The Dancing Wu Li Masters (24 page)

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 attraction 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 applica
ble 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.

As an object travels through the space-time continuum, it takes the easiest path between two points. The easiest path between two points in the space-time continuum is called a geodesic (geo dee’ sic). A geodesic is not always a straight line owing to the nature of the terrain in which the object finds itself.

Suppose that we are in a balloon looking down on a mountain that has a bright beacon on the top of it. The mountain rises gradually out of the plain, and becomes more and more steep as its elevation increases, until, close to the top, it rises almost straight up. There are many villages surrounding the mountain, and there are footpaths connecting all of the villages with each other. As the paths approach the mountain, all of them begin to curve in one way or another, to avoid going unnecessarily far up the mountain.

Suppose that it is nighttime and that, looking down, we can see neither the mountain nor the footpaths. All that we can see is the beacon and the torches of the travelers below. As we watch, we notice that the torches deflect from a straight path when they approach the vicinity of the beacon. Some of them curve gently around the beacon in a graceful arc some distance away from it. Others approach the beacon more directly, but the closer they get to it, the more sharply they turn away from it.

From this, we probably would deduce that some force emanating from the beacon was repelling all attempts to approach it. For example, we might speculate that the beacon is extremely hot and painful to approach.

With the coming of daylight, however, we can see that the beacon is situated on the top of a large mountain and that it has nothing whatever to do with the movement of the torch-bearers. They simply followed the easiest paths available to them over the terrain between their points of origin and destination.

This masterful analogy was created by Bertrand Russell. In this case, the mountain is the sun, the travelers are the planets, asteroids, comets (and debris from the space program), the footpaths are their orbits, and the coming of daylight is the coming of Einstein’s general theory of relativity.

The point is that the objects in the solar system move as they do not because of some mysterious force (gravity) exerted upon them at a distance by the sun, but because of the nature of the neighborhood through which they are traveling.

Arthur Eddington illustrated this same situation in another way. Suppose, he suggested, that we are in a boat looking down into clear water. We can see the sand on the bottom and the fishes swimming beneath us. As we watch, we notice that the fish seem to be repelled from a certain point. As they approach it, they swim either to the right or to the left of it, but never over it. From this we probably would deduce that there is a repellant force at that point which keeps the fish away.

However, if we should go into the water to get a closer look, we
would see that an enormous sunfish has buried himself in the sand at that point, creating a sizable mound. As fish swimming along the bottom approach the mound, they follow the easiest path available to them, which is around it rather than over it. There is no “force” causing the fish to avoid that particular spot. If all had been known from the first, that spot was merely the top of a large mound which the fish found easier to swim around than to swim over.

The movement of the fish was determined not by a force emanating from the mysterious spot, but by the nature of the neighborhood through which they were passing. (Eddington’s sunfish was called “Albert”) (really). If we could see the geography (the geometry) of the space-time continuum, we would see that, similarly, it, and not “forces between objects,” is the reason that planets move in the ways that they do.

It is not possible for us actually to see the geometry of the space-time continuum because it is four-dimensional and our sensory experience is limited to three dimensions. For that reason, it is not even possible to picture it.

For example, suppose that there existed a world of two-dimensional people. Such a world would look like a picture on a television or a movie screen. The people and the objects in a two-dimensional world would have height and width, but not depth. If these two-dimensional figures had a life and an intelligence of their own, their world would appear quite different to them than our world appears to us, for they could not experience the third dimension.

A straight line drawn between two of these people would appear to them as a wall. They would be able to walk around either end of it, but they would not be able to “step over” it, because their physical existence is limited to two dimensions. They cannot step off the screen into the third dimension. They would know what a circle is, but there is no way that they could know what a sphere is. In fact, a sphere would appear to them as a circle.

If they like to explore, they soon would discover that their world
is flat and infinite. If two of them went off in opposite directions, they would never meet.

They also could create a simple geometry. Sooner or later they would generalize their experiences into abstractions to help them do and build the things that they want to do and build in their physical world. For example, they would discover that whenever three straight metal bars form a triangle, the angles of the triangle always total 180 degrees. Sooner or later, the more perceptive among them would substitute mental idealizations (straight lines) for the metal bars. That would allow them to arrive at the abstract conclusion that a triangle, which by definition is formed by three straight lines, always contains 180 degrees. To learn more about triangles, they no longer would need actually to construct them.

The geometry that such a two-dimensional people would create is the same geometry that we studied in school. It is called Euclidean geometry, in honor of the Greek, Euclid, whose thoughts on the subject were so thorough that no one expanded on them for nearly two thousand years. (The content of most high-school geometry books is about two millennia old.)

Now let us suppose that someone, unbeknownst to them, transported these two-dimensional people from their flat world onto the surface of an enormously large sphere. This means that instead of being perfectly flat, their physical world now would be somewhat curved. At first, no one would notice the difference. However, if their technology improved enough to allow them to begin to travel and to communicate over great distances, these people eventually would make a remarkable discovery. They would discover that their geometry could not be verified in their physical world.

For example, they would discover that if they surveyed a large enough triangle and measured the angles that form it, it would have more than 180 degrees! This is a simple phenomenon for
us
to picture. Imagine a triangle drawn on a globe. The apex (top) of the triangle is at the north pole. The two lines intersecting there form a right angle. The equator is the base of the triangle. Look what happens. Both sides of the triangle, upon intersecting the equator, also form
right angles. According to Euclidean geometry, a triangle contains only two right angles (180 degrees), yet this triangle contains
three
right angles (270 degrees).

Remember that in our example, the two-dimensional people actually have surveyed a triangle on what they presumed was their flat world, measured the angles, and come up with 270 degrees. What a confusion. When the dust settles they would realize that there are only two possible explanations.

The first possible explanation is that the straight lines used to construct the triangle (like light beams) were not actually straight, although they seemed to be straight. This could account for the excessive number of degrees in the triangle. However, if this is the explanation that they choose to adopt, then they must create a “force” responsible for somehow distorting the straight lines (like “gravity”). The second possible explanation is that their abstract geometry does not apply to their real world. This is another way of saying that, impossible as it sounds, their universe is not Euclidean.

The idea that their physical reality is not Euclidean probably would sound so fantastic to them (especially if they had had no reason to question the reality of Euclidean geometry for two thousand years) that they probably would choose to look for forces responsible for distorting their straight lines.
^*

The problem is that, having chosen this course, they would be obligated to create a responsible force every time that their physical world failed to validate Euclidean geometry. Eventually the structure of these necessary forces would become so complex that it would be much simpler to forget them altogether and admit that their physical world does not follow the logically irrefutable rules of Euclidean geometry.

Our situation is parallel to that of the two-dimensional people
who cannot perceive, but who can deduce that they are living in a three-dimensional world. We are a three-dimensional people who cannot perceive, but who can deduce that we are living in a four-dimensional universe.

For two thousand years we have assumed that the entire physical universe, like the geometry that the ancient Greeks created from their experience with this part of it, was Euclidean. That the geometry of Euclid is universally valid means that it can be verified anywhere in the physical world. That assumption was wrong. Einstein was the first person to see that the universe is not bound by the rules of Euclidean geometry, even though our minds tenaciously cling to the idea that it is.