The Dancing Wu Li Masters

by Gary Zukav

  For all this, however, the special theory of relativity has one shortcoming. It is based on a rather uncommon situation. The special theory of relativity applies only to frames of reference that move uniformly, relative to each other. Most movement, unfortunately, is neither constant nor ideally smooth. In other words, the special theory of relativity is built upon an idealization. It is limited to and premised upon the special situation of uniform motion. That is why Einstein called it the “special,” or restricted, theory.

  Einstein’s vision was to construct a physics that is valid for all frames of reference, such as those moving with non-uniform motion (acceleration and deceleration) relative to each other, as well as those moving uniformly relative to each other. His idea was to create a physics which could describe events in terms of any frame of reference, no matter how it moves relative to any other frame of reference.

  In 1915, Einstein succeeded in achieving the complete generalization of his special theory. He called this achievement the general theory of relativity.

  1

  General Nonsense

  The general theory of relativity shows us that our minds follow different rules than the real world does. A rational mind, based on the impressions that it receives from its limited perspective, forms structures which thereafter determine what it further will and will not accept freely. From that point on, regardless of how the real world actually operates, this rational mind, following its self-imposed rules, tries to superimpose on the real world its own version of what must be.

  This continues until at long last a beginner’s mind cries out, “This is not right. What ‘must be’ is not happening. I have tried and tried to discover why this is so. I have stretched my imagination to the limit to preserve my belief in what ‘must be.’ The breaking point has come. Now I have no choice but to admit that the ‘must’ I have believed in does not come from the real world, but from my own head.”

  This narrative is not poetic hyperbole. It is a concise description of the major conclusion of the general theory of relativity and the means by which it was reached. The limited perspective is the perspective of our three-dimensional rationality and its view of one small part of the universe (the part into which we were born). The things that “must be” are the ideas of geometry (the rules governing straight lines, circles, triangles, etc.). The beginner’s mind was Albert Einstein’s. The long-held belief was that these rules govern, without exception, the entirety of the universe. What Einstein’s beginner’s mind realized was that this is so only in our minds.*

  Einstein discovered that certain laws of geometry are valid only in limited regions of space. This makes them useful since our experience physically is limited to very small regions of space, like our solar system. However, as our experience expands, we encounter more and more difficulty in trying to superimpose these rules upon the entire expanse of the universe.

  Einstein was the first person to see that the geometrical rules which apply to one small part of the universe as seen from a limited perspective (like ours) are not universal. This freed him to behold the universe in a way that no person had seen it before.

  What he saw is the content of the general theory of relativity.

  Einstein did not set out to prove anything about the nature of our minds. His interest was in physics. “Our new idea,” he wrote, “is simple: to build a physics valid for all co-ordinate systems.”1 The fact that he did illustrate something of importance about the way that we structure our perceptions is indicative of an inevitable trend toward the merger of physics and psychology.

  How did Einstein get from a theory of physics to a revolutionary statement of geometry? How did that lead to a significant insight into our mental processes? The answer to these questions is one of the least known, but one of the most important and intriguing intellectual adventures recorded.

  Einstein started with his special theory of relativity. As successful as it was, Einstein was not satisfied with it because it applied only to co-ordinate systems moving uniformly relative to each other. Is it possible, thought Einstein, to explain the same phenomenon as seen from two different frames of reference, one of them moving uniformly and the other of them moving non-uniformly, in such a way that there is a consistent explanation for the phenomenon in terms of both the uniformly moving frame of reference and the non-uniformly moving frame of reference. In other words, can we describe events which happen in a co-ordinate system which is moving non-uniformly in terms which are meaningful to an observer in a co-ordinate system which is moving uniformly, and the other way round? Can we create one physics that is valid for observers in both frames of reference?

  Yes, discovered Einstein, it is possible for observers in the two different frames of reference to relate in a manner which is both meaningful in terms of their own state of motion and in terms of the other’s state of motion. To illustrate this, he used another famous thought experiment.

  Imagine an elevator in an extraordinarily tall building. The cable which supports the elevator has snapped, and the elevator is plummeting downward. Inside the elevator are several physicists. They are not aware that the cable has broken, and, since there are no windows, they cannot look outside.

  The question is, what is the appraisal of this situation by the observers on the outside of the elevator (us) and by the observers on the inside of the elevator (the physicists)? Since this is an idealized experiment, we can disregard the effects of friction and the resistance of the air.

  To us, the situation is apparent. The elevator is falling and soon it will strike the earth and all of its inhabitants will be dead. As the elevator falls, it accelerates according to Newton’s law of gravity. The motion of the elevator is not uniform, but accelerated, because of the gravitational field of the earth.

  We can predict many things that might happen inside the elevator. For example, if someone inside the elevator dropped a handkerchief, nothing would happen. It would appear to the inside observers to float where it was released because it would be accelerating toward the earth at the same rate as the elevator and the people inside of it. Nothing really would be floating, everything would be falling, but, since everything would be falling at the same rate, there would be no change in their relative positions.

  To a generation of physicists born and brought up inside the elevator, however, things would appear quite differently. To them, dropped objects do not fall, they simply hang in midair. If someone gives a floating object a shove, off it goes in a straight line until it hits the side of the elevator. To the observers inside the elevator, there are no forces acting on any objects inside the elevator. In short, the observers inside the elevator would conclude that they are in an inertial co-ordinate system! The laws of mechanics are perfectly valid. Their experiments always produce results which agree exactly with theoretical predictions. An object at rest remains at rest. An object in motion remains in motion. Moving objects are deflected from their paths only by forces which are proportional to the amount of deflection. For every reaction there is an equal and opposite reaction. If we give a shove to a floating chair, it goes off in one direction, and we go off in the opposite direction with an equal momentum (although with a slower speed because of our greater mass).

  The inside observers have a consistent explanation for the phenomena inside the elevator: They are in an inertial co-ordinate system, and they can prove it by the laws of mechanics.

  The outside observers also have a consistent explanation for the phenomena inside the elevator: The elevator is falling in a gravitational field. Its passengers are unaware of this because, without being able to see outside the elevator, there is no way for them to detect it while they are falling. Their co-ordinate system is in accelerated motion, even though they believe that it is not moving at all.

  The bridge between these two explanations is gravity.

  The falling elevator is a pocket edition of an inertial co-ordinate system. A real inertial co-ordinate system is not limited in space or time.
The elevator edition is limited in both. It is limited in space because a moving object inside the elevator will not move in a straight line forever, but only until it reaches one of the walls of the elevator. It is limited in time because sooner or later the elevator and its passengers are going to collide with the earth, ending their existence abruptly.

  According to the special theory of relativity, moreover, it is significant that the elevator is limited in size because otherwise it would not appear to its inhabitants as an inertial co-ordinate system. For example, if the physicists inside the elevator simultaneously drop two baseballs, the baseballs float in the air exactly where they are released, and remain there. This, to the outside observer, is because they are falling parallel to each other. However, if the elevator were the size of Texas and the baseballs were as far apart when they were dropped as Texas is wide, the baseballs would not fall parallel to each other. They would converge, since each of them would be drawn by gravity to the center of the earth. The observers inside the elevator would notice that the baseballs, and any other floating objects in the elevator, move toward each other with the passage of time, as though there were a mutual attraction between them. This mutual attraction would appear as a “force” affecting the objects in the elevator, and the physicists inside hardly would conclude, under those circumstances, that they were in an inertial co-ordinate system.

  In short, if it is small enough, a co-ordinate system falling in a gravitational field is the equivalent of an inertial co-ordinate system. This is Einstein’s principle of equivalence. It is a telling piece of mental dexterity. Anything like an “inertial co-ordinate system” that can be “wiped out”2 (Einstein’s words) by the assumption of a gravitational field hardly deserves to be called absolute (as in “absolute motion,” and “absolute nonmotion”). While the observers inside the elevator experience a lack of motion and the absence of gravity, the observers outside the elevator see a co-ordinate system (the elevator) accelerating through a gravitational field.

  Now let us imagine a variation of this situation.

  Assume that we, the outside observers, are in an inertial co-ordinate system. We already know what happens in inertial co-ordinate systems; the same things that happened in the falling elevator. There are no forces, including gravity, to affect us. Therefore, let us assume that we are comfortably floating. Objects at rest remain at rest, objects in motion continue in a straight line forever, and every action produces an equal and opposite reaction.

  In our inertial co-ordinate system is an elevator. Someone has attached a rope to the elevator and is pulling it in the direction indicated.

  Since this is a thought experiment, it does not matter how this is done. The elevator is being pulled with a constant force, which means that it is in a state of constant acceleration in the direction of the arrow. How will observers outside the elevator and observers inside the elevator appraise this situation?

  As we float outside the elevator, we experience that our frame of reference is absolutely at rest and that there is no gravity affecting it. We see the elevator being pulled with a constant acceleration by the rope, and so we can predict certain things about it. Everything inside the elevator that is not attached quickly collides with the floor of the elevator. If someone in the elevator drops a handkerchief, the elevator floor rushes up to meet it. If someone in the elevator tries to jump off the floor, the floor, rushing upward, is instantly under his feet again. The floor of the elevator continually crashes into anything in its path as it accelerates upward.

  Inside the elevator, however, the appraisal of the situation is quite different. To a generation of physicists born and brought up inside the elevator, talk of acceleration upward is fantasy (remember, the elevator has no windows). To them, their co-ordinate system is quite at rest. Objects fall downward to the floor because of a gravitational field, just as objects on the earth fall downward to the floor because of a gravitational field.

  Both the observers inside the elevator and the observers outside the elevator have consistent explanations for the phenomena inside the elevator. We observers outside the elevator explain them by the accelerated motion of the elevator. The observers inside the elevator explain them by the presence of a gravitational field. There is absolutely no way to determine which of us is right.

  “Wait a minute,” we say, “suppose that we cut a small hole in one wall of the elevator and shine a light beam through it. If the elevator really were motionless, the light beam would strike the opposite wall of the elevator at a spot exactly opposite the hole. Since we can see that the elevator is accelerating upward, we know that the elevator wall will move upward slightly in the time it takes the light beam to cross the elevator. Therefore, the light beam will strike the far wall slightly below the spot just opposite the hole it entered through. In effect, it will seem to curve downward from the point of view of the people inside the elevator instead of traveling in a straight line. This should prove to them that their elevator is in motion.”

  “It does not prove anything of the sort,” says Jim de Wit, who, of course, is inside the elevator. “The light beams in this elevator do not travel in straight lines. How could they? We are in a gravitational field. Light is energy, and energy has mass. Gravity attracts mass, and a light beam traveling through our elevator will be drawn downward by our gravitational field exactly like a baseball thrown horizontally at the speed of light.”

  There is no way that we can convince de Wit that his co-ordinate system is in a state of accelerated motion. Everything that we can say to prove this to him he dismisses (accounts for) as a result of his “gravitational field.” There is absolutely no way of distinguishing between uniform accelerated motion and a constant gravitational field.

  This is another expression of Einstein’s principle of equivalence. In limited areas, gravity is equivalent to acceleration. We already saw that acceleration (falling) through a “gravitational field” is the equivalent of an inertial co-ordinate system. Now we see that a “gravitational field” is equivalent to accelerated motion. At last we are approaching a general theory of relativity, a theory valid for all frames of reference regardless of their states of motion.

  The bridge which links the explanations of the observers inside of the elevator and the explanations of the observers outside of the elevator is gravity. The clue which indicated to Einstein that gravity was the key to his general theory was as old as physics itself.

  There are two kinds of mass, which means that there are two ways of talking about it. The first is gravitational mass. The gravitational mass of an object, roughly speaking, is the weight of the object as measured on a balance scale. Something that weighs three times more than another object has three times more mass. Gravitational mass is the measure of how much force the gravity of the earth exerts on an object. Newton’s laws describe the effects of this force, which vary with the distance of the mass from the earth. Although Newton’s laws describe the effects of this force, they do not define it. This is the mystery of action-at-a-distance. How does the earth invisibly reach up and pull objects downward?

  The second type of mass is inertial mass. Inertial mass is the measure of the resistance of an object to acceleration (or deceleration, which is negative acceleration). For example, it takes three times more force to move three railroad cars from a standstill to twenty miles per hour (positive acceleration) than it takes to move one railroad car from a standstill to twenty miles per hour. Similarly, once they are moving, it takes three times more force to stop three cars than it takes to stop the single car. This is because the inertial mass of the three railroad cars is three times more than the inertial mass of the single railroad car.

  Inertial mass and gravitational mass are equal. This explains why a feather and a cannonball fall with equal velocity in a vacuum. The cannonball has hundreds of times more gravitational mass than the feather (it weighs more) but it also has hundreds of times more resistance to motion than the feather (its inertial mass). Its attrac
tion to the earth is hundreds of times stronger than that of the feather, but then so is its inclination not to move. The result is that it accelerates downward at the same rate as the feather, although it seems that it should fall much faster.

  The fact that inertial mass and gravitational mass are equal was known three hundred years ago, but physicists considered it a coincidence. No significance was attached to it until Einstein published his general theory of relativity.

  The “coincidence” of the equivalence of gravitational mass and inertial mass was the “clew,”3 to use Einstein’s word, that led him to the principle of equivalence, which refers via the equivalence of gravitational mass and inertial mass to the equivalence of gravity and acceleration themselves. These are the things that he illustrated with his famous elevator examples.

  The special theory of relativity deals with unaccelerated (uniform) motion.* If acceleration is neglected, the special theory of relativity applies. However, since gravity and acceleration are equivalent, this is the same as saying that the special theory of relativity is applicable whenever gravity is neglected. If the effects of gravity are to be considered, then we must use the general theory of relativity. In the physical world the effects of gravity can be neglected in (1) remote regions of space which are far from any centers of gravity (matter), and (2) in very small regions of space.

  Why gravity can be ignored in very small regions of space leads to the most psychedelic aspect of all Einstein’s theories. Gravity can be ignored in very small regions of space because, if the region is small enough, the mountainous terrain of space-time is not noticeable.*

  The nature of the space-time continuum is like that of a hilly countryside. The hills are caused by pieces of matter (objects). The larger the piece of matter, the more it curves the space-time continuum. In remote regions of space far from any matter of significant size, the space-time continuum resembles a flat plain. A piece of matter the size of the earth causes quite a bump in the space-time continuum, and a piece of matter the size of a star causes a relative mountain.