Surfaces and Essences

by Douglas Hofstadter

  In any case, whether it was a vertical or a horizontal mental move, Einstein’s extension of the Galilean principle of relativity wound up profoundly undermining much of the physics of the preceding three centuries. And yet this revolution emerged from the act of paying attention to the trivial-seeming similarity between experiments in a train (or any similar reference frame) that were limited to mechanics, and experiments that might also involve electromagnetism. Einstein’s intuition told him that any such distinction was unnatural, since, in the end, any conceivable experiment in any conceivable branch of physics belongs to the single unified tree of physics. Of course, this kind of sixth sense for how and when a category can be broadened is mysterious, and is one of the deepest of all arts.

  A Two-headed Flashlight Loses a Tiny Bit of its Mass

  The consequences of category-broadening by analogy, applied to the principle of Galilean relativity, were enormously deep and led Einstein to a rich network of ideas whose names are familiar to anyone interested in science today, such as the relativity of simultaneity, time dilation, the contraction of moving objects, the non-additivity of speeds, the twin paradox, and so on. But those ideas, fascinating though they are, are not our focus. We wish now to come back, as promised, to the equation E = mc2, which, oddly enough, was nowhere to be found in Einstein’s first article on relativity. That thirty-page article, published in the summer of 1905, contained plenty of other equations whose consequences were unprecedented and revolutionary, but it lacked the little tiny equation that became the indisputable emblem of relativity.

  For most people, experts and non-experts alike, the equation E = mc2 is so tightly linked with the notion of Einstein’s theory of relativity that imagining relativity with its signature equation completely absent would seem as strange as imagining the 1927 Yankees without Babe Ruth, or the town of Pisa in 1100 A.D., before its signature tower had ever been dreamt of. And yet the truth of the matter is that Einstein did not discover the now-celebrated equation until some months after his first relativity article appeared. He deemed his new finding interesting enough to warrant another article, which appeared in November of that same year (thus just barely squeezing in under the wire of the annus mirabilis), and which was just two pages long.

  Strictly speaking, the famous equation didn’t appear in that article, either, since the way Einstein saw fit to express his new discovery was through words rather than through an equation; however, those words were tantamount to saying “E = mc2”. And then, two years further down the pike, he realized that his second relativity paper concealed some highly important implications that he hadn’t at all suspected when it was published. And so in 1907, he published yet a third article, at last spelling out the full meaning of the symbols “E = mc2”. It was this article that grabbed the world’s imagination, because its conclusions not only were counterintuitive and scientifically far-reaching, but also had profound potential implications for society.

  We will consider these developments in chronological order, starting with the very short article of November, 1905. In it, Einstein imagined an object that could simultaneously emit two flashes of light in opposite directions (say east and west). So let us imagine a flashlight with bulbs at both ends. Since a flash of light possesses some energy, and since energy is always perfectly conserved by all physical processes (mechanical, electromagnetic, and so on), our two-headed flashlight will necessarily lose some energy — namely, the total energy carried off by the two departing flashes. From the point of view of energy, one has to pay for producing light! All this is quite obvious.

  The key step Einstein took here was the (nearly) trivial idea of looking at the two-headed flashlight from another frame of reference — specifically, a moving frame of reference, such as a train moving at 30 miles an hour, let’s say westwards. According to special relativity, observers sitting in the train have the right to consider themselves stationary and to claim that the flashlight is moving eastwards at 30 miles an hour (and always at that speed, since their frame of reference — the train — has a fixed speed). For the train’s passengers, the two flashes necessarily undergo the Doppler effect.

  For those readers unfamiliar with it, the Doppler effect merits a brief digression. It holds for any kind of wave, including light and sound waves. In the case of sound, it’s the shift that one hears each time an ambulance approaches, passes by, and then recedes into the distance: just at the moment it drives by, its siren seems suddenly to sink to a lower pitch. To those inside the ambulance, nothing changes, of course, but for people standing on terra firma, it’s quite another story. Why does this surprising sonic shift take place?

  Imagine a pond into which a stone has just been tossed. From the spot on the surface where the stone plunged into the pond and is now sinking, circular ripples go spreading out. Now toss in a cork floating somewhere on the pond’s surface. Soon enough the concentric ripples will reach the cork, one after the other, and they will start making it bob gently up and down at a regular frequency. This bobbing cork is analogous to the vibrating eardrum of a person who hears the siren from within the ambulance: the vibration clearly has a fixed frequency.

  But now imagine, by contrast, a toy motorboat speeding across these same circular ripples, first heading straight toward the center of the concentric circles (the waves’ source), and then continuing onwards towards the far bank. While it is moving toward the center as the ripples expand, the toy boat bobs up and down more frequently than the cork does (for the boat, the circles seem to be coming out to meet it), but once it has crossed the circles’ center (located just above the sinking stone), the toy boat has to catch up with the ripples that are now fleeing from it, and so it meets them less frequently than before, meaning that it bobs up and down less quickly than before. This is an aquatic Doppler effect: the felt frequency of the ripples suddenly falls, just at the moment when the boat passes their source.

  Likewise, in the ambulance situation, the perceived frequency (i.e., the pitch) of the siren suddenly falls as the ambulance rushes by the observer on the street. The Doppler effect generally says that, if an observer is moving with respect to the source of some waves (or conversely, if the source is moving with respect to the observer), then the frequency of the waves, as perceived by the observer, will depend on the relative speed of the two reference frames. Of course it was a nontrivial analogical extension to generalize the original effect from sound waves to other types of waves, such as light waves and ripples on a pond — but that’s another story. Suffice it to say that the Doppler effect as applied to electromagnetic waves was a fairly new idea at the turn of the twentieth century, and the third-class patent clerk in Bern, though he didn’t invent the notion, took great advantage of it.

  Indeed, Einstein calculated the Doppler effect for the double flashlight using his own theory of special relativity, freshly minted just a few months earlier. He imagined himself in the frame of reference where the flashlight was moving at a constant speed (in other words, the frame in which the train is stationary), and he carried out relativistic Doppler-effect calculations that gave him the energy of each of the two flashes of light that sped off simultaneously. By adding these energies together, he got the total energy lost by the flashlight. He was able to use this sum to calculate how much kinetic energy the moving flashlight had lost at the instant when the rays were emitted, which should have been exactly zero, since the flashlight had just kept on moving at a constant clip. But it wasn’t exactly zero — it was just a tiny bit different from zero. Einstein’s Doppler-shift calculations revealed to him that the moving flashlight had to have lost some kinetic energy by sending off two flashes of light.

  This result was extremely peculiar. It was obvious that to produce light, the flashlight had to give up some electrical energy (in its battery), but why would it also give up some of its energy of motion (which is given by the standard formula “Mv2/2”, the capital M of course denoting the flashlight’s mass, and v denoting its velocity)? We know th
at the flashlight doesn’t slow up in the least! By fiat, it is moving at a constant speed. (Recall that in the first frame of reference it is perfectly stationary; it’s only the observers on board the train who see it as moving, because their frame of reference is gliding down the tracks. And given that their frame is gliding at a perfectly constant speed, and that the flashlight is stationary with respect to the ground, the “moving” flashlight never loses or gains a speck of speed, as seen by train-bound observers.) How then can the steadily-moving flashlight have lost even the tiniest fraction of its energy of motion? Let’s devote a moment’s thought to this humble riddle, a humble riddle whose solution shook the world.

  If, upon releasing the two flashes of light that carry total energy E, the flashlight loses even the tiniest amount of its kinetic energy, then the just-cited formula for kinetic energy “Mv2/2” tells us that either the flashlight’s mass M or its velocity v must have suddenly diminished at the moment of emission. But as we just mentioned, the train has a constant speed, which means that the flashlight, as seen from the train, also has a constant speed. Thus v is unchanged. We therefore have no choice: the only thing that could possibly have become smaller is M, the flashlight’s mass, and according to Einstein’s Doppler-shift computations (which we will not spell out here), the tiny amount of mass that the flashlight loses, which we’ll denote by lowercase m, is equal to E/c2. (It’s crucial not to confuse the flashlight’s total mass M with the negligible quantity of mass m that it loses when it gives off the two rays of light.)

  The Definition of the Concept of Energy

  Anyone who follows Einstein’s (rather simple) calculations must agree with him that an object that gives off electromagnetic radiation will necessarily lose some mass — namely, an inconceivably tiny quantity of mass that depends on the amount of energy E carried off by the radiation. Why tiny? Because the energy of the light itself (E, which is the fraction’s numerator) is negligible, and the fraction’s denominator c2 is incredibly huge — after all, it’s the square of the speed of light, which is to say, the square of 299,792 kilometers per second (that is, the square of 1,079,252,849 kilometers per hour). And when one divides an already microscopic energy — that of the two flashes — by this gigantic quantity, the result will necessarily be infinitesimal.

  The fact that one is multiplying a mass by a speed squared here (mc2) might surprise a nonscientist, but it doesn’t surprise physicists, for ever since Galileo, Kepler, and Newton, physicists have grown accustomed to the idea that the laws of nature involve algebraic expressions — often powers (most often squares or cubes) of quantities that are directly observed. Indeed, to anyone who has ever taken any physics at all, the formula “K.E. = Mv2/2”, giving the kinetic energy of a moving object with mass M and velocity v, is both familiar and unsurprising.

  So let’s come back to the quantity E/c2, which Einstein had just identified as being relevant to this situation. What is surprising in this quantity, then, is not the nature of the algebraic expression itself, featuring an energy divided by a velocity squared, the result of which will necessarily have the units of mass — but its meaning. The amazing thing is firstly that this m represents the mass lost by our energy-emitting object, no matter what its original mass M was, and secondly that the relationship between the sizes of m and E is mediated by a special and universal constant of nature — namely, the speed of light. This is what was truly new and strange, not the algebraic structure of the formula (an energy divided by a velocity squared), which in itself contains no surprises. In summary, it’s the idea suggested by this formula — the idea of energy possessing mass — that should catch people totally off guard, not its mere algebraic form, which is rather ho-hum to anyone who realizes, as physicists already had realized for three centuries, that energy always has the units of mass times velocity squared.

  What is the formula actually telling us, then? Well, we are now going to try to reconstruct Einstein’s own thinking process on this subject, starting with his first article on the famous formula, which came out in the fall of 1905, and in which he described just the tip of the iceberg, and finishing with the publication of his follow-up article in 1907, in which he finally revealed the iceberg’s entirety.

  Energy and Mass

  In his two-page article in the fall of 1905, Einstein showed that any object that emits energy in the form of light loses thereby a small — in fact, unimaginably small — amount of mass. This conclusion caught physicists off guard, but the public at large paid it no attention at all, since infinitesimal changes in an object’s mass, whether counterintuitive or not, have no potential use to society. To return momentarily to our caricature analogy involving Pisa and its tower, the appearance of this first article about E = mc2 was like the appearance, in 1173, of an elegant new stone tower in the center of Pisa — a tower that stood straight up, just as towers should. In those days, an Italian town with a tall tower gained a bit of prestige, but not an enormous amount of it. Though towers were impressive structures, they were pretty commonplace. Likewise, the two-page article in the fall of 1905 didn’t attract huge amounts of attention.

  We shall come back to Pisa and its tower very shortly, but in the meantime, let us consider what happens to the tiny bit of mass that a radiating object loses. Does it just poof out of existence without a trace, or do the departing flashes of light carry it away with them? It is tempting to localize the missing mass in the rays, and thus to conclude that the light in flight weighs something. (By this, we mean that if one were to catch the light inside a box with mirrored inner walls between which it will bounce, and then if one were to place the box on a scale, one would obtain a microscopically higher reading than for an identical box with no light in it.) But such a conclusion is based on the idea that if some mass seems to have vanished, then it must have gone somewhere. In other words, the conclusion that the fleeing light rays must be carrying off some mass with them follows from the belief that mass is indestructible, or, stated another way, that in all physical processes, there is a law of conservation of mass, just as there is a law of conservation of energy. (Notice the words “just as”, which suggest that mass and energy behave in analogous ways. This analogy will become crucial to our discussion.) If there is such a law for mass, then clearly the departing flashes of light would have to be carrying off the mass lost by the object. (Where else could the mass go? Isn’t loot likely to be carried off by the thief?) But this is rather puzzling to a human being, because we are all imbued with the image of light as an insubstantial, ghostly entity — in some ways as the diametric opposite of matter. How, then, could light weigh anything?

  In any case, any process of radiation inevitably entails a loss of mass by the radiating object, the precise amount of which is given by Einstein’s famous formula. Once again we stress that the heart of Einstein’s first discovery linking energy and mass is not the precise value for the loss, which is specified by his mathematical formula, but rather the statement in italics, above. But this was just the first act; it was not this initial finding but other, deeper meanings of the equation, discovered in the ensuing two years, that finally rendered it so enormously famous.

  Banesh Hoffmann’s Special Way of Looking at Einstein

  The physicist and mathematician Banesh Hoffmann was a collaborator of Einstein’s during the 1930’s, and in 1972 he published an exemplary biography of Einstein. That book, Albert Einstein: Creator and Rebel, is remarkable for the limpid fashion with which it conveys the inner workings of the mind of the great thinker. Certain passages in it give a sense for the subtlety of the analogies with which Einstein gradually homed in on the essence of this discovery that is expressed by just five symbols. Rather paradoxically, the essence of the discovery is also masked by those five symbols, because an equation in physics is not self-sufficient, in the sense of explaining itself; an equation just sits mutely on a page. It’s up to physicists to decipher its meaning, or rather, its various meanings at different levels, because there can be
several levels of meaning, even for a very tiny equation.

  For example, the equation “E = mc2” is often stated without any clear context. In such a situation, what do the letters “E” and “m” stand for? What energy and what mass are meant? Are they always attached to the same spot and the same moment of time? To be more precise, does the equals sign mean that the mass is accompanied by a certain energy, or that it actually is an energy, or that it yields an energy, or that it results from an energy? Does this equation mean that some energy can transform into some mass (or vice versa, or both)?

  The answers to questions of this sort are by no means self-evident. They do not effortlessly jump off the page, nor is mathematical skill the magic key to their answers. Even today, very few non-scientists know how to interpret these symbols, and there are a good many physicists whose understanding of them is at times a bit shaky as well; the fact is, this simple-seeming equation’s meaning is elusive. Even its discoverer had to mull it over for a couple of years in order to fathom its full depth.

  In order to try to understand Einstein’s intellectual pathway between 1905 and 1907, let us begin with the following passage by Banesh Hoffmann, which describes a key moment in the article that Einstein published in the fall of 1905: