Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (113 page)

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.

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 = mc
²
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.

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 = mc
²
”
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 nonscientists 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:

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/c
²
.

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/c
²
.
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 newfound 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/c
²
.”
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 = mc
²
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?

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.