Surfaces and Essences

by Douglas Hofstadter

  With his instinctive sense of cosmic unity he now tosses off a penetrating and crucially important remark: that the fact that the energy is in the form of light “evidently makes no difference”.

  In other words, once Einstein had formally derived the counterintuitive result, he was perfectly happy to ignore his own derivation and to jump to the conclusion that it must also hold in far more general circumstances than those that allowed him to discover it. In particular, Einstein wrote that exactly the same result must hold in any situation in which an object releases energy in any form at all — thermal energy, kinetic energy, sound waves, and so forth.

  Now this is a classic vertical category leap by Einstein, supposedly justified by the modest word “evidently” (which, incidentally, would have been better translated by Hoffmann as “obviously”). However, Einstein’s calling it “evident” or “obvious” does not legitimize the leap, for it is an extremely bold leap of generalization, owing nothing to logical or mathematical reasoning or to algebraic calculations. This leap comes solely from a physical intuition that all processes of energy release have so much in common with each other that if a given result has been rigorously established for one type of process, then it must hold for all such processes. In other words, it comes from an analogical belief that, in this type of situation, all forms of released energy are equivalent. This first broadening by Einstein of the meaning of his equation was thus the idea that any object, whenever it releases an energy E of any type whatsoever, loses a minute amount of mass equal to E/c2.

  Actually, before this, Einstein made one prior extension of his equation’s meaning. It came from a smaller, more modest leap — a leap involving a conceptual reversal. He declared that any object, whenever it absorbs an amount E of incoming energy, gains an amount of mass equal to E/c2. This mental turnaround constituted a nontrivial analogy: the object, instead of giving off some energy, absorbs some, and instead of losing some mass, gains some. In other words, Einstein saw that there was not a profound difference between the new-found phenomenon running forwards in time and running backwards in time. Such a conceptual reversal, although it may seem extremely simple, doesn’t just step forward all by itself; somebody has to imagine it. And even such a simple mental turnaround can on occasion elude deep thinkers, even “Einsteins” (we’ll give an example very shortly) — but this particular conceptual reversal did not elude this particular Einstein.

  These descriptions of the process of emission or absorption can be seen as a causal interpretation of the equation. As we said earlier, in his first article on these ideas, Einstein didn’t write out the now-famous equation with algebraic symbols; he expressed his discovery solely in words by saying, “If a body emits energy E in the form of radiation, then its mass diminishes by E/c2.” This sentence describes an event (emission or absorption of some energy) that inevitably gives rise to a consequence (loss or gain of mass). Much as in the case of some equations discussed in the previous chapter, this sentence amounts to an asymmetric reading of the equation, in which one side is seen as the reason behind the other side, but where causality running in the reverse direction is not imagined.

  This, in broad brushstrokes, is the meaning that Einstein saw in E = mc2 in 1905. That meaning, although already a very surprising and provocative idea, is not nearly as far-reaching as the final understanding that he reached in 1907. In the course of his ponderings between 1905 and 1907, the equation itself didn’t change in any way; all that changed was the interpretation that Einstein attached to its five symbols. In principle, any physicist of the time could have read Einstein’s 1905 article, could have reflected on it for two years, and could have arrived at all of its consequences — and yet, to no one else did these ideas occur. What went on that was so different and special in Einstein’s mind?

  A New and Strange Type of Mass

  In order to understand the mental obstacles that Einstein had to overcome, one must try to enter into the mindset of the physicists of that period. The existence of atoms was still not certain in 1905, and if they existed at all, their nature was entirely mysterious. Einstein believed in them with near-certainty, just as he believed in the vibrations of the atoms in a solid as the explanation of heat, even if he wasn’t able to envision what the atoms themselves were like. But how could Einstein (or any other physicist of his time) imagine a radiating object, such as a flashlight, losing some of its mass? How could such a bizarre event possibly happen?

  For example, would it lose some of its constituent atoms? If so, where would they go? Or would they just suddenly cease to exist? Or else, could some (or all) of its constituent atoms become a smidgen less massive while staying the same in number? In that case, by what mechanism could a single atom lose some of its mass? Was it possible that the fundamental particles (like electrons, which had just been discovered in 1897 by the English physicist J. J. Thompson) might have variable masses rather than fixed ones? On the other hand, if the object lost none of its atoms, and if each constituent atom retained all of its original mass, then how could the whole object possibly lose any of its mass? This was a genuine enigma.

  All such questions hinged, of course, on how physicists in those days imagined mass. And as to that, there isn’t any doubt: they saw it just as we all do intuitively, even today, over a hundred years later — namely, as a fixed property of any material object, ranging from clocks to clouds to dust motes to atoms, but not applicable to an intangible notion like a jiggle, a ripple, a rumble, or a tumble, because such verb-like phenomena are merely patterns of motion of some matter, and have no weight. As mass was considered a fixed property of a material object, it certainly couldn’t just poof into or out of existence. Indeed, an object’s mass couldn’t change at all — unless bits of it broke off and sailed away, like a cigarette giving off tiny particles of smoke that invisibly disperse into the surroundings. But even then, the sum of all the little invisible masses would have to equal the starting mass; this seemed (and still seems) self-evident. The total mass couldn’t grow or shrink; it was an invariant, and thus conserved, quantity.

  Einstein’s new equation put him in a sticky wicket, therefore, because everyone grasps the distinction between material objects and immaterial phenomena, and yet the new equation seemed to be saying that a material object could — in fact, had to — lose or gain mass as a result of losing or gaining energy, despite the fact that, to all appearances, energy is anything but a material object. An example of what the new equation implied would be a hot object that is cooling off, giving off a bit of its heat to its surroundings. This object must also, according to Einstein, be losing a bit of its mass. This amounts to saying that some mass is associated with heat, but let’s recall that for Einstein, as for most physicists of the time, “heat” was synonymous with “vibration of atoms”, which meant that he was forced by his own beliefs to the surrealistic idea that the vibration of atoms inside a solid contributes to the solid’s mass, with the bulk of its mass residing, of course, in the atoms themselves, seen as material objects.

  We are thus led to a dichotomy, in the mind of Einstein and anyone who accepts his conclusions, concerning the notion of mass: on the one hand, there is the familiar type of mass, which we will henceforth refer to as normal mass, and which corresponds to the standard everyday notion of “mass of a material object”, and on the other hand, there is another type that we will call strange mass, which corresponds to the counterintuitive new notion revealed by the famous equation. (Einstein himself made the same distinction in his 1907 article, using the terms “ ‘true’ mass” and “ ‘apparent’ mass”.) This breakdown of mass into two types, although unexpected, is imposed on us by the equation; it cannot be avoided. A cloud and a clock obviously possess normal mass, since they are both made of stuff (that is, atoms), while light, sound, and heat have none; the latter three, however, all possess strange mass. Of course, the cloud and the clock will also possess some strange mass, because they contain some heat, and as we said above, heat
is imbued with strange mass.

  Any ordinary material object thus possesses lots of normal mass and a tiny bit of strange mass, and their relative proportions can change with time. For example, a flashlight that gives off light for a long time will gradually lose its strange mass, thus becoming ever so slightly lighter. What then happens when it has fully exhausted its strange mass (or “runs out of juice”, in colloquial terms), and all that’s left is its normal mass? Well, as we all know, at that point the flashlight will no longer be able to emit any electromagnetic radiation, unless a new battery is installed. The fact that it still contains plenty of normal mass would seem utterly irrelevant to its flash-producing capability, because there is no interconvertibility between normal and strange mass. That is to say, it would seem that we can’t draw on a flashlight’s abundant reserves of normal mass to get it to give off light. All of one’s experience leads one to think that only by drawing on its very small supply of strange mass can an object emit energy (such as light, in the case of a flashlight).

  The following allegory may help us to convey more clearly the distinction between these two types of mass. Jan has the wherewithal to purchase the most essential items in her life, but her modest bank account does not allow her to indulge in luxuries. Some time ago she inherited a huge mansion with palatial grounds, worth at least several million dollars — but in her mind, this type of possession doesn’t belong to the same category as her day-to-day money; no matter what its official value might be in dollars, she doesn’t conceive of her residence as a spendable liquid resource — she sees it merely as a frozen, solid one. For Jan, the two types of possession have nothing to do with each other; it’s as if there were a rigid mental barrier separating the concepts. It would never occur to her to sell a few acres of her gardens, let alone her mansion, in order to purchase a luxury item or to take a vacation. In Jan’s mind, whereas her meager liquid assets flow easily (indeed, that’s why they are called “liquid assets”), her real-estate assets are completely frozen and inaccessible; if she needs money, she never thinks of the latter at all. But then one day, several months after her non-payment of a substantial bill, Jan receives a worrisome letter stating that in a few days some of her belongings will be forcibly seized by law enforcement officers. And then, all at once, something clicks in her mind…

  One can easily translate our allegory into the language of mass and the conceptual dichotomy Einstein had discovered. The idea is simply that the seemingly uncrossable mental barrier between strange mass (= liquid assets) and normal mass (= frozen assets) is not in fact uncrossable after all, but can be crossed provided that there is enough pressure (= the threat of a repossession) to make the idea leap to mind.

  However, for Albert Einstein in 1905 and for the readers of his first article on the idea of E = mc2, the conceptual barrier between normal mass and strange mass was completely impenetrable. How could it not have been so? Consider a boulder, for instance (a quintessential example of normal mass). It’s one thing to imagine that the boulder’s internal stock of heat (a quintessential example of strange mass) will gradually diminish as the boulder emits infrared radiation that warms up its environment. But who would ever have suspected that the boulder itself could all at once vanish from the universe, resulting in the shooting-off of much more intensive rays of light? If one is not under severe pressure, one does not jump to embrace wild and woolly scenarios such as that; one does not spontaneously offer a warm welcome to notions that violently clash with a lifetime of experience, not to mention with the collective wisdom of humanity.

  In the Copycat analogy problem “abc ⇒ abd; xyz ⇒ ???”, no one thinks of the elegantly symmetric answer wyz without first having been lured down the pathway of taking the successor of z and banging up against that barrier. Only after all one’s initial simple and intuitive ideas have failed does one start trying out more radical ideas. In short, it takes a great deal of mental pressure in order to trigger a radical conceptual slippage, and not least among the contributing pressures is one’s sense of esthetics.

  Here is what Banesh Hoffmann wrote about this very subtle transitional stage in Albert Einstein’s thought processes:

  In his paper of 1905 Einstein said that all energy of whatever sort has mass. It took even him two years more to come to the stupendous realization that the reverse must also hold: that all mass, of whatever sort, must have energy. He was led to this by æsthetic reasons.

  Why should one make a distinction in kind between the mass that an object already has and the mass it loses in giving off energy? To do so would be to imagine two types of mass for no good reason when one would suffice. The distinction would be inartistic and logically indefensible. Therefore all mass must have energy.

  This passage is eloquent and insightful, but it’s also rather curious, for its two halves almost seem to contradict each other. Whereas the first paragraph states that it would be very hard (even for Einstein) to transcend the intuition that there are two different types of mass, the second paragraph (which is attempting to give us a privileged, first-person view from within Einstein’s own mind) claims that there would be no good reason for believing in a distinction between these two types of mass. But actually there was a very strong reason for such a belief; it was in fact Einstein’s own equation that had created a schism within the formerly monolithic concept of mass. Viewed in this way, Hoffmann’s short passage constitutes a perfect summary of Einstein’s inner mental trajectory over two years. Its first paragraph alludes to Einstein’s initial glimpse of a new kind of mass in 1905, as well as his lack of full understanding of it; its second paragraph indicates that this imperfect understanding gave rise to such serious tension in Einstein’s mind that he was eventually forced to make a daring esthetics-driven extension of that initial notion (that is, of strange mass), getting rid of the conceptual schism and thereby re-establishing conceptual unity, thus leading to a harmonious new state of understanding from which all mental tension had been banished.

  We shall now try to put this mental trajectory under a magnifying glass. The fact that light, heat, and sound (etc.) all possess mass (even if it’s just an extremely slight amount) implies that we are dealing with a new type of mass of which no one had ever dreamt before. An object that gives off energy and in so doing loses a tiny amount of its mass does not lose any of its solid or normal constituents; it loses something radically new — it loses a different type of mass. In short, for anyone who understands the equation E = mc2 (and this certainly includes its discoverer), there is a very intense pressure to imagine two extremely different types of mass — the familiar type (normal) and the new type (strange), which is associated with outgoing or incoming energy. To be more specific, the tiny mass carried off by the light rays comes from strange mass that was lost by the battery. No particles in the battery were lost or destroyed, however; every one of its corpuscles (atoms, molecules, whatever) remained intact. Since the normal mass of the battery is never affected by any process of emission (or absorption) of light, this gives rise to the image of a rigid barrier between these two types of mass (informally put, “they don’t talk to each other”).

  Even Albert Einstein took two years to arrive at the conclusion that the impermeable conceptual barrier that was suggested — indeed, forced — by his equation did not actually exist. His “instinctive sense of cosmic unity”, as Hoffmann dubbed it, eventually led him to the radical notion that nature’s internal consistency — that is, the uniformity and simplicity of the laws of physics — required that any material object (i.e., any normal mass), whether an electron or a cannonball, should be able to “melt” into strange mass carried off by escaping light rays, much as does the inert energy stored in a battery, or much as the frozen assets latent in an estate might turn into liquid cash.

  This was truly a shocking idea, because it meant not only that solid, massive physical objects could literally dematerialize and vanish (or, if we run the scenario in reverse, that such objects could materialize out o
f nowhere), but also that any such metamorphosis would necessarily be accompanied by the sudden, simultaneous appearance (or disappearance) of a phenomenal amount of energy. Indeed, it was the phenomenal amounts of energy involved that made the newly-revealed full meaning of Einstein’s equation stunning and even surrealistic.

  What Machinations Took Place Behind the Scenes in Einstein’s Mind?

  What went on in the hidden recesses of Einstein’s mind that brought him, after two years of thought, to this most disorienting idea, for which there was, at the time, no experimental evidence at all?

  To begin with, there is every reason to believe that Einstein saw the light rays leaving the flashlight as carrying not only energy but also mass (both of which had been “subtracted” from the flashlight). This amounts to the idea that the strange mass, rather than just poofing out of existence when it left the flashlight, mutated from one form to another. Before the two flashes were produced, the strange mass resided in the chemical bonds inside the flashlight’s battery, whereas after the flashes’ release, it resided in the vibrations of the electromagnetic waves making them up (and thus, if one were to weigh a mirror-lined box in which the rays had been captured and were bouncing back and forth, one would find that the box weighed ever so slightly more than it had before the rays were captured in it).

  This fluidity of strange mass — the fact that strange mass can glide back and forth between different forms — could not have failed to remind Einstein of the fluidity of energy (for energy, likewise, is constantly gliding from one form to another), and such a connection would have come to his mind all the more easily given that his equation had revealed an unexpected link between mass and energy. But even if a given type of strange mass could easily mutate into other types of strange mass, it still seemed totally self-evident that normal mass could not mutate. As we stated above, one never sees boulders or other solid objects (or liquids or gases, for that matter) simply vanishing into flashes of light, or springing magically out of them. Material things are made of tangible stuff, and as such they seem to belong to a different class of things from intangible energy and its “strange mass”. This sharp distinction makes for a rigid, uncrossable barrier inside the concept of mass, as described earlier, dividing it into two subspecies that are not interconvertible.