Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (109 page)

This analogy between a black body and a swimming pool will in fact be very useful to us. If you toss an object — a wedding ring, a bowling ball, a grand piano — into a swimming pool, the splash will create circular waves having various wavelengths, which will then be reflected off the pool’s walls. Roughly speaking, the lighter the object, the daintier will be the waves (both in their amplitude, or height, and in their wavelength). A physicist might well ask, “Given the mass of the object thrown into the water, what will be the dominant wavelength of the waves it causes?” Along the same lines but more sophisticated would be the question, “Given the mass of the thrown object and also some particular wavelength, how much of the total energy of the agitated water do the waves at that wavelength have?” One easily imagines that for each different object
tossed in, there will be a specific wavelength that will be dominant in the pool, and that waves having much longer or much shorter wavelengths will hardly be triggered at all by the splash. A graph showing the distribution of energy among different wavelengths would constitute the spectrum of waves on the pool.

Much the same holds for a black body, except that there is no palpable medium that wiggles up and down or back and forth, like the water in the pool; the vacuum through which the electromagnetic waves course, crisscrossing each other all the time, contains no matter (whether solid, liquid, or gas), by the very definition of the word “vacuum”. The waves themselves are ghostlike entities, consisting of electric and magnetic fields oscillating in space in time, which have the ability to make charged particles move. An electromagnetic wave is thus much like a fluctuating gravitational field; it is something that pervades empty space and whose value at any point in space changes according to a “wave equation”, or possibly a set of intertwined equations — in this case, Maxwell’s equations, discovered in the early 1860’s.

The black body’s temperature corresponds, in our analogy, to the size of the object tossed into the pool, which gives rise to the ripples on the surface. The hotter the black body’s walls are kept, the more electromagnetic energy there will be in the cavity, and, perhaps surprisingly, the shorter will be the dominant wavelength of the waves. This may seem counterintuitive, since it contradicts what one would guess on the basis of the swimming-pool analogy. After all, a sofa heaved into a pool will make very long-wavelength ripples, while a pebble will make short-wavelength waves. But we won’t worry about the conflict with the water-wave situation; physicists at the time were well aware of these phenomena in blackbody radiation. They were very familiar with the bell-curve shape of the graph of the blackbody spectrum, as determined through many careful experiments. They simply had no coherent explanation for that shape.

Admittedly, in 1900, the German physicist Max Planck had discovered an elegant mathematical formula that, for any given temperature, exactly reproduced the experimentally found graph of the blackbody spectrum, but at the outset he had no justification for his formula. It was as if he had pulled it out of his hat, like a rabbit. Then, after several months of obsessive toil, he showed how one could deduce his formula from a strange and arbitrary-seeming assumption about the atoms in the walls of the cavity — namely, that their vibrational energies were limited to taking on just certain special values, determined by a new constant of nature, to which Planck gave the name
“h”
, which later was dubbed “Planck’s constant”. An atom, instead of being able to vibrate with any arbitrary amount of energy (as all physicists would have expected), could only assume energies that were exact multiples of a certain very small chunk or “quantum” of energy, as Planck called it. The idea of energy coming in such chunks struck physicists, even Planck himself, as very weird; however, everyone realized that Planck’s formula, based on this idea, matched the experimental results perfectly.

For macroscopic creatures such as humans, the fact that the atoms in the walls couldn’t take on arbitrary amounts of vibrational energy would be unnoticeable, as the magnitude of
h
was so tiny. Indeed, if
h
were equal to zero, then all possible energies would be allowed and the quantum restriction would not hold. However, the graph
that could be calculated from the assumption that
h
equaled 0 didn’t look at all like the true blackbody spectrum, as found in experiments; contrariwise, Planck’s theoretical graph, which he had calculated using his tiny but non-zero constant, matched the mysterious blackbody spectrum with total accuracy. These developments certainly had a promising feel to them, but for most physicists of the time, Planck’s theory of the “quanta of excitation” of the atoms in the cavity’s walls seemed too arbitrary, and for that reason almost no one took it seriously. Even Planck felt very uncomfortable. Although he had led himself to the quantum pond, he did not want to sip from it.

This lull provided a most opportune moment for another genius to step into the scene — a young clerk in the Swiss patent office in Bern. The great masterstroke of Albert Einstein, for indeed he is the person we mean, was to spot a key parallel between a black body and a different system that also has a spectrum determined by its overall temperature. In particular, we are alluding to an
ideal gas
trapped inside a container. Why did Einstein make this analogy, which no one else — or almost no one else — had imagined? And why did he trust it so deeply? We will shortly give some speculations.

Curiously enough, until the
annus mirabilis
, no totally convincing proof had been found of the existence of atoms (whether in solids, liquids, or gases). To be sure, certain audacious scientific spirits, such as Ludwig Boltzmann, from Austria, and James Clerk Maxwell, from Scotland, had conjectured that all gases consisted of myriads of small particles constantly bashing into each other, as well as bouncing off the walls that contained them, and from this “ideal gas” assumption they had been able to derive certain formulas that matched the empirical observations of
real
gases to an incredible degree of precision. Although this was a strong piece of evidence in favor of the atomic hypothesis, a good number of physicists, chemists, and philosophers remained skeptical.

It will help us here to draw another explanatory analogy — this time between an ideal gas and a frictionless pool table. On the pool table we imagine hundreds of tiny balls that have been set in motion by a violent explosion (such as the “break” at the beginning of a pool game), and that now are all madly bouncing off the table’s walls and off of one another. If someone were to tell us the amount of energy of the initial explosion, then we might wonder what the dominant speed (
i.e.
, the most common speed) of the balls on the table would be, once things had settled down into a more or less stable state. More ambitiously, we might ask what the
distribution
of speeds of the balls will be. It might seem surprising,
a priori
, that there is a precise answer to such questions, but in fact there is. And in an analogous fashion, there is a precise formula — the so-called “Maxwell–Boltzmann distribution” — that gives, for any particular value of kinetic energy you wish, the percentage of molecules of gas that will have that energy (given the temperature of the gas). The location of the peak of this graph reveals the dominant kinetic energy of the gas molecules.

Albert Einstein had a hunch that these two types of system — the black body and the ideal gas — were deeply related despite their surface-level dissimilarities. In both cases, there is a container filled with energy, but beyond that, what would make anyone suspect that these two systems were deeply linked? Let’s shift the question back to the more familiar territory of our two analogues — the swimming pool and the pool table.
It then becomes a question about the likelihood of there being a profound connection between the rippling surface of a swimming pool agitated by a splash, and hundreds of tiny balls bouncing about in a frenzied manner on a pool table, all set in motion by a sudden explosion. Both situations are filled with random motion and take place on horizontal, flat surfaces, but those very superficial facts would hardly seem to add up to a strong reason to make anyone suspect that a deep relationship links them.

Thus, for the vast majority of physicists at that time, Einstein’s analogy between the ideal gas (here likened to a seething billiard table) and the black body (here likened to a natatorium’s undulating surface) seemed utterly implausible. So why did Einstein see things differently? First of all, as he stated in the first of his articles of 1905, he had noticed a curious
mathematical
similarity linking the two formulas giving the energy distributions (for the blackbody spectrum, Einstein used a formula discovered by the German physicist Wilhelm Wien before Planck found his more precise one, and for the ideal-gas spectrum, he used the formula of Maxwell and Boltzmann), and this suggested to him that the
physical
similarity of the two systems might easily go well beyond the surface. All one can say here is that Einstein had an eagle eye; he almost always knew how to put his finger on just what mattered in a situation in physics.

It is fascinating to note that Wilhelm Wien, in his search for a formula for the blackbody spectrum in the mid-1890s, had had the excellent intuition — closely related to Einstein’s intuition some ten years later — to try using an analogy he had “sniffed”, linking the blackbody spectrum to Maxwell and Boltzmann’s ideal-gas spectrum. It is thus no coincidence that Einstein refound Wien’s analogy when he looked at the two formulas at the same time, for Wien’s formula was rooted in the Maxwell–Boltzmann formula. For Wien, however, the analogy between the two systems was solely
formal
; it did not suggest to him that the two systems had a deep
physical
link, and so in his mind he did not pursue it nearly as doggedly or as profoundly as did Einstein.

One other factor that might have contributed to Einstein’s faith in his analogy between the physics of the ideal gas and that of the black body (not just between the mathematical formulas for their spectra) was the fact that only a few months earlier, he had found and deeply exploited an analogy between an ideal gas and another physical system — namely, a liquid containing colloidal particles whose nonstop, apparently random hopping-about could be observed through a microscope. This analogy had allowed him to argue persuasively for the existence of extremely tiny invisible molecules that were incessantly pelting the far larger colloidal particles (like thousands of gnats bashing randomly into hanging lamps) and giving them their mysterious hops, known as “Brownian motion”. It is thus probable that two distinct forces in Einstein’s mind — the mathematical similarity of the formulas and also his recent Brownian-motion analogy — gave him great trust in his analogy between a black body and an ideal gas.

In any case, building on the bedrock of his latest analogy, Einstein undertook a series of computations, all based on thermodynamics, the branch of physics that he thought of as the deepest and most reliable of all. First he calculated the entropy of each of the systems and then he transformed the two entropy formulas so that they would look as similar as possible to each other; in fact, at the end of his ingenious
manipulations, they wound up exactly identical except for the algebraic form of one simple exponent. This provocative maneuver made it clear that the two systems were far more intimately related than Wilhelm Wien had ever suspected.

In the key spot in the formula for the ideal gas’s entropy, the letter “
N
” appeared, standing for the number of molecules in the gas; in “the same” spot in the formula for the black body’s entropy, the expression
“E/hν”
appeared. (The letter “
h
” stands for Planck’s constant, and the Greek letter “
ν
” — “nu” — for the frequency of the electromagnetic waves, always inversely proportional to their wavelength.) Einstein had thus compressed the entire distinction between these two vastly different physical systems down into one tiny but telling contrast: an integer
N
in one case, and the simple expression
“E/hν”
in the other.

But what did this precision pinpointing of their difference mean? Well,
E/hν
represents the act of dividing up the total energy
E
(a large number of ergs, an erg being a standard energy unit) into many minuscule chunks all having energy
hν
(a tiny fraction of one erg). This ratio tells how many small chunks make up the larger chunk; and thanks to the cancellation of the units (ergs in both numerator and denominator), it is a “pure” number: its value is independent of the system of units used. Einstein’s analogy now plays a key role, telling us that this number in the blackbody system maps onto the number of molecules
N
in the ideal gas. The dividing-up of
E
into identical pieces all having size
hν
(a “measurement” of
E
, to echo the term used in
Chapter 7
for one of the naïve analogies to division), was an unmistakable clue, for Einstein, that the radiation in the cavity was composed, as is a gas, of discrete particles. For any given wavelength, all “light chunks” carried the same tiny load of energy.

Even for its finder, this was a monumentally shocking idea, because to him, just as to all physicists of his day,
electromagnetic radiation
was synonymous with
light
(along with light’s cousins having longer and shorter wavelengths), and Einstein was very aware, as were all his colleagues, that the ferocious battle between advocates of
light as corpuscles
and advocates of
light as undulations
had finally been conclusively won, a century earlier, by the undulatory side. Furthermore, ever since then, thanks especially to Maxwell’s fundamental equations, discoveries in physics had reinforced over and over the view of light as continuous waves and not as discrete particles. How, then, could a corpuscular view of light possibly stage a comeback a hundred years after its demise? And yet, this is exactly what seemed to be happening, thanks to a very simple analogy.