Surfaces and Essences

by Douglas Hofstadter

  Like all physicists of the time, Einstein was intimately familiar with the principle of conservation of energy — the solidly confirmed fact that energy can change form but without ever increasing or decreasing. Countless experiments had shown that heat (thermal energy) could be converted into movement (kinetic energy) of macroscopic objects (for example, of a piston in a cylinder) and vice versa (rubbing something warms it up), and that chemical energy in a battery can be converted into electromagnetic energy, and so on. The technology of the day relied on this fundamental principle.

  Einstein had an unswayable faith in the law of conservation of energy; now, all of a sudden, he found himself face to face with a similar new conservation law — namely, the conservation of strange mass. That is to say, strange mass, much like energy, could apparently glide from one form to another without increasing or decreasing. For example, if a crystal absorbed some radiation, a bit of electromagnetic strange mass (that is, a ray of light) would suddenly go out of existence and at the same moment a bit of thermal strange mass would instantly come into existence; likewise, in an act of radiation, the reverse transformation could take place. We can therefore imagine that in Einstein’s mind, thanks to the analogy between the laws of conservation of energy and conservation of strange mass, there was starting to exist a tight analogical link between the concepts of energy and strange mass.

  So far, we have completely neglected to mention an extremely important type of energy — namely, potential energy, suggested in 1799 by the French physicist Pierre Simon de Laplace. This is perhaps the most peculiar and unintuitive form of energy, since it depends solely on the positions of objects relative to each other, but peculiar or not, it plays a crucial role in the conservation of energy. For example, a ball rolling down a hill gains kinetic energy while losing potential energy (which is proportional to its altitude), and vice versa: if it rolls uphill, then it loses speed (and therefore kinetic energy) and all the while it gains potential energy. Another example of how potential energy plays a key role in the conservation of energy is furnished by a spring. In its neutral state (neither stretched or compressed), a spring has no potential energy, but the act of compressing or stretching it gives it some. As long as one holds the spring tight, preventing it from snapping back to its neutral state, its energy remains potential, but at the moment of release, this positional energy is converted into kinetic energy, with the total energy remaining perfectly constant at every instant of the process.

  Thus energy, like mass, seems to come in two very different varieties: on the one hand, there are all the dynamic forms of energy that have to do with movement — heat (jiggling of molecules), waves, rotation, movement through space, etc. — and on the other hand, there is static or potential energy, which seems very different, because it has nothing to do with movement, just with position. Recalling our financial allegory, we might be inclined to dub the first variety liquid energy, since it always involves something that flows, whereas the second variety, potential energy, exists in the absence of any kind of motion, which could encourage us to dub it frozen energy.

  This splitting-up of the concept of energy into two varieties — liquid and frozen — can’t help but remind us of our splitting-up of the concept of mass into two varieties — strange and normal (and we mustn’t forget that this second dichotomy was imposed on us by Einstein’s equation). The analogy is clear; indeed, it cries out to be made. But despite its salience, this analogy leads us to a problem, for already in Einstein’s day, people had known for roughly a century that the two varieties of energy (dynamic and static) are fully interconvertible (otherwise conservation of energy would not hold), whereas we have just been insisting that the two varieties of mass (strange and normal) are not interconvertible. If they were, then an iron atom or a pearl or a boulder could just poof out of existence, provided that it left in its wake the proper amount of strange mass — that is, some heat, some sound, some light… But that never happens — or at least this is what any sane person would naturally think. In sum, then, as far as mass is concerned, it seems that there has to be a watertight partition between the two varieties, while as far as energy is concerned, there is no partition at all between the two varieties. And therefore, our budding mass–energy analogy goes up in smoke. What a shame!

  But here is where Einstein’s “instinct for cosmic unity” comes into play. As Banesh Hoffmann put it, to insist on the existence of a watertight partition between strange and normal mass “would be to imagine two types of mass for no good reason when one would suffice. The distinction would be inartistic and logically indefensible.” If we take Hoffmann’s word for it, then, Einstein must have said to himself in 1907, in essence, “My unflagging faith in nature’s uniformity leads me to conclude that it must be possible for an ordinary lump of matter possessing normal mass to be converted into a quantity of strange mass or vice versa, even though nothing of the sort has ever been seen anywhere.” This moment of deep inspiration for Einstein, triggered by his esthetic of simplicity, would be analogous to Jan’s epiphany when, faced with the threat of repossession, she broke the invisible financial barrier and imagined the previously unimaginable idea of converting her frozen assets into liquid assets.

  But what could have prodded Einstein to break the analogous barrier in the concept of mass, which had seemed so definite and so firm? What metaphorical “repo person” came knocking one fine day and put sufficiently intense mental pressure on him? An esthetics-based longing for cosmic unity alone couldn’t have done it, because as we said above, it was simply self-evident that there was no interconvertibility between the two varieties of mass. Clocks, blocks, and rocks never evaporate into flashes of light, sound waves, or anything else. They just sit there, inert and immutable. All this was clear as day. What, then, might have led Einstein to see things otherwise?

  Recall how Banesh Hoffmann summarized Einstein’s state of mind concerning mass and energy in 1907. If we rephrase that quote using the terminology of this chapter, Einstein would be thinking essentially the following: “Normal mass somehow has to possess energy because it is essentially the same thing as strange mass, and the latter, according to the equation I derived two years ago, possesses energy. Analogy thus forces me to generalize, and so I conclude that all types of mass possess energy.” The analogy clearly resides in the words “it is essentially the same thing as”, but once again we have to wonder why Einstein would have been confident of such an analogy, given the vast difference between how one conceives of normal mass and strange mass, and given that there was nary a shred of experimental evidence for the idea that locked up inside every single piece of ordinary, innocent-seeming matter were vast hidden reserves — indeed, inconceivably enormous reserves — of energy.

  The key hint for Einstein could well have been potential energy, for as we pointed out above, potential energy is reminiscent of normal mass. While other forms of energy involve movement, potential energy is inert. Likewise, while strange mass involves movement, normal mass is inert.

  Up to this point, the analogy between potential energy and normal mass is strong, but in Einstein’s mind, it would have been weakened by the fact that all forms of energy, including potential energy, are interconvertible, whereas for mass, the notion of interconvertibility applies only to one side of the partition, pointedly excluding normal, “frozen” mass. This is a most disturbing asymmetry — but for that very reason, it is most provocative! Why should there be an impermeable membrane separating strange from normal mass, if the analogous membrane within the concept of energy is perfectly permeable? This is the key question leading to the key breakthrough.

  It would clearly be a wild leap in the dark to propose that normal, “frozen” mass also can participate in the fluidlike phenomenon of conservation of total mass. It would lack any justification except a deep esthetic desire for unification, reinforced of course by a suggestive analogy — namely, the fact that potential energy participates in the conservation of total energy. But no ma
tter how suggestive the analogy, to make such a leap would be reckless, because it would oblige one to believe in the wild idea of lumps of mass poofing into and out of existence — an unheard-of kind of event at that time.

  Furthermore — and this made the notion even more surrealistic — Einstein was most aware that, because of the enormous multiplicative constant c2 in his equation, the metamorphosis of even the most insignificant quantity of normal mass into strange mass would make an inconceivably huge amount of energy materialize seemingly out of nowhere (although it would actually have always been there, just hidden out of sight in innocent-seeming lumps of matter, thus strongly analogous to chemical potential energy lurking silently and invisibly in chemical bonds). This release of gigantic reserves of hidden energy would allow the development of stupendous sources of energy, not to mention stupendous weapons. If one day this kind of metamorphosis could be carried out, the world would be profoundly changed.

  In sum, the new interpretation of E = mc2 amounted to a daring leap into wild science-fiction scenarios. And yet, though it was based on nothing but an intuitive esthetics-based analogy, this is exactly the leap that Einstein made in print in 1907, thereby opening the door to a revolutionary vision according to which a material object having normal mass could be converted into other, intangible forms of mass, thereby freeing up vast amounts of hidden energy that had been locked up inside it as a kind of potential energy. 1907 is thus the year in which the metaphorical new tower of Pisa started to lean, thereby attracting a great deal of attention. From that moment on, the soon-to-be-cliché phrase “Einstein’s relativity theory” became inseparable, in the public’s imagination, from the equation E = mc2.

  In 1907, however, there didn’t exist the tiniest shred of experimental evidence for Einstein’s extension of the original meaning of his equation. Only many years later — in the year 1928 — thanks to a subtle fusion of relativity and quantum mechanics, was the idea of antiparticles (such as the positron, antiparticle of the electron) proposed on theoretical grounds by the English physicist P. A. M. Dirac, and a few years after that, the sudden and total mutual annihilation of two stationary lumps of matter — specifically, an electron and a positron — was experimentally observed in a process that gave rise to just two photons (the elegant and indispensable word “photon” had finally been coined, in 1926, by Gilbert Lewis) zipping away from each other at the speed of light (by definition!), and undulating with exactly the amount of electromagnetic energy that, when divided by c2, equaled the sum of the two late particles’ normal masses. In other words, this experimental discovery showed that ordinary matter having normal mass (in this case, the electron and the positron, which could be thought of as mutually annihilating “nano-boulders”, so to speak) could indeed suddenly cease to exist, as long as it was simultaneously supplanted by a burst of radiation energy possessing exactly the same amount of strange mass. Thus, after twenty-five years had passed, experimental confirmation finally arrived for Einstein’s risky leap that had been based solely on an analogy grounded in esthetics.

  We find it revelatory, as does Banesh Hoffmann, that Einstein’s dramatic conclusion in 1907 (namely, that mass always contains energy) was nothing but the flip side of his 1905 discovery (namely, that energy always possesses mass). It’s as if, after writing down his equation, he at first read it in only one direction (“m = E/c2” — that is, “an inconceivably tiny amount of mass is possessed by any standard-size portion of energy”), and then finally realized, after two years, that it could be read in the other direction as well (“E = mc2” — that is, “an inconceivably huge amount of energy lurks hidden in any standard-size portion of mass”). This shows that even for the most audacious of spirits, it sometimes takes a great deal of time and intense concentration, not to mention analogy-driven cognitive dissonance, to carry out what might seem, after the fact, to be the most elementary of conceptual reversals.

  From 1905 to 1907 in a Nutshell

  Below we offer a summary of the many-voiced symphony of ideas about energy and mass in Einstein’s mind that eventually led to his breakthrough in 1907, resulting in a far deeper understanding of the meaning of the equation that he had first written down in his annus mirabilis.

  Ideas inherited from previous eras…

  •There are two fundamental varieties of energy: dynamic energy, due to the movement of objects and to the oscillation of waves, and static (or potential) energy, due to the relative positions of objects.

  •Either variety of energy can be converted into the other.

  •All physical processes conserve the total energy in the given system; the same holds for the system’s total mass.

  Ideas that Einstein came up with in 1905…

  •Whenever any object emits a ray of light, it loses not only a quantity of energy E but also a microscopic quantity of mass, which is given by the equation m = E/c2. Analogously, if a ray of light is absorbed by an object, the object acquires not only some energy but also some mass, given by the same equation.

  •A ray of light carrying some energy E must also carry some mass m, once again given by the same equation.

  •Conjecture by analogy: not only electromagnetic waves but any form of dynamic energy possesses mass. Thus, whenever an object acquires (or loses) a quantity of dynamic energy E, it acquires (or loses) an infinitesimal quantity of mass m, once again given by the same equation.

  •Conjecture by analogy: this holds not only for dynamic energy but also for static energy.

  •The mass of an object consists of two fundamental varieties: its normal mass, which is due to the matter the object is made up of, and its strange mass, which is due to the energy it contains.

  •Since the basic particles composing an object do not mutate during the emission or absorption of energy, the object’s normal mass never varies.

  •All the energy contained in an object possesses strange mass; conversely, any strange mass contains energy, the exact amount being given by the equation E = mc2. By contrast, the normal mass of an object plays no role in the mass–energy relation, and so the equation E = mc2 applies only to strange mass.

  A mass–energy analogy starts to form…

  •Mass and energy are alike in that both of them are conserved by all physical processes; moreover, the equation E = mc2 connects a given quantity of energy to a corresponding quantity of mass in a simple, natural fashion. Mass and energy are thus analogous entities — indeed, they are intimately related.

  •There is a very inviting resemblance between static energy and normal mass (since both are unrelated to movement), and likewise there is an inviting resemblance between dynamic energy and strange mass (since both are due to movement). These two resemblances constitute the heart of the incipient mass–energy analogy.

  At the same time, a lack of symmetry gives rise to cognitive dissonance…

  •Energy (since it is not composed of particles) is endowed with strange mass, but it has no normal mass. Also the reverse holds: any object’s strange mass is endowed with invisible energy, sitting quietly in reserve until it is released, but this does not hold for the normal mass of the same object (that is, normal mass possesses no energy).

  •There is thus an “internal partition” in the concept of mass, separating normal mass from strange mass; because of this partition, the two are not interconvertible. However, this internal partition in the concept of mass, keeping two varieties forever apart, has no counterpart as far as energy is concerned (all forms of energy being interconvertible). This mass–energy mismatch is a serious blight on the incipient analogy linking the two concepts.

  Thanks to a hypothesis that restores “cosmic unity”, the cognitive dissonance is dissipated…

  •Since there is no partition separating different types of energy, and since there is a promising analogy linking energy to mass, then if one truly believes in this analogy, it becomes conceivable that mass, just like energy, might not be divided by an internal partition, but that its two va
rieties (normal and strange) might be interconvertible.

  •This idea, if true, would imply that normal mass, no less than strange mass, constitutes a reservoir of energy, and that (under special circumstances of an unclear nature) it can transform into strange mass (or vice versa). This would imply that an object could (under these special circumstances) completely poof into thin air, as long as its normal mass were instantly transformed into an equal quantity of strange mass.

  •The amount of energy associated with the “poofing out of existence” of an object having mass m (or more precisely, the conversion of normal mass into strange mass) is given by the equation E = mc2, and would therefore be astonishingly large, even if the object itself were extremely lightweight.

  Clearly, this is a very subtle story, and in our attempt to make all of its many stages vivid, we struggled hard. A big part of the challenge was to find the optimal pair of contrasting English adjectives to convey the key dichotomy between the two varieties of mass. We entertained quite a few possibilities, including “corpuscular / vibrational”, “permanent / volatile”, “unusable / usable”, “solid / liquid”, “concrete / abstract”, “tangible / intangible”, “classical / Einsteinian”, “hard / soft”, “corporeal / ghostly”, and even “lumpy / wiggly”. We also considered Einstein’s own terms (“true mass” and “apparent mass”), but as he used those terms only one single time, they were far from canonical. And so in the end we settled on “normal mass” and “strange mass”. This was a difficult decision, because each of the contrasting pairs that we tried on for size had both virtues and defects: that is, each pair suggests (or comes from) a slightly different analogy with familiar situations, and thus it brings out certain subtleties of this mysterious distinction that are not brought out by other pairs.