the center of the universe, and that the planets, Sun and stars rotated about it on ‘celestial spheres’. This system, referred to as geocentrism, was championed by Aristotle in 5th century B.C. Greece and the astronomer Ptolemy in 2nd century B.C. Egypt. The first major challenge to this system came about in 1543 with the publication of Nicolaus Copernicus‘ book On the Revolutions of the Heavenly Spheres. Copernicus argued, and backed up with observations, that the Sun was the actual center of the solar system (heliocentrism).

A number of arguments were made against this Copernican system. One of the most powerful, at least at the time, was that if the Earth was in fact moving, we should be able to ‘feel’ the effects of this motion. Contemporary opponents of Galileo and Copernicus would argue that, if the Earth is spinning on its axis, a cannonball dropped from a tower should land some distance to the west, since the Earth would have moved some distance to the east as the cannonball fell. Similarly, you should be able to throw a ball farther to the west than to the east, since the ground is ‘moving towards’ the ball in the first case, and ‘moving away’ from the ball in the second. This argument is based on Aristotle’s flawed idea that the natural state of an object is to come to some sort of ‘absolute rest,’ i.e. an object put into motion will inevitably slow down and stop moving.

Such an idea seems reasonable in our everyday perceptions of the world. If you roll a ball along the ground, it will gradually come to a stop. If you take your foot off the gas pedal in a car, the car will gradually slow down and stop. We understand today that this slowing down occurs because of frictional forces between the car and ground and wind resistance between the car and the air, but to the early scientists it seemed eminently reasonable that the natural state of things is to remain motionless. From that viewpoint, one would expect that all things would be naturally dragged to the west if the Earth was in fact in motion.

Galileo refutes this argument and numerous others in his book Dialogue Concerning the Two Chief World Systems, in which he compares the Copernican and Ptolemic cosmologies. The text is written in the form of a dialogue between three men: Salviati (pro-Copernican), Sagredo (initially neutral), and Simplicio (pro-Ptolemic). In response to the arguments stated above, Salviati presents the following ‘thought experiment’:

Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though there is no doubt that when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than toward the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump…

In other words, the laws of physics work the same regardless of your (constant) rate of motion. There is no physical way to detect the presence, or absence, of absolute motion. The only thing that matters is relative motion between objects. This is relativity, in a nutshell: there is no such thing as absolute motion, only relative motion matters. On Earth, we call ourselves moving if we can see ourselves moving relative to the ground, or can feel ourselves moving relative to the wind; if you take away these considerations, as Galileo did by imagining himself in the cargo hold of a ship, there is no physical way to tell that you’re moving.

Of course, Galileo’s arguments (and his book) in favor of a Copernican solar system got him in trouble with the Catholic Church, and in 1633 (a year after the book’s publication) he stood trial for heresy in front of the Inquisition. He was forced to recant his theories, and his book was banned. This ban lasted until 1741, when a slightly censored version of the text was allowed to be published. (Oddly, as I’ve chronicled previously, there are still Biblical champions of the geocentric theory.)

The next step in developing the theory of relativity came from Isaac Newton (1643-1727), pictured below.

Newton made numerous contributions to the sciences, including important studies in optics, gravitation, and the development of calculus. He also made fundamental contributions to the study of mechanics (the study of how objects move when subjected to forces) which were compiled into his Mathematical Principles of Natural Philosophy (Philosophiae Naturalis Principia Mathematica). Here, Newton compiled his now famous three laws of motion, taken (in translated form) from his own words:

1. The law of inertia: “Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.”
2. The law of acceleration: “The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed.” (F = ma)
3. The law of reciprocal actions: “To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.”

We will not for now spend time discussing these laws. Newton, however, was aware of the works of Galileo and Copernicus before him, and it is to be noted that relativity is implicitly built into them. Before stating his laws, Newton spends quite some time discussing notions of “absolute” and “relative” space and time. To quote him again,

Hitherto I have laid down the definitions of such words as are less known, and explained the sense in which I would have them to be understood in the following discourse. I do not define time, space, place and motion, as being well known to all. Only I must observe that the vulgar conceive those quantities under no other notions but from the relation they bear to sensible objects. And thence arise certain prejudices, for the removing of which, it will be convenient to distinguish them into absolute and relative, true and apparent, mathematical and common.

…

IV. Absolute motion is the translation of a body from one absolute place into another; and relative motion, the translation from one relative place to another. Thus in a ship under sail, the relative place of a body is that part of a ship which the body possesses; or that part of its cavity which the body fills, and which therefore moves together with the ship: and relative rest is the continuance of the body in the same part of the ship, or of its cavity. But real, absolute rest, is the continuance of the body in the same part of that immovable space, in which the ship itself, its cavity, and all that it contains, is moved. Wherefore, if the earth is really at rest, the body, which relatively rests in the ship, will really and absolutely move with the same velocity which the ship has on the earth. But if the earth also moves, the true and absolute motion of the body will arise, partly from the true motion of the earth, in immovable space; partly from the relative motion of the ship on the earth; and if the body moves also relatively in the ship; its true motion will arise, partly from the true motion of the earth, in immovable space, and partly from the relative motions as well of the ship on the earth, as of the body in the ship; and from these relative motions will arise the relative motion of the body on the earth.

…

But because the parts of space cannot be seen, or distinguished from one another by our senses, therefore in their stead we use sensible measures of them. For from the positions and distances of things from any body considered as immovable, we define all places; said then with respect to such places, we estimate all motions, considering bodies as transferred from some of those places into others. And so, instead of absolute places and motions, we use relative ones…For it may be that there is no body really at rest, to which the places and motions of others may be referred.

There’s a lot to digest in those excerpts, but we can summarize them relatively concisely.

The first excerpt observes that understandings of space, time and motion that arise from everyday experience lead one to certain ‘prejudices’, or misunderstandings, about the nature of these quantities. In particular, there is a confusion concerning the ideas of ‘absolute’ and ‘relative’ motion. These seem to be the same misconceptions we discussed earlier in the context of Galileo’s work.

The second excerpt first discusses ‘relative’ and ‘absolute’ motion. Using what is likely a Galileo-inspired example of an object in a ship’s hold, Newton points out that the ‘absolute’ motion of an object inside a moving ship is quite complicated: one must account for the motion of the object within the ship, the motion of the ship on the Earth, and the motion of the Earth with respect to some ‘absolute’ reference point.

In the third excerpt Newton observes that, even though there may be an ‘absolute’ reference point in space from which all motion should be measured, we do not know where that point is and must therefore work only with relative motions of objects.

This ‘Newtonian principle of relativity’ is the observation that Newton’s laws work just as well for any observer moving with constant velocity. To put it in more physical terms, all observers agree on the forces involved in some sort of motion. If two cars collide on the road, measurements of the force of impact by an observer standing by the side of the road and by an observer driving by on the opposite side of the road will be in agreement.

Newton’s observation that the laws of physics work equally well for all observers moving at constant speed held for two centuries. In Newton’s time, however, the only clearly established laws of physics were Newton’s formulas which described how forces related to motion. By the late 1800s, a new set of formulas had been established, by James Clerk Maxwell (1831-1879), pictured below.

Before Maxwell’s time, it was known that electric and magnetic forces were related to one another, but a complete description of their relationship was lacking. In 1864, this scruffy-bearded scientist presented a set of equations to the Royal Society which gave a complete description of electromagnetic phenomena. Furthermore, Maxwell demonstrated that his equations predicted the existence of electromagnetic waves, and further suggested that light is an electromagnetic phenomena, a suggestion which is now well-known to be true. In Maxwell’s own words,

The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.

For completeness, we present Maxwell’s equations in free space, in modern mathematical parlance, below:

,

,

,

.

(There are, of course, multiple forms of Maxwell’s equations, depending on the system of units chosen. I’m using the theorist-friendly Gaussian system.)

A problem arises when applying Newton’s version of relativity to Maxwell’s equations: two people observing the same electromagnetic phenomena with different speeds will evidently have different opinions on the forces involved! We can illustrate this with a simple thought experiment: suppose we have a single electric charge q situated a distance d from an electrically charged wire with charge density (charge/meter) λ:

The observer on the left will see an electric force of magnitude repelling the point charge and the wire. Under Newtonian relativity, the moving observer on the right will see the same electric force. However, for the moving observer, the point charge and the line charge appear to be moving.  The line charge will therefore also appear to be carrying a current I. This means there should also be a magnetic force between the line and the point also repelling them. The combination of Newtonian relativity and Maxwell’s equations suggests that the moving observer and the stationary observer come to different conclusions about the amount of force exerted between the wire and the point charge! This is a very unusual conclusion; returning to our earlier example of a car accident, it is comparable to saying that a stationary witness to the accident sees both cars totaled while the moving witness claims it was only a fender-bender!

Both observers cannot be right. Physicists at the time were faced with a choice: either Newtonian relativity, and implicitly Newton’s laws, were incorrect or incomplete, or relativity simply didn’t hold for Maxwell’s equations. This latter option brings back the idea that there is a ‘special’ frame of motion in which Maxwell’s equations are true: an observer moving in that frame of motion has the ‘correct’ view of electromagnetic waves, and everyone else is ‘wrong’.

This didn’t seem quite so outlandish at the time. Once it had been determined that light was an electromagnetic wave, it was natural to ask what medium carried that wave. After all, water waves are carried in water, and sound waves are carried in air. By analogy, one would expect that there exists some previously undiscovered medium whose vibrations created light waves. This medium was dubbed ‘the aether‘, and it was assumed that the ‘correct’ frame of reference was an observer standing still with respect to the aether.

The next logical step, if the aether existed, would be to measure its properties somehow. If the Earth moves through the aether, the speed of light as measured on Earth should depend on how fast it moves relative to the aether. Experimentalists began to try to measure this motion, and their inability to detect any ‘aether drift’ led to Einstein’s scientific revolution.

In the next post, we’ll discuss experimental techniques to measure the speed of light, including the famous Michelson-Morley experiment. Once we have seen the outcome of their experiment, we’ll be well positioned to understand why Einstein needed to develop a new theory of relativity.