Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (118 page)

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.

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 permeates 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.

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:

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.

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.

If a nonmoving frame of reference in a gravitational field is indistinguishable
in every way
from an accelerating frame, then any effect that can be observed in an accelerated frame will also be observable in a lab on
terra firma
(since such a lab is obviously immersed in a gravitational field). The generalized equivalence principle thus told him that since a light ray in an accelerating lab in gravitation-free space follows a curved trajectory, then so must a light ray released by an observer standing still on the earth.

Einstein realized that this conclusion allowed him to predict certain celestial phenomena that had never been dreamt of, such as a tiny amount of bending of lightbeams coming from a distant star as they pass by our sun, whose gravitational field is, of course, very strong. However, for reasons too technical to go into here, this effect would be observable only during a total solar eclipse, and so, already in 1907, he urged that this effect be sought by astronomical observers. The German astronomer Erwin Finlay-Freundlich carefully examined many hundreds of photos of solar eclipses to find evidence of the minuscule effect, but found none. In fact it turned out to be necessary to wait twelve years longer, until 1919, for the confirmation of this prediction during a total eclipse observed by an English team led by the physicist Arthur Eddington from two islands in the south Atlantic Ocean.

The global effect of Eddington’s team’s confirmation was phenomenal. Not only did Einstein’s prediction hit the bull’s-eye, but the world, just emerging from under the dark pall cast by the “Great War”, was thrilled that an English team had confirmed a fantastic prediction made by an “enemy” scientist (even if Einstein had renounced his German citizenship and become Swiss in order to distance himself from German militarism); indeed, many people saw Eddington’s confirmation of Einstein’s prediction as a moment of great glory for humanity as a whole. Soon Einstein would watch helplessly as he was transformed overnight into a world-famous celebrity.

To conclude this tour of some of the many analogies that undergird general relativity, we will give a capsule description of the key breakthrough that brought to light the appropriate branch of mathematics for Einstein’s conception of gravitation. As we have just seen, all of Einstein’s first thought experiments about how gravity and
acceleration are related had to do with
linear
acceleration — scenarios in which a reference frame is moving in a fixed direction but with a speed that is changing. But another and equally important form of acceleration takes place whenever an object in motion undergoes a change in
direction
(though not necessarily in speed). The simplest and most canonical example is a disk spinning at a fixed number of revolutions per second. Each point on such a disk is accelerating because at every instant it is changing its direction of motion. Although applying the principle of equivalence to the rotating-disk scenario proved to be considerably harder than applying it to the linear scenario, this did not in any way discourage Einstein, who felt compelled to explore in depth the alluring case of the rotating disk.