Chasing a Beam of Light:
Einstein's Most Famous Thought Experiment

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh, Pittsburgh PA 15260
Homepage: www.pitt.edu/~jdnorton
This page (with animated figures) is available at www.pitt.edu/~jdnorton/goodies

Einstein recalled how, at the age of 16, he imagined chasing after a beam of light and that the thought experiment had played a memorable role in his development of special relativity. Famous as it is, it has proven difficult to understand just how the thought experiment delivers its results. It fails to generate serious problems for an ether based electrodynamics. I propose a new way to read it that fits it nicely into the stages of Einstein's discovery of special relativity. It shows the untenability of an "emission" theory of light, an approach to electrodynamic theory that Einstein considered seriously and rejected prior to his breakthrough of 1905.

For more details, see:

"Chasing the Light: Einstein's Most Famous Thought Experiment," prepared for Thought Experiments in Philosophy, Science and the Arts, eds., James Robert Brown, Mélanie Frappier and Letitia Meynell, Routledge. Download.

Sections 5-6 of "Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905," Archive for History of Exact Sciences, 59 (2004), pp. 45­105. Download.

1. The Puzzle

How could we be anything but charmed by the delightful story Einstein tells in his Autobiographical Notes of a striking thought he had at the age of 16? While recounting the efforts that led to the special theory of relativity, he recalled

"...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 thought is simplicity itself. Here is light, a waveform propagating at c:

If the young Einstein were to chase after it at c, he would catch up with the wave and be moving with it, like a surfer riding the wave. He would see a frozen lightwave.

The untenability of that thought led to the downfall of the great achievement of nineteenth century physics, the ether, which then provided the basis for all electromagnetic theory.

The trouble is that it is quite unclear just how this thought creates difficulties for the ether. Einstein gave three reasons and each of them could be answered readily by an able ether theorist.

So what are we to make of the thought experiment? Perhaps it is no more than the recording of the visceral hunches of a precocious 16 year old who did not even study Maxwell's theory until two years later. This is a possibility we cannot rule out. If it is correct, then we need not puzzle any further over how the thought experiment works, for there is little more to be found that illuminates Einstein's pathway to special relativity.

But then we must ask why the thought experiment merits pride of place in Einstein's defining autobiography? Does it have a cogency that extends beyond Einstein's final high school year? That Einstein mentions Maxwell's equations in the thought experiment suggests their relevance to the operation of the thought experiment and thus that this operation was pertinent to Einstein's later thought, after he had learned Maxwell's equations.

While we cannot know on the evidence available if the thought experiment truly had cogency beyond the mullings of Einstein's 16th year, we can ask if there are plausible accounts of Einstein's pathway to special relativity in which the thought experiment figures more significantly.

2. A Proposed Solution

There is a way of understanding how the thought experiment could have a significance that extended well beyond the confines of Einstein's final year at high school. The key is not to relate the thought experiment to ether theories of electromagnetism. Rather, we know that Einstein devoted some effort during the years leading up to his discovery of 1905, to so-called "emission" theories of light and electromagnetism. Einstein eventually found such theories objectionable and untenable.

I propose that Einstein's thought experiment provided an especially cogent way of formulating those objections and thereby supported Einstein in his final decision: it give up an emission theory in favor of retaining the celebrated Maxwell-Lorentz theory, but with a radically altered theory of space and time.

3. An Emission Theory of Light and Electromagnetism

On many later occasions, Einstein recalled that, prior to his discovery of special relativity, he had investigated emission theories, indicating a similarity in his approach to that used by Walter Ritz. In the then standard electrodynamics of Maxwell and Lorentz, electromagnetic action always propagated at c with respect to the ether. The simplest example was the propagation of a lightwave. But it held equally for the action of one charge upon another. It was this fact that made it seem impossible to conform the principle of relativity to electromagnetism. The ether supplied a preferred state of rest essential to the theory, but incompatible with the idea that all inertial states of motion are equivalent.

So Ritz in 1908, and Einstein sometime before 1905, tried to modify electromagnetic theory in such a way that electromagnetic effects are always propagated at c with respect to the source of the effect. If such a theory could be found, it would no longer require an ether state of rest and it would reasonable to expect that it could conform to the principle of relativity.

The animation below displays the difference. On the left, in the Maxwell-Lorentz theory, electromagnetic action propagates from a fixed point in the ether. So when two charges moving together act on each other, the source of the effect felt by one is a fixed point in the ether left behind by the moving source. Since the effect propagates from a point left behind by the moving charges, an observer moving with the charges can use this fact to determine that the charges are moving.

On the right, we see the corresponding process in a modified "emission" theory, such as devised by Ritz and Einstein. The motion of the source is added to the propagating effect. So now the effect propagates isotropically from a point that moves with the source. To see this, notice how the expanding spherical shells remain centered on the moving positive charge that is their source, just as would happen if the two charges were at rest. The propagation of electromagnetic effects can no longer be used by observers moving with the two charges to detect their absolute motion; the principle of relativity is no longer threatened.

The simplest electromagnetic action is the propagation of light. So in this theory, the velocity of the emitter--the source--is added to the velocity of the light emitted. For this reason it is known as an "emission" theory.

Promising as this must initially have seemed to an Einstein intent on restoring the principle of relativity, the emission theory was ultimately rejected by Einstein. His later correspondence and papers are littered with remarks on the problems the theory faced. Two will return as our story unfolds.

- In a letter to Paul Ehrenfest of June 1912 (and elsewhere), Einstein remarked that an emission theory ran afoul of an elementary result of optics: the physical state of a ray of light is determined completed by its intensity and color (and polarization).

- In an interview with R. S. Shankland in the 1950s, Einstein remarked that the theory could not be formulated as a local field theory that is, in terms of differential equations.

In a local field theory, we reconstruct how a field evolves over time by taking its state at one instant and consulting the theory's differential field equations. These equations take the present state of the fields and tell us how rapidly they are changing. From these rates of change we can then infer the states of the field at future times. (A similar analysis tells us how the field will alter at different parts of space.)

4. Einstein's Thought Experiment in the Context of an Emission theory of Light

Let us now return to Einstein's thought experiment and imagine that its target has become an emission theory of light. We immediately see that the three objections Einstein's reports present serious obstacles to an emission theory. Let us take the three objections in order.

1. The first objection was that we don't actually experience frozen light. That is a puzzle in an emission theory of light. We must presume that there are light sources with all sorts of velocities around us. A light source moving rapidly away from us will emit a lightwave that propagates slowly with respect to us. The most extreme case is of light source moving away from us at c. That source will leave a frozen light wave behind in space, as the animation shows:

So, if an emission theory is the correct theory of light, we should expect eventually to run into frozen lightwaves, emitted by rapidly receding sources. But we experience no such thing.

2. The second objection was that frozen light was incompatible with Maxwell's equations. Why should this be a problem for an emission theory when such a theory does not employ Maxwell's equations? It will be a problem, but it takes a few steps to arrive at the conclusion.

First note that an emission theory allows frozen light in ordinary circumstances; we don't need to be moving at c to find it. That means that a frozen light wave must be a part of electrostatics and magnetostatics, the theories of static electric and magnetic fields. Now Maxwell's electrodynamics evolved over the course of half a century and built on a long series of experiments in electricity and magnetism. An emission theory must adjust the theory, but it cannot alter it too radically on pain of incompatibility with those experiments. The one part of Maxwell's theory that seems most secure is its simplest part, its treatment of static electric and magnetic fields. So we would expect a successful emission theory to agree with Maxwell's theory in this simplest and most secure part.

Now we have a problem: An emission theory allows the existence of frozen light waves. But the emission theory must agree closely with the treatment of static fields in Maxwell's theory and Maxwell's theory does not admit the static fields that corresponds to frozen light waves.

3. In his third objection, Einstein lamented for the observer catching the light beam, "...For how should the first observer know or be able to determine, that he is in a state of fast uniform motion?" Of course, in the context of an emission theory, the "state of fast uniform motion" must be read as "fast uniform motion with respect to the source of the light."

At first it is not clear why it should matter at all whether the observer catching the light beam can make this judgment. It turns out to be important if the overall emission theory of light is to be deterministic; that is, if the present state of fields and the like in space are to be able to determine how they will develop in the future. Einstein's worry is that determinism will fail. To see why, imagine that you are an observer given a waveform, but all you know of it is its state at the present instant.

Light wave at an instant

Would you be able to tell whether the waveform is one that is frozen in space;

One possible future: a frozen light wave

or whether it is one that is propagating past you?

Another possible future: a propagating light wave

Both are possible in an emission theory. Which is the case depends upon your velocity with respect to the light's source. If you are moving at c with respect to the source, the wave is frozen. If you are at rest with respect to the source, the wave is propagating at c.

Can you tell which case you have by merely looking at the waveform at an instant? You cannot. Einstein's earlier remark about light is now decisive. A light wave is fully characterized by its color, intensity and polarization and both cases agree on these properties. The waveform has no property at an instant that would enable you to tell what its future time development would be. This is indeterminism. The present state of the wave does not determine its future time development.

While this circumstance might just be just an odd incompleteness of our knowledge, it becomes a crisis if we imagine that we are not human observers but the differential equations of a local field theory. For, as we saw above, a basic function of those field equations is to take the present state of the fields and from them infer the rates of change of the field. Those rates of change then determine the time development of the waveform--whether it propagates or not and how fast it propagates. This essential function will not be possible in an emission theory, for the instantaneous state of the lightwave does not determine the rates of change of the field.

Hence, thanks to Einstein's thought experiment, we infer that an emission theory cannot be formulated as a local field theory.

We can summarize the problems brought by Einstein's thought experiment to an emission theory:

5. Conclusion

When Einstein abandoned an emission theory of light, he had also to abandon the hope that electrodynamics could be made to conform to the principle of relativity by the normal sorts of modifications to electrodynamic theory that occupied the theorists of the second half of the 19th century. Instead Einstein knew he must resort to extraordinary measures. He was willing to seek realization of his goal in a re-examination of our basic notions of space and time. Einstein concluded his report on his youthful thought experiment:

"One sees that in this paradox the germ of the special relativity theory is already contained. Today everyone knows, of course, that all attempts to clarify this paradox satisfactorily were condemned to failure as long as the axiom of the absolute character of time, or of simultaneity, was rooted unrecognized in the unconscious. To recognize clearly this axiom and its arbitrary character already implies the essentials of the solution of the problem."

Copyright John D. Norton, December 2004. Rev. February 15, 2005. Reformatted April 14, 2005 on a transatlantic flight returning to Pittsburgh from the Israel Academy of Science and Humanities conference on Einstein.