Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (116 page)

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.

Intangible flavors of this sort are what guide a physicist instinctively toward new hypotheses. In this particular situation, Albert Einstein was pushed and pulled in various directions by numerous unspoken, probably largely unconscious, analogies, and finally, after two years, he imagined what he had been unable to imagine in 1905. The allegory of Jan, who, under intense pressure, suddenly had the breakthrough realization that her massive, solid mansion was in principle every bit as lithe and liquid as her bank account was, is an explanatory caricature analogy that helped us to convey, in highly concrete terms, the flavor of Einstein’s 1905–1907 esthetics-permeated ponderings.

To sum up, there are different and interconvertible types of mass, just as there are different and interconvertible types of energy; we thus learn that mass is every bit as protean and as ever-changing as is energy. This link between mass and energy is the astonishing analogy lying hidden in the five symbols of this most celebrated equation.

We earlier alluded to a special quality of Einsteinian analogies, which is that they often turned out not just to take advantage of similarities but to create deep unifications. Consider how Einstein discovered special relativity. The key step was making a trivial-seeming analogy from mechanics to all other branches of physics. Probably most physicists of Einstein’s day, had they been handed the principle of Galilean relativity on a silver platter and told, “Generalize this principle!”, would have been able to make the same analogy and would have reached Einstein’s generalization of it (although whether they would have realized its far-reaching consequences is another question). But the key fact is that physicists back then were
not
mulling over the limits of the principle of Galilean relativity, so no silver platter was proffered to them. The idea of generalizing Galileo’s principle had to be coaxed out of the woodwork. Einstein saw that this simple and fundamental centuries-old principle lay at the crossroads of a number of important problems in physics, and that it was crying out for generalization. In contrast, other physicists were focusing on what
distinguished
electromagnetism from mechanics, instead of seeking a shared essence that could
unite
those two branches of physics.

Similar remarks can be made concerning Einstein’s 1907 analogy linking two seemingly different types of mass. Some people might even object that this was not an analogy, because (at least according to the common stereotype) analogies are always just partial and approximate truths, whereas in the case of
E = mc
²
, the link Einstein found between what we’ve called “strange” and “normal” types of mass turned out to be a complete and precise truth, revealing them to be merely two facets of one single phenomenon. Well, such an objection might seem tempting, but it is off base.

The fact is, the idea started out life as a typical analogy, tentative and shaky — the fruit of a long and patient quest by Einstein to unify two concepts that, in the minds of his very few colleagues who took these kinds of questions seriously, were clearly distinct. Two decades later, however, experimental findings showed that this apparent distinction had to be dropped in favor of a single, more extended concept. The reason is that Einstein’s irrepressible instinct of cosmic unity had hit the bull’s-eye once again, revealing a new, broader concept built deeply into nature.

We will now take a look, albeit brief, at general relativity, at whose basis there are, once again, some Einsteinian analogies that were initially perceived as bold leaps of an idiosyncratic intuition, if not as wild speculations, but which later, once they had been repeatedly shown to be correct, were retroactively perceived as
eternal truths of nature
rather than as merely one individual’s subjective and uncertain speculations about some kind of similarity.

What greatly troubled Einstein, as he looked back at his theory of special relativity (which at the outset was not called “special”, since at that time it was not part of a more general theory), was that it applied only to frames of reference that were moving constantly and smoothly — that is, without any acceleration. His extension of Galileo’s principle of relativity, proposed in 1905, and later dubbed by him the Principle of Special Relativity, said that
for certain types of frames of reference
, it is impossible to tell, using internal experiments, whether one is at rest or not. These frames of reference were those moving at a constant velocity. But internal experiments
can
distinguish, without any problem, between
some
kinds of frames of reference — namely, between accelerating and constant-velocity frames. For example, if you are inside a car that is accelerating rapidly, you
cannot
pour a glass of water just as you would do in your kitchen or in an airplane flying at a constant speed, because the water, as seen by people sitting in the car, will not fall vertically downwards, but will follow a curved arc whose shape is determined by the direction of, and the strength of, the car’s acceleration. Anyone inside the car can conduct this very simple experiment, which clearly reveals that the car is not moving with a fixed speed but is accelerating.

Most physicists of Einstein’s day would have said that all this shows is that the principle of relativity has its limits and cannot aspire to cover
all
frames of reference — just frames moving with constant velocities. But Einstein could not stop scratching his head over this frustrating situation. He felt that his newly extended principle of
relativity should somehow be able to be extended yet further, so as to cover more frames of reference than just those that were not accelerating — in fact, he felt that a truly general principle should be able to cover
all
frames of reference. This strange faith of Einstein’s, which flew in the face of the most self-evident facts about the way the world works, was rooted in a profound and nearly inexplicable intuition.

In his mind, Einstein imagined an infinite universe that was totally empty but for one sole sentient observer. This perceiving being considers itself to be stationary, and thus has no sensation of dizziness. After all, as it looks around, it sees nothing but empty space. On the other hand, what if the being were
spinning
in this completely empty universe — would it feel dizzy? Well, what does it mean to speak of “something spinning in a totally empty universe”? Or conversely, what if our observing being was
not
spinning, but the rest of the universe was spinning around it, like a merry-go-round spinning around a stationary pillar? Would the observing being then feel dizzy? And finally, is it in theory possible to distinguish between these two scenarios? Are they identical or are they different? Such thought experiments force us to ask whether it is possible to determine which of the two — the observer or the rest of the universe — is spinning. It would seem that for the two scenarios to be distinguishable, there would have to be a
preferred
frame of reference, sometimes called an
absolute
frame of reference, or “God’s frame of reference”, or the frame of reference of a hypothetical “ether”.

In his early years, Einstein was constantly haunted by philosophical questions like these, and in the wake of his discovery of special relativity, he found the idea of an absolute frame of reference so distasteful that he rejected it out of hand. (Einstein was particularly inspired by the writings of the Austrian philosopher and physicist Ernst Mach — ironically, one of the staunchest disbelievers in atoms! — in which Mach dreamt up hypothetical universes of this sort and carefully studied their consequences, which had led Mach to the conviction that the idea of absolute motion — a notion due to Newton — makes no sense.) Einstein nourished the hope of discovering some way to incorporate even accelerating frames of reference into his principle — in other words, he wanted to show that acceleration, much like speed, is not absolute but depends on the frame of reference that one chooses.

Special relativity implied that what was perceived as motion at a constant speed by the observers in one frame of reference could validly be perceived as perfect immobility by the observers in some other properly chosen frame. Einstein wanted to generalize this idea; he wanted an analogous principle to hold for all types of motion, including accelerated motion. His hope was that if, from the viewpoint of one set of observers in one particular frame of reference, some object was accelerating, it would be possible to find
another
frame of reference in which all observers would say the object was perfectly still. To put it in another way, he hoped that the laws of nature as perceived by an accelerating observer would be identical to the laws of nature as perceived by an observer who was at rest.

Despite this idealistic hope, Einstein was keenly aware of all sorts of phenomena, such as the pouring of a glass of water inside a speeding-up or slowing-down car, that
do
allow one to distinguish accelerating frames of reference from non-accelerating ones.
The undeniable conflict between the unbending reality of nature and his strong intuitions pushed Einstein to focus intently on the nature of acceleration. Very few physicists are driven to seek answers to such pithy and essential questions as “What is acceleration?” or, more specifically, “Must it be the case that something accelerating for
one
observer will be accelerating for
all
observers?” But it was typical of Einstein to do just that — to tackle with unbounded stubbornness questions that concern matters that seem so primordial and so pervasive that nearly anyone else would have wondered what use it could possibly be to worry about them.

Whenever one studies classical mechanics (as Einstein did at ETH, the Swiss Federal Polytechnical School in Zürich), the mathematical form of the laws of physics from the vantage point of an accelerating reference frame is always covered at least briefly. The most salient fact one learns in such an overview is that any observer in an accelerated frame who insisted that the frame was
not
accelerating would have to posit a mysterious “extra force” acting on all objects. Without such an extra force, there would be no way to account for the anomalous movements of the objects in the frame.

Imagine, for instance, that the passengers in a city bus collectively decided, on some odd whim, to declare that their often-accelerating, often-decelerating bus never moved but was always perfectly still. In order to account for the strange phenomena that they witness (
e.g.
, water following a curve rather than falling straight down when poured from a pitcher, not to mention their own frequent sensation of being jerked forwards or backwards), they would have to posit a mysterious extra force sometimes pulling things towards the front of the bus, other times towards the rear. Of course, they wouldn’t need to refer to any such force if they acknowledged that their bus is sometimes speeding up and sometimes slowing down, but if they refused to adopt that point of view, then this extra force would have to be included in the laws of physics that they formulate to explain the phenomena that they observe.

Forces like this, which show up only in accelerating frames of reference, are called
fictitious forces
, and it happens that all fictitious forces have a special mathematical property — namely, if such a force acts on an object that has mass
m
, then it will necessarily be proportional to
m
, no matter how the frame is accelerating and no matter how the object is moving (think of a tennis ball being dropped inside a car just as the driver slams very hard on the brakes). This proportionality to mass gives rise to an interesting consequence, which is that if one releases several objects at the same instant in an accelerating frame of reference, they will all follow perfectly parallel trajectories. For instance, if a ball, a bell, and a bowl are all released into the air at exactly the same moment inside a car that is accelerating, then all three of them will describe identical-looking curves as they fall. This means that if they start out in a tight cluster, then they will remain in a tight cluster; at the end of their fall, their cluster will be just as compact as it was at its start.