by Gary Zukav
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 d
istorting 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.
Although we cannot perceive the four-dimensional space-time continuum directly, we can deduce from what we already know of the special theory of relativity that our universe is not Euclidean. Here is another of Einstein’s thought experiments.
Imagine two concentric circles, one with a small radius and one with a very large radius. Both of them revolve around a common center as shown.
Imagine also that we, the observers, are watching these revolving circles from an inertial co-ordinate system. Being in an inertial co-ordinate system simply means that our frame of reference is at rest relative to everything, including the revolving circles. Drawn over the revolving circles are two identical concentric circles which are in our co-ordinate system. They are not revolving. They are the same size as the revolving circles and have the same common center, but they remain motionless. While we and our nonrevolving circles are motionless, we are in communication with an observer who is on the revolving circles. He actually is going around with them.
According to Euclidean geometry, the ratio of the radius to the circumference of all circles is the same. If we measure the radius and the circumference of the small circle, for example, the ratio of these two measurements will be the same as the ratio of the radius to the circumference of the large circle. The object of this thought experiment is to determine whether this is true or not for both the observers on the stationary circles (us) and the observer on the revolving circles. If the geometry of Euclid is valid throughout the physical universe, as it should be, we should discover that the ratio between the radius and the circumference of all the circles involved is identical.
Both we and the observer on the revolving circles will use the same ruler to do our measuring. “The same ruler” means that either we actually hand him the same ruler that we have used, or that we use rulers that have the same length when at rest in the same co-ordinate system.
We go first. Using our ruler, we measure the radius of our small circle, and then we measure the circumference of our small circle. Then we note the ratio between them. The next step is to measure the radius of our large circle and then the circumference of our large circle. Then we note the ratio between them. Yes, it is the same ratio that we found between the radius and the circumference of our small circle. We have proved that Euclidean geometry is valid in our co-ordinate system, which is an inertial co-ordinate system.
Now we hand the ruler to the observer on the revolving circles as he passes by us. Using this ruler he first measures the radius of his small circle and finds that it is the same as ours, since our circles are drawn directly over his circles. Next he measures the circumference of his small circle. Remember that motion causes rulers to contract in the direction that they are moving. However, since the radius of the small circle is so short, the velocity of the ruler when it is placed on the circumference of the small circle is not fast enough to make the effect of relativistic contraction noticeable. Therefore, the observer on the revolving circles measures the circumference of his small circle and finds it to be the same as the circumference of our small circle. Naturally, the ratio between them also is the same. So far so good. The ratios between the radius and the circumference of three circles have been determined (our small circle, our large circle, and his small circle) and they are all identical. This is exactly what should happen according to high-school geometry books across the country. Only one more circle to go.
The observer on the revolving circles measures the radius of his large circle and finds it to be the same length as the radius of our large circle. Now he comes to the last measurement, the circumference of his large circle. However, as soon as he puts his ruler into position to make a measurement on the circumference of the large revolving circle, his ruler contracts! Because the radius of his large circle is much larger than the radius of his small circle, the velocity of the circumference of the large revolving circle is considerably faster than the velocity of the circumference of the small revolving circle.
Since the ruler must be aligned in the direction that the circumference is moving, it becomes shorter. When the revolving observer uses this ruler to measure the circumference of the large revolving circle, he finds that it is larger than the circumference of our large circle. This is because his ruler is shorter. (Contraction also affected his ruler when he measured the radius of his large circle, but since it then was placed perpendicular to the direction of motion, it became skinnier, nor shorter).
This means that the ratio of the radius to the circumference of the small revolving circle is not the same as the ratio of the radius to the circumference of the large revolving circle. According to Euclidean geometry, this is not possible, but there it is.
If we want to be old-fashioned about it (before-Einstein) we can say that this situation is nothing unusual. By definition, the laws of mechanics and the geometry of Euclid are valid only in inertial systems (that is what makes them inertial systems). We simply don’t consider co-ordinate systems which are not inertial. (This was really the position of physicists before Albert Einstein.) This is exactly what seemed wrong to Einstein. His idea was to create a physics valid for all co-ordinate systems, since the universe abounds with the non-inertial as well as the inertial kind.
If we are to create such a universally valid physics, a general physics, then we must treat both the observers in the stationary (inertial) system and the observer on the revolving circles (a non-inertial system) with equal seriousness. The person on the revolving circles has as much right to relate the physical world to his frame of reference as we have to relate it to ours. True, the laws of mechanics as well as the geometry of Euclid are not valid in his frame of reference, but every deviation from them can be explained in terms of a gravitational field which affects his frame of reference.
This is what Einstein’s theory allows us to do. It allows us to express the laws of physics in such a way that they are independent of specific space-time co-ordinates. Space and time co-ordinates (measurements) vary from one frame of reference to another, depending upon the state of motion of the frame of reference. The general theory of relativity allows us to universalize the laws of physics and to apply them to all frames of reference.
“Wait a minute,” we say, “how can anyone measure distance or navigate in a co-ordinate system like the one on the revolving circles? The length of a ruler varies from place to place in such a system. The farther we go from the center, the faster the velocity of the ruler, and the more it contracts. This doesn’t happen in an inertial co-ordinate system, which, in effect, is a system that is at rest. Because there is no change of velocity throughout an inertial co-ordinate system, rulers do not change length.
“This allows us to organize inertial systems like a city, block by block. Since rulers do not change length in inertial systems, all the blocks that are laid out with the same ruler will be the same
length. No matter where we travel, we know that ten blocks is twice the distance of five blocks.
“In a non-inertial system the velocity of the system varies from place to place. This means that the length of a ruler varies from place to place. If we used the same ruler to lay out all the city blocks in a non-inertial co-ordinate system, some of them would be larger than others depending upon where they were located.”
“What is wrong with that,” asks Jim de Wit, “as long as we still can determine our position in the co-ordinate system? Imagine a sheet of india rubber on which we have drawn a grid so that it looks like a piece of graph paper (first drawing, next page). This is a co-ordinate system. Assuming that we are at the lower left corner (we can start anywhere) let us say that a party Saturday night is being held at the intersection marked ‘Party.’ To get there we have to go two squares to the right and two squares up.
“Now suppose that we stretch the sheet of rubber so that it looks like the second drawing.
“The same directions (two squares right and two squares up) still bring us to the party. The only difference is that unless we are familiar with this part of the co-ordinate system, we cannot calculate the distance that we have to travel as easily as we could if all of the squares were the same size.”
According to the general theory of relativity, gravity, which is the equivalent of acceleration, is what distorts the space-time continuum in a manner analogous to our stretching the sheet of rubber. Where the effects of gravity can be neglected, the space-time continuum is like the sheet of rubber before we stretched it. All of the lines are straight lines and all of the clocks are synchronized. In other words, the undistorted sheet of rubber is analogous to the space-time continuum of an inertial co-ordinate system and the special theory of relativity applies.