Surfaces and Essences

  One Einsteinian Analogy Bites the Dust and Another One Replaces it

  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.

  In summary, thanks to the memory of fictitious forces in classical mechanics that bubbled up in his brain, Einstein was able to breathe new life into his beloved principle of relativity, and he did so in a situation where any other physicist would have been sure that there was no hope of doing so.

  The Principle of Equivalence (First Draft)

  To the new and radically extended version of the principle of relativity that he had just found, Einstein gave the name “Equivalence Principle”. By this, he meant that there was an equivalence or indistinguishability between gravity, on the one hand, and acceleration, on the other. In order to explore the consequences of his simple but counterintuitive hypothesis more deeply, he imagined another scenario involving his laboratory floating in space. This time, the lab is not out in the middle of nowhere, but is hovering 100 miles above the earth, where a magical angel is holding it up and perfectly still. The people inside have no qualms about using words like “floor”, “ceiling”, “up”, and “down”, because the earth’s gravitational pull pe
rmeates their laboratory exactly as it would permeate a lab sitting on the surface of the earth, the only difference being that gravity is slightly weaker at an altitude of 100 miles than at sea level. If one ignores this detail, the people inside the lab could easily imagine themselves as being on the earth. But all at once the angel is stung by a randomly buzzing interplanetary bumblebee and lets go of the lab, which starts plummeting down towards the earth. What do the people inside it feel?

  As soon as their cube starts its earthward descent, all the objects inside it also start to fall, and recall that, as Galileo first showed, they will all fall in the same way (recall his experiments at the Tower of Pisa). This means that any object that had previously been sitting on the floor no longer feels itself held down; it is suddenly free to float about in the lab. In fact, for the people inside the lab, there is no floor any longer, nor is there any ceiling; in a flash, the words “floor”, “ceiling”, “up”, and “down” have been deprived of their meanings. The lab’s inhabitants are experiencing weightlessness: that is, the sensation of zero gravity. Since everything in the lab is falling earthwards at exactly the same rate, its denizens have the curious impression that nothing is falling; for them, everything (themselves included) is now floating about inside their big room.

  For Einstein, this realization was one of the most beautiful moments of his life, for as he started to glimpse it, a wonderfully fertile analogy suddenly sprang to his mind. He had been musing about electromagnetism for a long time, and he knew intimately that if one moves from one reference frame to another, the apparent values of the electric and magnetic fields change throughout all of space. For example, an observer who moves a magnet in a lab will be able to detect an electric field in the neighborhood of the magnet, and the more quickly the magnet is moved, the more intense will be the measured value of the electric field. (If the magnet is simply kept at rest, then it gives rise to no electric field, of course.) This important effect is called “electromagnetic induction”, and was first observed in 1831 by the English physicist Michael Faraday. Now let us imagine a second observer sitting tightly attached to the magnet (which means the observer is in the magnet’s reference frame). By definition, for this person, the magnet is perfectly at rest, and so Faraday’s induction law says there is only a magnetic field. Put otherwise, from this person’s viewpoint, there is no moving magnet to give rise to any electric field through electromagnetic induction. So we see that in electromagnetism, merely by changing viewpoint (i.e., frame of reference), one can make an electric field completely disappear (or appear out of nowhere). (The same holds for a magnetic field, although we haven’t described this case.) Even in his early adolescence Einstein had already been struck and fascinated by this mysterious effect.

  Shortly after the memory of this effect in electromagnetism bubbled up, Einstein expressed his great joy thus: “At that moment the happiest thought of my life occurred to me — namely, the gravitational field, just like the electric field generated by a moving magnet, has an existence that is only relative.” (“Relative” here meant that its existence depended on the frame of reference in which one was located, and in particular that in at least one frame, it didn’t exist at all.)

  And indeed, with his new scenario of the earthwards-plunging laboratory whose occupants feel that they now are experiencing no gravity, Einstein had found a situation where a perfectly real gravitational field in one reference frame can be made to totally vanish merely by jumping to another one. To an outside observer — say, someone on earth — the falling laboratory is still permeated by the earth’s gravitational field (which is why everything in it is falling earthwards); and yet to the people inside it, there simply is no gravitational field at all, and nothing is falling.

  This scenario is in some sense the flip side of the scenario featuring the laboratory in remote outer space being pulled by the powerful rocket, since in the latter scenario, the people inside feel, observe, and measure a gravitational field, while outside observers claim that there is no gravitational field — all they see is a rocket that is making the lab go faster and faster, with respect to the far-off stars.

  Einstein Seeks and Finds a Deeper Analogy

  We come now to a decisive moment in the story of general relativity. Above, we described Einstein’s new principle as asserting that an accelerating reference frame is completely indistinguishable from a non-accelerating reference frame immersed in a gravitational field. However, in putting it this way we jumped the gun, because his original principle was significantly more limited than that — namely, it asserted that an accelerating reference frame should be indistinguishable, by means of mechanical experiments, from a non-accelerating reference frame immersed in a gravitational field. Einstein was keenly aware of the fact that the analogies that had led him to his newfound principle — the “happiest thought” of his life — involved only the mechanical behavior of imagined objects in various different imagined laboratories. That is, when he had imagined his various spacebound laboratories, he had considered only scenarios that involved concepts such as speed, acceleration, rotation, gravity, friction, orbits, collisions, springs, pendula, vibration, tops, gyroscopes, etc. — just the concepts of classical mechanics. He had not considered what might happen in the case of an electromagnetic experiment in an accelerating laboratory — say, an experiment that used light rays or electric or magnetic fields.

  This was because he knew that he did not have the requisite knowledge, be it theoretical or experimental, that would allow him to predict what would happen in such a case. And that was why he knew he had arrived at a critical crossroads. His theoretical knowledge and his gift for imagining the consequences of various idealized physical circumstances (his famous “thought experiments”), even when aided by the cleverest reasoning, simply would not allow him to go any further. He had reached a crucial spot where he would have to take yet another daring step, once again a step that would rely solely on an esthetic motivation, a step grounded solely in his belief in the deep unity of the laws that govern the universe — that is to say, to his unshakable faith in the existence of very simple, general, and elegant principles.

  As readers of this chapter are well aware, back in 1905 Einstein had already found himself, by chance, in an analogous situation — namely, when he had chosen to broaden the Galilean principle of relativity on esthetic grounds (his faith in the unity of the laws of physics), by replacing the phrase “any kind of mechanical experiment” with the phrase “any kind of physical experiment”. In other words, he had already visited this region in the world of ideas, had already dared once earlier to make just this leap of analogical faith, and on that occasion his intuition had been richly rewarded — and so, why not do “exactly the same thing” in this analogous new situation?

  Einstein thus extended his principle to run as follows: “An accelerating reference frame cannot be distinguished, no matter what kind of physical experiment one might use, from a non-accelerating reference frame immersed in a gravitational field.” Once again we point out that from a certain point of view, replacing the word “mechanical” by the word “physical” (in other words, noting the analogy between mechanics and any other branch of physics) was the most trivial step Einstein could have taken, since that same analogical extension had already worked with flying colors one time earlier in his life (special relativity had already been confirmed by a good number of experiments) — and yet, from another point of view, it was an extremely audacious analogical leap into an utterly unknown world.

  Let us listen once again to the words of Banesh Hoffmann on the subject of this jump that carried Einstein from the restricted principle of equivalence to the extended principle of equivalence:

  [The new principle] had artistic unity: for why should he needlessly assume one type of relativity for mechanical effects and a different one for the rest of physics?

  Once again, we see an analogy-based conceptual leap that was apparently minuscule and elementary, and yet on the
other hand turned out to be gigantic and brilliant. All of this came to him courtesy of his “instinct for cosmic unity”, which was an almost inexhaustible font of rich analogies.

  Consequences of the Extended Principle of Equivalence

  We will give here an example of the unexpected consequences of this daring leap towards a truly general principle of relativity. Einstein imagined that there was, in his celestial laboratory being pulled through deepest outer space by the rocket, a perfectly horizontal flashlight (i.e., parallel to the lab’s “floor”) that emitted a light ray. Observers outside the lab will say that this ray is moving in a fixed direction with respect to the distant stars, while at the same time the lab surrounding it is “rising” at an ever greater velocity. From this constant “vertical” acceleration of the lab, it follows that observers inside the lab will perceive the light ray as descending towards the floor ever more rapidly as it crosses the lab at a fixed horizontal speed. In a word, for them, the light ray will follow a curve rather than a straight line. To be sure, the discrepancy from a horizontal trajectory will be tiny, because of the enormous ratio of the speed of light to the modest speed of the lab, but no matter; no sooner has the light ray emerged horizontally from the flashlight than its trajectory starts to bend downwards. At this juncture Einstein makes use of his newly conjectured equivalence principle generalized outwards so as to include electromagnetic scenarios. He reasons as follows.