Einstein's Pathway to Special Relativity

Background reading: J. Schwartz and M. McGuinness, Einstein for Beginners. New York: Pantheon.. pp. 1 - 82.

We have now reviewed the developments in the physics of moving bodies, of light, of electricity and magnetism that brought the physics that Einstein found when he began to think about ether, electricity, magnetism and motion.

It was pondering these developments that led Einstein to discover the special theory of relativity in 1905. The discovery was not momentary. The theory was the outcome of, in Einstein's own reckoning, seven and more years of work. He even places one of his early landmarks in a thought experiment he had at the age of 16, in 1896, nine years before the year of miracles of 1905. Unfortunately we have only fragmentary sources to document the years of this struggle. Below I identify a few of the major ones.

The story of Einstein's discovery of special relativity has exercised an almost irresistible fascination on many, in spite of the dearth of sources. So, if you read more widely, you will see much speculation over how to fill in the blanks between the known landmarks and even over which are the important landmarks. Some of it is responsible; some is not.

Chasing a beam of light

Einstein in high school

Writing a half century later in 1946 in his Autobiographical Notes, Einstein recounted a thought experiment conducted while he was a 16 year old student in 1896 that marked his first steps towards special relativity.

"...a paradox upon which I had already hit at the age of sixteen:
If I pursue a beam of light with the velocity c (velocity of light in a vacuum), I should observe such a beam of light as an electromagnetic field at rest though spatially oscillating.
There seems to be no such thing, however, neither on the basis of experience nor according to Maxwell's equations.
From the very beginning it appeared to me intuitively clear that, judged from the standpoint of such an observer, everything would have to happen according to the same laws as for an observer who, relative to the earth, was at rest. For how should the first observer know or be able to determine, that he is in a state of fast uniform motion?
One sees in this paradox the germ of the special relativity theory is already contained."

The basic thought is clear. If Einstein were to chase after a propagating beam of light at c

he would see a frozen light wave

and that Einstein deemed impossible.

At first it seems that is will be simple to figure out just what is worrying Einstein. He states a few simple reasons. I don't want to go into them here since they actually turn out to be rather hard to disentangle. My best effort to disentangle them is given at "Chasing a Beam of Light: Einstein's Most Famous Thought Experiment," http://www.pitt.edu/~jdnorton/Goodies/Chasing_the_light

Magnet and conductor

Einstein's thinking evolved from this early, youthful flight into richer and technically more detailed scrutiny of motion in Maxwell's electrodynamics. Einstein initially took the idea of an ether state of rest seriously and conceived experiments that were designed to reveal the earth's motion through the ether.

These thoughts eventually took a very different turn with Einstein deciding that the ether state of rest had no place in electrodynamics and that the principle of relativity was to be upheld. The decisive moment seems to have come with a thought experiment, the magnet and conductor, that is recounted in the opening paragraph of Einstein's 1905 paper.

This is a version of that thought experiment that is modified slightly from the way Einstein sets it up. (Caution!)

The simple idea behind the thought experiment is that Maxwell's electrodynamics treats a magnet at rest in the ether very differently from one that moves in the ether. A magnet at rest is surrounded by a magnetic field only.

However, if the magnet moves through the ether, things are very different. In addition to the magnetic field, a new entity comes into being around the magnet, an induced electric field.

The creation of the electric field draws on details of Maxwell's theory that need not distract us here. Briefly, as the magnet moves past a fixed point in the ether, the magnetic field strength changes with time at that point. That change in field strength, according to Maxwell's theory, creates an electric field.

This difference between the two cases seems to provide an unequivocal marker of motion through the ether--or so it would seem. To determine if a magnet is moving absolutely through the ether or not, one merely needs to look for that induced electric field. That is easy to do. An electric field accelerates electric charges, such as the conducting electrons in a piece of wire, a conductor. So all that has to be done is to place a conductor near the magnet, as the figures show, and to look for an induced electric current. If there is one, then there is an induced electric field and magnet is moving; if there isn't one, then the magnet is at rest in the ether.

It all seems so straightforward. But it doesn't work. The simplest situation arises if we attach the conductor to the magnet so that it moves or rests with the magnet. If the magnet is at rest in the ether, then there will be no current in the conductor. So far, it is as expected. But if the magnet and conductor move together an extra complication enters. Because the conductor is now moving absolutely in a magnetic field, another part of Maxwell's theory tells us that a second electric current will be induced in the conductor. Remarkably that second current flows in the opposite direction to the one produced by the electric field and it turns out to cancel it out exactly.

The upshot is that checking for an electric current in the conductor fails as a means of distinguishing the absolute rest of the magnet from its motion. In both cases, the current is the same--no current at all. So an Einstein riding with an absolutely moving magnet, would detect no current and find the situation to be indistinguishable from absolute rest as far as the observable currents were concerned.

More curiously, it is as if the electric field just isn't there for an observer moving with the magnet. But one at rest in the ether would say there is an electric field present.

Einstein later described how this realization had affected him quite profoundly:

"In setting up the special theory of relativity, the following ... idea concerning Faraday’s magnet-electric induction [experiment] played a guiding role for me... [magnet conductor thought experiment described]. ...The idea, however, that these were two, in principle different cases was unbearable for me. The difference between the two, I was convinced, could only be a difference in choice of viewpoint and not a real difference. Judged from the [moving] magnet, there was certainly no electric field present. Judged from the [ether state of rest], there certainly was one present. Thus the existence of the electric field was a relative one, according to the state of motion of the coordinate system used, and only the electric and magnetic field together could be ascribed a kind of objective reality, apart from the state of motion of the observer or the coordinate system. The phenomenon of magneto-electric induction compelled me to postulate the (special) principle of relativity. [Footnote] The difficulty to be overcome lay in the constancy of the velocity of light in a vacuum, which I first believed had to be given up. Only after years of [jahrelang] groping did I notice that the difficulty lay in the arbitrariness of basic kinematical concepts."

Einstein, Albert (1920) “Fundamental Ideas and Methods of the theory of Relativity, Presented in Their Development,” Collected Papers of Albert Einstein, Vol. 7, Doc. 31. Einstein in 1920

In sum Einstein's lesson was this. Maxwell's theory employed an ether state of rest; but that state of rest could not be revealed by observation. So somehow the principle of relativity needed to be upheld.

In retrospect, this relativity of the induced electric field had, in effect, committed Einstein to the relativity of simultaneity, although he certainly did not know it at the time. A simple thought experiment shows that it can only be reconciled with Maxwell's electrodynamics if we give up the absoluteness of simultaneity. See From the Magnet and Conductor to the Relativity of Simultaneity on my "Goodies" page.

And a second moral was an unexpected relativity. Prior to Einstein, it had been thought that whether an electric field is present at some place is an absolute fact. Einstein now concluded that it is observer dependent: some observers will judge an electric field to be present; others in a different state of motion will not. This was the first of Einstein's reorganization of our ideas of which quantities are absolute and which relative.

Emission theories of light

The magnet and conductor thought experiment marked the way forward for Einstein. He was to uphold the principle of relativity in electrodynamics. The only obvious way of doing that was to modify electrodynamical theory. As the concluding footnote in Einstein's quote from 1920 above suggests, Einstein could already know one element that must be in the modification. According to Maxwell's theory, light always propagates at c with respect to the ether. That result must change if the theory conforms to the principle of relativity since there will no longer be an ether state of rest against which the motion of the light can be judged.

Walther Ritz

We know from later recollections what one of Einstein's modified versions of electrodynamics looked like. In that version, the velocity of light is a constant, not with respect to the ether, but with respect to the source that emits the light. Such a theory is called an "emission" theory of light and, if the other parts of the theory are well behaved, will satisfy the principle of relativity.

Einstein later recalled that the theory he developed was essentially that developed later by Walther Ritz in 1908. In Ritz's theory--and thus probably also in Einstein's theory--all electrodynamic action, not just light, propagated in a vacuum at c with respect to the actions source. The essential change is shown in the animation:

For experts: the way to built the theory was actually very easy. If Maxwell's theory is formulated in terms of retarded potentials, one needs only to tinker with the formula for the retardation time to bring the whole theory into the form of an emission theory. Everything else can stay the same.

In Maxwell's theory, all electrodynamic action, generated by a source charge at some moment, propagates at c from the fixed point in the ether occupied by the source at that moment.

In a Ritz-style emission theory, all electrodynamic action, generated by a moving source, propagates at c from a point that moves at uniform velocity with the source.

Here is a non-animated version:

My own best effort to reconstruct of the details of Einstein's theory can be found in "Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905," Archive for History of Exact Sciences, 59 (2004), pp. 45-105.

Crisis: the relativity of simultaneity

It was a lovely theory. But it didn't work. We can only guess what the problems were. But we know he found many. Indeed Einstein seems to have expended considerable energy trying to figure out if any emission theory might work. His later recollections are littered with different reasons for why no emission theory at all could do justice to electrodynamics.

My own conjectures on how these arguments may have worked are discussed in part in my"Chasing a Beam of Light:Einstein's Most Famous Thought Experiment,"

An emission theory fails. So Einstein would have found himself in an impossible position. The speed of light cannot vary with the speed of the emitter; presumably it must be a constant, as Maxwell's theory had urged all along. Yet in addition, Einstein was convinced that the principle of relativity must obtain in electrodynamic theory. How can both obtain? They require the speed of light to be the same for all inertial observers?

The footnote already quoted above points us to Einstein's next step.
"The difficulty to be overcome lay in the constancy of the velocity of light in a vacuum, which I first believed had to be given up. Only after years of [jahrelang] groping did I notice that the difficulty lay in the arbitrariness of basic kinematical concepts."

The key to the puzzle is the relativity of simultaneity. If Einstein gives up the absoluteness of simultaneity, then the principle of relativity and the constancy of the speed of light are compatible after all. The price paid for the compatibility is that we must allow that space and time behaves rather differently than Newton told us.

More importantly for Einstein's struggles of that time is an extra bonus: it turns out that within the new theory of space and time of special relativity, Maxwell's electrodynamics does not need to be modified at all. It turns out to be compatible with principle of relativity just as it is. That would have been a very satisfactory outcome for Einstein.

Einstein recounted later the moment of discovery. In a lecture in Kyoto on December 14, 1922, he is reported by Ishiwara, who took notes in Japanese, to have said:

"Why are these two things inconsistent with each other? I felt that I was facing an extremely difficult problem. I suspected that Lorentz’s ideas had to be modified somehow, but spent almost a year on fruitless thoughts. And I felt that was puzzle not to be easily solved. But a friend of mine living in living in Bern (Switzerland) [Michele Besso]helped me by chance. One beautiful day, I visited him and said to him: ‘I presently have a problem that I have been totally unable to solve. Today I have brought this “struggle” with me.’ We then had extensive discussions, and suddenly I realized the solution. The very next day, I visited him again and immediately said to him: ‘Thanks to you, I have completely solved my problem.” My solution actually concerned the concept of time. Namely, time cannot be absolutely defined by itself, and there is an unbreakable connection between time and signal velocity. Using this idea, I could now resolve the great difficulty that I previously felt. After I had this inspiration, it took only five weeks to complete what is now known as the special theory of relativity."

Einstein taking sake A portrait of Einstein by the cartoonist Okamoto Ippei (1886-1948), done in December of 1922 in Sendai, Miyagi Prefecture, Japan

David Hume

This moment of recognition of the relativity of simultaneity is one of the great moments of discovery in science and, at this moment philosophical reflections played a key role. Absolute simultaneity seems an uncontroversial part of the world. How could we give it up? Einstein had been reading many philosophers, including Hume and Mach. They had stressed that concepts are our servants, not our masters, and they are warranted only in so far as they might be grounded in experience. So was absolute simultaneity grounded properly in experience? Einstein began to think about the experiences that we use to establish simultaneity of events and he realized that it was not. Reading these philosophers gave him the courage to continue and to abandon absolute simultaneity. In its place came the relativity of simultaneity.

Ernst Mach

For an account of how reading Hume and Mach helped, see my "How Hume and Mach Helped Einstein Find Special Relativity."

The turn to principles

The moment of the recognition of the relativity of simultaneity came, in the above account, 5 weeks prior to Einstein's completion of the 1905 paper (and in another 5 to 6 weeks). In these five to six weeks in which he pulled together the pieces of the finished theory, Einstein made one more very significant methodological advance that would forever color how we see relativity theory.

Einstein's pathway to discovery amounted to the recognition that if you take Maxwell's electrodynamics seriously you have to see that built into it is both the principle of relativity and a new kinematics of space and time that supports it. Yet Einstein does not simply argue it that way in the finished paper.

The reason is not hard to see. Prior to, just a few months before completing his 1905 special relativity paper, Einstein had published a paper in which he had foreshadowed the demise of Maxwell's electrodynamics! In his earlier light quantum, Einstein had advanced the astonishing assertion that sometimes light does not behave like a wave as Maxwell's theory demanded; sometimes it behaved like a spatially localized collection of energy.

So how could Einstein now base a new theory of space and time on Maxwell's theory? He knew something was very right about Maxwell's theory. There was also something very wrong about it. How could one theorize in such an unstable environment. The answer came to Einstein, as he reported in his Autobiographical Notes, in a distinction of what he called constructive theory from theories of principle.

"Reflections of this type made it clear to me as long ago as shortly after 1900, i.e., shortly after Planck's trailblazing work, that neither mechanics nor electrodynamics could (except in limiting cases) claim exact validity. Gradually I despaired of the possibility of discovering the true laws by means of constructive efforts based on known facts. The longer and the more desperately I tried, the more I came to the conviction that only the discovery of a universal formal principle could lead us to assured results. The example I saw before me was thermodynamics. The general principle was there given in the theorem: The laws of nature are such that it is impossible to construct a perpetuum mobile (of the first and second kind). How, then, could such a universal principle be found?"

In effect, what Einstein saw was that he did not really need all of Maxwell's theory for his new account of space and time. He needed only a few core ideas robust enough to survive the coming quantum revolution. Following the model of thermodynamics, these few core ideas would be advanced as principles from which the entire theory could be deduced.

What could those principles be? The principle of relativity itself was an obvious choice. He also needed something that distilled the relevant essence of Maxwell's electrodynamics. What about the hardest won lesson of his years of work towards the final theory: the recognition that an emission theory of light must fail? That is, that Maxwell's theory was right after all in demanding that that light always propagates at c, no matter how fast the emitter may be moving? That became the second principle, light postulate. Those two principles proved to be sufficient to allow the entire theory to be deduced. Einstein laid out both as his postulates and the theory adopted its now familiar form.

Three Components

We have seen three components in Einstein's discovery:

• Astute analysis of new and surprising experiments.
• Deeply reflective philosophical analysis of the nature of time and physical theories.
• Solving an incongruous and overlooked problem in the foundations of electricity and magnetism.

While all three had a role in Einstein's discovery, the last was the most decisive. Unfortunately this is often overlooked in accounts of the origins of Einstein's theory. Einstein's engagement with current experiments and his facility in philosophical analysis are important. However special relativity would not have come about at all were it not for the particular problems in electrodynamics addressed by Einstein and which demanded a radical solution.

Einstein's 1905 "On the Electrodynamics of Moving Bodies"

Einstein arrived at his "On the electrodynamics of moving bodies," which is my best candidate for the most famous scientific paper ever written.

An online version of this paper is here. Beware of a famous misrendering in this standard edition as noted in this version of the first two sections.

The paper has several parts. First there is an introduction. It commences with the recounting of the magnet and conductor thought experiment. It then announces the project of solving the resulting problem with a new theory of space and time based on the principle of relativity and the light postulate.

In the first "Kinematical Part" of the paper, Einstein develops the parts of the theory devoted only to space and time. Its first section, "Definition of Simultaneity," Einstein gives his celebrated analysis of the relativity of simultaneity. It is one of the most celebrated conceptual analyses of the century and a model very many others tried to follow.

The second "Electrodynamical Part" proceeds to what must have seemed for Einstein in 1905 to be the real benefit of the paper. He proceeded to show how Maxwell's electrodynamics was already a theory that conformed to the principle of relativity and noted that this fact made solution of some problems in electrodynamics very easy. For a problem concerning moving systems, such as the reflection of light off a moving mirror, was really the same as another much easier problem with resting bodies, such as the reflection of light off a resting mirror. If you could solve the easy problem, then the principle of relativity let you write down a solution to the harder one almost immediately, just by transforming your viewpoint from one frame of reference to another.