Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (117 page)

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.

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/d
²
”, 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
/
d
²
”, 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 “disaster” 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 offered 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.

Although this analogy with electromagnetism was both natural and attractive, its discoverer, who was at the same time his own most severe critic, soon became aware that the formula that it gave for gravitational force had a fatal flaw: the force between two objects would no longer be proportional to the product of their masses (the numerator “m
x
M
” that we saw above). That property of gravity was so well established and seemed so central to the very nature of gravity that violating it was
intuitively repugnant to Einstein. As a result, he abandoned the lines of work that came from this first analogy and began searching for a different analogy that would link gravitation and relativity.

In this new quest, he focused not only on gravitation, whose most stable and defining feature is its proportionality to the mass of each of the two objects pulling each other, but also on accelerated frames of reference, which are so fundamentally different from frames at rest (and ones that have a constant speed). Indeed, because the fruit of his first analogy had failed to respect gravity’s proportionality to mass, this fact about gravity became one of the primary desiderata for a better theory.

This was the point at which some of the old ideas from courses at the ETH in Zurich started bubbling up — ideas involving accelerated frames of reference. In particular, the memory of fictitious forces came back to Einstein, for any fictitious force is likewise
proportional to the mass
of the entity it is acting on. Let us try to relive from a first-person perspective this Einsteinian mental process: “Hmm… Gravity
reminds me
of a fictitious force… Gravity
acts like
a fictitious force. Gravity
is analogous to
a fictitious force… Might, then, gravity actually
be
a fictitious force?” Here is a remarkably smooth mental glide that starts out with an innocent little case of reminding but that ends up being, once again, nothing less than a cosmic unification.

To see more clearly the implications of this idea, let’s consider the canonical example of an accelerated frame of reference that Einstein himself used in order to explain his new analogy. Instead of imagining that we’re in an accelerating bus or car, let’s jump on board a cube-shaped interplanetary laboratory floating somewhere in empty space, far removed from any star, quite literally out in the middle of nowhere. And now let’s throw into the mix a powerful rocket that starts to pull the lab by means of a cable attached to one of its six exterior walls. If the rocket pulls with a constant force, this will give rise to a constant acceleration of the lab (after all, as Newton told us,
F = ma
— that is, a constant force induces a constant acceleration). Before the rocket started firing, people inside the lab were floating about between its six walls, and there was no reason to single out one particular wall and call it “the floor”; however, the moment the rocket started to pull, one of the six walls started approaching the lab-bound celestial travelers, and all of them banged into it and remained stuck against it, because the lab’s constant acceleration broke the symmetry, putting an end to the possibility of floating in it. This particular wall thus became the lab’s “floor”.

Moreover, if one of the lab’s denizens tossed a pencil into the air, the latter would “fall down” onto the “floor”, just as Newton’s famous (if apocryphal) apple fell down onto his head. Why would this be the case? Seen from
outside
the lab, it’s clear as day: the floor is constantly moving towards the pencil (until they collide). By contrast, the people who are
inside
the lab and are ignorant of the rocket conceive of their lab as sitting absolutely still (or as moving at a constant speed) in the middle of outer space, and so for them the pencil is falling because a
gravitational pull
suddenly and inexplicably permeated their lab, affecting all people and objects in it, and of course that brand-new pull singled out a particular direction in space (which was then baptized “down”), and it made things
fall
in that direction.

This means, among other things, that if some Galileo copycat in the lab were to “stand up” on the new “floor” and were to “drop” two very different objects, such as a pencil and a cannonball, these items would start to “fall” side by side and would bang onto the “floor” at precisely the same instant. And why would this be? Once again, for external observers, it’s obvious: the two objects aren’t moving at all; rather, it’s the floor that is moving “up” to greet them. So of course it will hit them at exactly the same instant. But from the viewpoint of the people inside the laboratory, the phenomenon arises because gravity has that key property that Galileo, creatively exploiting the leaning Tower of Pisa, was the first to demonstrate — namely, that all objects fall in the same way (
i.e.
, with the same acceleration), whatever their masses might be.

This “force of gravity” perceived by the denizens of the lab is a quintessential example of a fictitious force, but for them there is nothing fictional about it — for them, it is a
real
force; for them, there is a
real
floor and a
real
ceiling; for them, there is a
genuine
distinction between
up
and
down.
Unless they somehow manage to sneak a peek
outside
of their lab (which would violate the premises of Einstein’s thought experiment), these voyagers have no way to tell apart the rocket-made gravitation from normal earthly gravitation, which they have known ever since their childhood. This means they have no way to tell whether their lab is constantly accelerating in empty space or is sitting still in the earth’s gravitational field. The two situations are indistinguishable.

If you are picking up echoes of Galileo’s principle of relativity, you’re not mistaken. This is exactly what Einstein was up to. Like a top-drawer magician, he had started with the idea, obvious to everyone, that an accelerating frame of reference is
easily distinguishable
from a stationary frame, only to arrive at the diametrically opposite conclusion: that an accelerating frame of reference is
completely indistinguishable
from a stationary reference frame immersed in a gravitational field. What a fantastic trick! Galileo would surely have loved this ingenious combination of two of his greatest ideas.

Moreover, Einstein soon saw that there was a broad spectrum of indistinguishable labs; to get the flavor of them, you need merely imagine a lab on the surface of the moon (whose gravitational pull is much weaker than earth’s), which at some point starts to be pulled upward by a slightly less powerful rocket than the previous one. Now the combination of the moon’s feeble gravitational pull and this weaker rocket’s pulling adds up to a result that, for people inside the lab, is exactly like the case we described earlier. They feel as if they are now in the
earth’s
gravitational field. And it’s obvious that we could twist the knobs that control (1) the strength of the “real” gravitational field in which the lab is sitting, and (2) the power of the rocket that is pulling the lab, in such a manner that in each case the resultant experience of gravity will be identical to the earth’s gravitational field for the people inside the lab. All of these imaginary labs are indistinguishable from one another by any kind of mechanical experiment at all, as long as it is carried out entirely
within
the lab.