Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (119 page)

The key turned out to be the phenomenon of
length contraction
, a subtle consequence of special relativity according to which the length of any object, when measured in a frame moving with respect to the object, is shorter than when measured in the object’s
rest
frame. Roughly put, moving objects, when their dimensions are measured by observers who are
not
moving, undergo longitudinal but not transverse shrinking.

And turnabout is fair play: since an observer in either frame can validly consider
their
frame to be stationary and the
other
frame to be in motion, each one will see objects in the other frame — but not objects in their own — as being longitudinally contracted. This seems paradoxical at first, but Einstein showed why it is not. The following caricature analogy will help to suggest the subtle flavor of his resolution.

Chinese children might be tempted to say that Chilean children walk around upside down; symmetrically, Chilean children might be tempted to say that Chinese children walk around upside down. So who’s right and who’s wrong? Well, we who know that gravity pulls toward the earth’s center understand that the direction called “down” is not the same direction everywhere on earth, despite what the naïve analogy would suggest; rather, it is relative — that is,
down
depends on where the observer is located. In analogous fashion, an object’s
length
turns out not to be the same for all observers but dependent on the relative speed of object and observer; this is why two differently-moving observers of a given object will come up with disagreeing measurements of its length. But length, like down-ness, is relative, not absolute (that is, the naïve analogy is wrong in both cases), so the surface-level disagreement, though puzzling at first, does not constitute a true contradiction.

And with that, let’s return to the rotating disk and general relativity. When a disk rotates, each point on it carries out motion that is circumferential but not radial; in other words, it moves round and round but not in and out. Special relativity then tells us that there is longitudinal but not transverse length contraction: this means there is contraction along the circumference but not perpendicularly (
i.e.
, along the radii). And so the disk’s
circumference
that we measure, from our nonrotating vantage point, will be different in length from the one measured by observers in the frame of the spinning disk; on the other hand, our measurements of the disk’s
diameter
will agree with theirs.

This idea was reached only very slowly and with great difficulty, for it profoundly violated intuition, but nonetheless Einstein stuck to his faith and proceeded to think it through with great care. And in so doing, he was led to a most peculiar finding: when a
disk spins, non-spinning observers who measure its circumference-to-diameter ratio will discover that it is not equal to
π.
Moreover, the size of that ratio will depend on how fast the disk is spinning, with a disk that spins faster having a ratio that deviates further from
π
.

Even for Albert Einstein, who by this time was pretty used to coming up with intuition-defying ideas, this effect seemed very weird, and he and his colleagues referred to it as “Ehrenfest’s paradox” (after the Dutch physicist who had first pointed it out), and they worked extremely hard to try to resolve it. It took several years, though. The key breakthrough came at last toward the end of the summer of 1912. Here is how Einstein described this special burst of clarity:

(By “this theory”, he meant his incipient theory of gravitation, which was equivalent — thanks to his well-named equivalence principle! — to the theory of accelerating frames of reference.)

To make sense of what he had just found about rotating frames, Einstein once again drew on long-dormant memories from his student days at ETH in Zürich, some twelve years earlier. At that time, he had taken a seminar on the geometry of two-dimensional curved surfaces, and in it he had learned that in this kind of non-Euclidean geometry, developed by Karl Friedrich Gauss and Bernhard Riemann in the nineteenth century, the ratio of the circumference of a circle to its diameter can be arbitrarily different from
π.
To get a feeling for how this can be, consider the earth’s surface as a curved two-dimensional space. If we take the equator to be our sample circle, then a radius will be any line of longitude running due north from the equator all the way up to the North Pole. Such a line’s length is one quarter of an equator long, and so a diameter equals
half
an equator, meaning that for observers who inhabit this curved space, the equator’s circumference-to-diameter ratio will equal 2, not
π.

At some point in late 1912, this theorem about non-Euclidean circles, which had long been buried in Einstein’s memory, unexpectedly popped to mind — an old memory suddenly resuscitated by an event — much as had happened a few years earlier with the memory of fictitious forces. This very welcome new connection suggested to him that he could directly borrow all the formulas of Gauss’s and Riemann’s geometry of curved surfaces to characterize in mathematical terms the physics of an accelerating frame of reference, and even better — thanks to his equivalence principle — the physics of a world immersed in a gravitational field.

To be sure, events in our universe take place not in a two-dimensional plane but in three-dimensional space, and they take time to take place (so to speak). This amounts to four dimensions (three spatial ones, plus a temporal one). Although today the term “four-dimensional space-time” is very familiar — indeed, it has become a hackneyed cliché — the notion, when it was first proposed by the German mathematician
Hermann Minkowski, was a strange and momentous new revelation. Minkowski, who had been one of Einstein’s professors at ETH, had noticed a striking analogy between the equations of Galilean relativity and those of Einsteinian special relativity, and this analogy (which, ironically, Einstein himself had somehow missed!) led him rapidly to the idea of four-dimensional space-time.

In developing
general
relativity, Einstein realized that what he needed to do was to take the two-dimensional theorems of Gaussian geometry that he’d studied years earlier at ETH (and which, back then, he’d considered purely mathematical and of no relevance to physics) and adapt them to Minkowski’s more abstract space having three spatial dimensions and one temporal one. As he did so often, he was exploiting to the hilt an intuitively sniffed analogy — namely, he was “copying” in four space–time dimensions the ideas that his professors in Zürich many years earlier had taught him in two purely spatial dimensions — and what this analogy brought him was astonishingly rich and novel. It led him to the surreal notion of a
four-dimensional curved space
— and as if that weren’t enough, the concept of
curvature
was now no longer limited to space but was being extended to the dimension of time. The idea of “curved time” was certainly pushing at the very limits of the human imagination.

The prolific set of analogies that we have just recounted — first, the analogy between a gravitational field and an accelerating frame of reference; second, the analogy (redeployed in a new context) between the laws of mechanics alone and the laws of physics as a whole; third, the analogy between rotating frames of reference and two-dimensional non-Euclidean geometries; and fourth, the analogy between two-dimensional and four-dimensional non-Euclidean geometries — provided Einstein with the crucial clue he had long sought, pointing the way to the tools with which to handle gravity mathematically. And so, over a period of several years, he was finally able to “tame” gravity, bringing it at last into the family of mathematical laws of physics. The resulting theory of general relativity was the most complex and daring accomplishment of Einstein’s career, the crowning glory of all his astonishing discoveries.

An entire book could easily be devoted to the fecundity of Albert Einstein’s great analogies. Our goal in this chapter was more modest — it was merely to offer a sampling of them, in order to show that major advances in physics are not the result of virtuoso acts of stand-alone mathematical deduction and formal manipulation of equations, but that, quite to the contrary, they emerge as the fruit of analogies intuited by individuals who have the gift of seeing a unity where others see only diversity, individuals who have a keenly honed instinct for spotting the deep identity of phenomena that look extremely different from each other on the surface, individuals who trust their inchoate faith in such analogical links even more than they trust the imposing mathematical fortresses erected by prior generations, even if it means that extremely well-established, once rigorously worked-out ideas may possibly have to be uprooted and completely thought through anew.

Our characterization of Einstein’s way of thinking portrays him as the polar opposite of the clichéd mathematical genius who launches an extraordinarily powerful calculating and deducing machine, which proceeds like a steamroller to mow down every obstacle that it encounters en route. Quite to the contrary, our discussion has shown that Einstein was not guided by phenomenal computational or reasoning skills. His brain did not house an enormous, lightning-fast supercomputer. Rather, he was driven by an unstoppable desire to seek out profound conceptual similarities, beautiful hidden analogies. Indeed, Einstein’s primary psychological driving force was the quest for beauty, and he was spurred on by his certainty that the laws of nature are pervaded by the deepest, most divine beauty of all.

This kind of mindset would seem to belong to the Age of Enlightenment, to a thinker in the grand tradition of Leonardo da Vinci, Newton, Pascal, and Leibniz. Some might even suppose that Einstein must have been the last member of that special tribe of scientific geniuses that flourished in the era before today’s hyper-specialization made earth-shaking insights just a relic of the past. But nothing could be further from the truth. We are dealing with phenomena that are much stabler than that. The same cognitive style is in fact the hallmark of the most outstanding scientists, whatever their discipline and whatever era they may live in. Today’s most gifted researchers, even if their areas of research are so arcane that it would be impossible to explain them to a lay audience, are driven by the same inner fire as drove Albert Einstein, which is to say, by an unrelenting quest to uncover deep analogical links between phenomena.

For example, the French mathematician Cédric Villani, recent Fields medalist, in his book
The Living Theorem
, paints a portrait of his mathematical style that is nearly word-for-word identical with the portrait we have painted of Einstein’s style. The excerpts below are in perfect resonance with our characterization of the great physicist:

Banesh Hoffmann, in his biography of Einstein, expressed a quite similar thought, although his passage exudes a flavor that is somewhat more mystical:

This paragraph by Hoffmann portrays science as advancing solely through quasi-magical mental processes that are unlikely ever to be explained, and a reader of it might well be led to the conclusion that scientific genius, such as Einstein’s, is an unfathomable mystery shrouded in the elusive depths of the Freudian unconscious. While we greatly respect Hoffmann’s ideas, we do not have as mystical a viewpoint as the one above. We have, rather, tried to show that the sudden bolts from the blue that change the face of physics are invariably analogies, often even “tiny” analogies — that is, leaps that seem obvious once they have been pointed out.

Lifting the veil on Einstein’s mental processes, as we have striven to do in this chapter, does not in any way diminish the depth of his genius or of his discoveries, because his lifelong obsession with “cosmic unity” pushed him to discover analogies at such a deep level that in the end he saw one single category of phenomena where even the other top scientists of his era had seen only unrelated things.