Surfaces and Essences

  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.

  The Analogies of Einstein and the Categories of Physics

  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 = mc2, 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.

  The Principle of Relativity and Accelerated Frames of Reference

  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, includin
g 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.

  Einstein had understood all this at ETH, just as did all his classmates. But the moment that their doctoral exams were over, all the young physicists were happy to let all of this book-learning sink into oblivion, since the formulas involving fictitious forces in accelerating frames are, as a rule, quite convoluted; indeed, there is no reason to carry out calculations in such a frame, since one can always first describe the situation from the point of view of another observer moving at a constant speed and then do the calculations in that frame of reference. (In the case of the accelerating bus or car, this observer could be a smoothly-riding bicyclist, or a pedestrian standing at a corner.) It’s easy to see why young physicists gladly left all these complexities behind. And yet, whenever someone is obsessed by some idea, even the most deeply-buried of memories can suddenly bubble up, almost magically. In order to tell the story of one such sudden retrieval in Einstein’s mind, we first need to say a few words about gravitation.

  Applying Relativity to Gravity by Analogy

  One of Einstein’s obsessions, once he had discovered special relativity, was to integrate it somehow with gravity, which, for him, as for any physicist of the period, was a force that bore a tight analogy to the electrical force between two charged particles. One key difference was that for gravity, the force is always attractive, while for two electrically charged objects, it can be either attractive or repulsive, depending on whether the electrical charges of the objects are both positive, both negative, or of opposite signs. Like charges repel, and unlike charges attract. Despite this difference, the analogy between the two forces was clear and compelling. In the case of gravitation, the force between two unmoving objects having masses m and M and separated by distance d is given by the famous Newtonian formula “m x M/d2”, whereas for electricity, the strength of the force between two unmoving objects having charges q and Q and separated by distance d is given by the formula “q x Q/d2”, which was discovered exactly a century later by the French physicist Charles Coulomb. (We have left out multiplicative constants since they aren’t relevant in this context.) These formulas are identical, except that charges in the latter replace masses in the former. The analogy seems flawless on first sight, but Newton’s formula gave rise to a serious problem concerning gravity. To illustrate it, we consider an extreme scenario.

  Suppose our sun were to suddenly cease existing. It would take eight minutes for this disastrous piece of news to reach us on Earth (and for once the word “dis-aster” would be living up to its etymology); only after that period of time had elapsed would the sky go completely dark. The reason for the eight-minute delay is, of course, the finiteness of the speed of light, and this speed can be calculated directly from Maxwell’s equations for electromagnetism. Unfortunately, the tight analogy between gravitational and electrical forces does not involve Maxwell’s equations; it involves just Newton’s law and Coulomb’s law. Nothing comparable to Maxwell’s equations had yet been found for gravitation. Newton’s law — the lone equation that was then known to apply to gravity — did not predict that gravity could propagate across space. Thus, far from emerging as a consequence of known equations, the “speed of gravity” was an unheard-of notion; to suggest that gravity had a speed was to verge on spouting absurdities.

  For this reason, a physicist of that era might well have declared, “It wouldn’t be necessary to wait eight minutes for the bad news to reach us. Mother Earth would react immediately to the sudden dematerialization of the sun. After all, it would have no reason to continue to follow its quasi-circular orbit around a star that had ceased to exist and thus would no longer be exerting any tug on it. The earth would be like a dog whose leash had suddenly been cut: instant freedom!” On the other hand, another physicist of the era might well have argued the exact opposite — namely, that it would take time to detect the far-away sun’s demise, a conclusion based on the intuitive belief that no event can have an instant effect on objects arbitrarily far away from it.

  In any case, no experimental results or theoretical ideas were available to back up either side. And in the less disastrous (and more plausible) scenario of the sun’s center of mass suddenly moving a little bit (perhaps on account of some kind of internal explosion), exactly the same sorts of questions (“How long would it take for the earth to ‘find out’ that the sun had moved?”) could be asked, but to such questions no answer was offer
ed by the physics of the day. In short, while gravity deeply resembles the electrical force in some ways, there are other ways in which the two forces seem to be profoundly different, and in those days no one had any idea how to write down a set of equations that would fully capture the phenomenon of gravitation.

  In essence, the problem was to figure out how changes in gravity’s intensity are propagated across space, and at what speed — finite or infinite? — such “news” travels from one point to another. This boils down to asking “What is the formula for the gravitational attraction between two moving objects?” This was a question that had already been of serious concern to Isaac Newton, the first person to offer a quantitative theory of gravitation, but in all the years since him, no one had yet solved the problem. Einstein, in an attempt to answer this riddle, turned to the analogy between Newton’s formula for static gravitational attraction and Coulomb’s formula for static electrical attraction (the two formulas we mentioned above), and threw in an extra term that seemed exceedingly natural, and which came from his theory of (special) relativity. This new term elegantly extended the analogy between gravity and electromagnetism so that it included objects moving relative to each other. This small but very tempting addition yielded a new Einsteinian theory of gravitation that had a Maxwellian flavor, in that gravity now had wavelike behavior, and among the new theory’s consequences was that gravity propagated through space at a finite speed — in fact, at exactly the same speed as did light. According to Einstein’s new theory, then, the earth would “learn”, so to speak, that the sun had ceased to exist (or had suddenly moved a little bit) at exactly the same moment as our eyes would see it. Einstein’s new theory of gravity meshed well with special relativity.