Curved spaces with two dimensions
According to Newton everything attracts everything else with a force inversely proportional to the square of the distance from it, and objects respond to forces with accelerations proportional to the forces. They are Newton’s laws of universal gravitation and of motion. As you know, they account for the motions of balls, planets, satellites, galaxies, and so forth.  Einstein had a different interpretation of the law of gravitation. According to him, space and time—which must be put together as space-time—are curved near heavy masses. And it is the attempt of things to go along “straight lines” in this curved space-time which makes them move the way they do. Now that is a complex idea—very complex. It is the idea we want to explain in this chapter. Our subject has three parts. One involves the effects of gravitation. Another involves the ideas of space-time which we already studied. The third involves the idea of curved space-time. We will simplify our subject in the beginning by not worrying about gravity and by leaving out the time—discussing just curved space. We will talk later about the other parts, but we will concentrate now on the idea of curved space—what is meant by curved space, and, more specifically, what is meant by curved space in this application of Einstein. Now even that much turns out to be somewhat difficult in three dimensions. So we will first reduce the problem still further and talk about what is meant by the words “curved space” in two dimensions. In order to understand this idea of curved space in two dimensions you really have to appreciate the limited point of view of the character who lives in such a space. Suppose we imagine a bug with no eyes who lives on a plane, as shown in Fig. 42–1. He can move only on the plane, and he has no way of knowing that there is anyway to discover any “outside world.” (He hasn’t got your imagination.) We are, of course, going to argue by analogy. We live in a three-dimensional world, and we don’t have any imagination about going off our three-dimensional world in a new direction; so we have to think the thing out by analogy. It is as though we were bugs living on a plane, and there was a space in another direction. That’s why we will first work with the bug, remembering that he must live on his surface and can’t get out. As another example of a bug living in two dimensions, let’s imagine one who lives on a sphere. We imagine that he can walk around on the surface of the sphere, as in Fig. 42–2 but that he can’t look “up,” or “down,” or “out.” Now we want to consider still a third kind of creature. He is also a bug like the others, and also lives on a plane, as our first bug did, but this time the plane is peculiar. The temperature is different at different places. Also, the bug and any rulers he uses are all made of the same material which expands when it is heated. Whenever he puts a ruler somewhere to measure something the ruler expands immediately to the proper length for the temperature at that place. Wherever he puts any object—himself, a ruler, a triangle, or anything—the thing stretches itself because of the thermal expansion. Everything is longer in the hot places than it is in the cold places, and everything has the same coefficient of expansion. We will call the home of our third bug a “hot plate,” although we will particularly want to think of a special kind of hot plate that is cold in the center and gets hotter as we go out toward the edges (Fig. 42–3). Now we are going to imagine that our bugs begin to study geometry. Although we imagine that they are blind so that they can’t see any “outside” world, they can do a lot with their legs and feelers. They can draw lines, and they can make rulers, and measure off lengths. First, let’s suppose that they start with the simplest idea in geometry. They learn how to make a straight line—defined as the shortest line between two points. Our first bug—see Fig. 42–4—learns to make very good lines. But what happens to the bug on the sphere? He draws his straight line as the shortest distance—for him—between two points, as in Fig. 42–5. It may look like a curve to us, but he has no way of getting off the sphere and finding out that there is “really” a shorter line. He just knows that if he tries any other path in his world it is always longer than his straight line. So we will let him have his straight line as the shortest arc between two points. (It is, of course an arc of a great circle.) Finally, our third bug—the one in Fig. 42–3—will also draw “straight lines” that look like curves to us. For instance, the shortest distance between $A$ and $B$ in Fig. 42–6 would be on a curve like the one shown. Why? Because when his line curves out toward the warmer parts of his hot plate, the rulers get longer (from our omniscient point of view) and it takes fewer “yardsticks” laid end-to-end to get from $A$ to $B$. So for him the line is straight—he has no way of knowing that there could be someone out in a strange three-dimensional world who would call a different line “straight.” We think you get the idea now that all the rest of the analysis will always be from the point of view of the creatures on the particular surfaces and not from our point of view. With that in mind let’s see what the rest of their geometries looks like. Let’s assume that the bugs have all learned how to make two lines intersect at right angles. (You can figure out how they could do it.) Then our first bug (the one on the normal plane) finds an interesting fact. If he starts at the point $A$ and makes a line $100$ inches long, then makes a right angle and marks off another $100$ inches, then makes another right angle and goes another $100$ inches, then makes a third right angle and a fourth line $100$ inches long, he ends up right at the starting point as shown in Fig. 42–7(a). It is a property of his world—one of the facts of his “geometry.” Then he discovers another interesting thing. If he makes a triangle—a figure with three straight lines—the sum of the angles is equal to $180^\circ$, that is, to the sum of two right angles. See Fig. 42–7(b). Then he invents the circle. What’s a circle? A circle is made this way: You rush off on straight lines in many many directions from a single point, and lay out a lot of dots that are all the same distance from that point. See Fig. 42–7(c). (We have to be careful how we define these things because we’ve got to be able to make the analogs for the other fellows.) Of course, its equivalent to the curve you can make by swinging a ruler around a point. Anyway, our bug learns how to make circles. Then one day he thinks of measuring the distance around a circle. He measures several circles and finds a neat relationship: The distance around is always the same number times the radius $r$ (which is, of course, the distance from the center out to the curve). The circumference and the radius always have the same ratio—approximately $6.283$—independent of the size of the circle. Now let’s see what our other bugs have been finding out about their geometries. First, what happens to the bug on the sphere when he tries to make a “square”? If he follows the prescription we gave above, he would probably think that the result was hardly worth the trouble. He gets a figure like the one shown in Fig. 42–8. His endpoint $B$ isn’t on top of the starting point $A$. It doesn’t work out to a closed figure at all. Get a sphere and try it. A similar thing would happen to our friend on the hot plate. If he lays out four straight lines of equal length—as measured with his expanding rulers—joined by right angles he gets a picture like the one in Fig. 42–9. Now suppose that our bugs had each had their own Euclid who had told them what geometry “should” be like, and that they had checked him out roughly by making crude measurements on a small scale. Then as they tried to make accurate squares on a larger scale they would discover that something was wrong. The point is, that just by geometrical measurements they would discover that something was the matter with their space. We define a curved space to be a space in which the geometry is not what we expect for a plane. The geometry of the bugs on the sphere or on the hot plate is the geometry of a curved space. The rules of Euclidean geometry fail. And it isn’t necessary to be able to lift yourself out of the plane in order to find out that the world that you live in is curved. It isn’t necessary to circumnavigate the globe in order to find out that it is a ball. You can find out that you live on a ball by laying out a square. If the square is very small you will need a lot of accuracy, but if the square is large the measurement can be done more crudely. Let’s take the case of a triangle on a plane. The sum of the angles is $180$ degrees. Our friend on the sphere can find triangles that are very peculiar. He can, for example, find triangles which have three right angles. Yes indeed! One is shown in Fig. 42–10. Suppose our bug starts at the north pole and makes a straight line all the way down to the equator. Then he makes a right angle and another perfect straight line the same length. Then he does it again. For the very special length he has chosen he gets right back to his starting point, and also meets the first line with a right angle. So there is no doubt that for him this triangle has three right angles, or $270$ degrees in the sum. It turns out that for him the sum of the angles of the triangle is always greater than $180$ degrees. In fact, the excess (for the special case shown, the extra $90$ degrees) is proportional to how much area the triangle has. If a triangle on a sphere is very small, its angles add up to very nearly $180$ degrees, only a little bit over. As the triangle gets bigger the discrepancy goes up. The bugs on the hot plate would discover similar difficulties with their triangles. Let’s look next at what our other bugs find out about circles. They make circles and measure their circumferences. For example, the bug on the sphere might make a circle like the one shown in Fig. 42–11. And he would discover that the circumference is less than $2\pi$ times the radius. (You can see that because from the wisdom of our three-dimensional view it is obvious that what he calls the “radius” is a curve which is longer than the true radius of the circle.) Suppose that the bug on the sphere had read Euclid, and decided to predict a radius by dividing the circumference $C$ by $2\pi$, taking \begin{equation} \label{Eq:II:42:1} r_{\text{pred}}=\frac{C}{2\pi}. \end{equation}  Then he would find that the measured radius was larger than the predicted radius. Pursuing the subject, he might define the difference to be the “excess radius,” and write \begin{equation} \label{Eq:II:42:2} r_{\text{meas}}-r_{\text{pred}}=r_{\text{excess}}, \end{equation}  and study how the excess radius effect depended on the size of the circle.  Our bug on the hot plate would discover a similar phenomenon. Suppose he was to draw a circle centered at the cold spot on the plate as in Fig. 42–12. If we were to watch him as he makes the circle we would notice that his rulers are short near the center and get longer as they are moved outward—although the bug doesn’t know it, of course. When he measures the circumference the ruler is long all the time, so he, too, finds out that the measured radius is longer than the predicted radius, $C/2\pi$. The hot-plate bug also finds an “excess radius effect.” And again the size of the effect depends on the radius of the circle.  We will define a “curved space” as one in which these types of geometrical errors occur: The sum of the angles of a triangle is different from $180$ degrees; the circumference of a circle divided by $2\pi$ is not equal to the radius; the rule for making a square doesn’t give a closed figure. You can think of others. We have given two different examples of curved space: the sphere and the hot plate. But it is interesting that if we choose the right temperature variation as a function of distance on the hot plate, the two geometries will be exactly the same. It is rather amusing. We can make the bug on the hot plate get exactly the same answers as the bug on the ball. For those who like geometry and geometrical problems we’ll tell you how it can be done. If you assume that the length of the rulers (as determined by the temperature) goes in proportion to one plus some constant times the square of the distance away from the origin, then you will find that the geometry of that hot plate is exactly the same in all details1 as the geometry of the sphere. There are, of course, other kinds of geometry. We could ask about the geometry of a bug who lived on a pear, namely something which has a sharper curvature in one place and a weaker curvature in the other place, so that the excess in angles in triangles is more severe when he makes little triangles in one part of his world than when he makes them in another part. In other words, the curvature of a space can vary from place to place. That’s just a generalization of the idea. It can also be imitated by a suitable distribution of temperature on a hot plate. We may also point out that the results could come out with the opposite kind of discrepancies. You could find out, for example, that all triangles when they are made too large have the sum of their angles less than $180$ degrees. That may sound impossible, but it isn’t at all. First of all, we could have a hot plate with the temperature decreasing with the distance from the center. Then all the effects would be reversed. But we can also do it purely geometrically by looking at the two-dimensional geometry of the surface of a saddle. Imagine a saddle-shaped surface like the one sketched in Fig. 42–13. Now draw a “circle” on the surface, defined as the locus of all points the same distance from a center. This circle is a curve that oscillates up and down with a scallop effect. So its circumference is larger than you would expect from calculating $2\pi r_{\text{meas}}$. So $C/2\pi$ is now greater than $r_{\text{meas}}$. The “excess radius” would be negative.  Spheres and pears and such are all surfaces of positive curvatures; and the others are called surfaces of negative curvature. In general, a two-dimensional world will have a curvature which varies from place to place and may be positive in some places and negative in other places. In general, we mean by a curved space simply one in which the rules of Euclidean geometry break down with one sign of discrepancy or the other. The amount of curvature—defined, say, by the excess radius—may vary from place to place.  We might point out that, from our definition of curvature, a cylinder is, surprisingly enough, not curved. If a bug lived on a cylinder, as shown in Fig. 42–14, he would find out that triangles, squares, and circles would all have the same behavior they have on a plane. This is easy to see, by just thinking about how all the figures will look if the cylinder is unrolled onto a plane. Then all the geometrical figures can be made to correspond exactly to the way they are in a plane. So there is no way for a bug living on a cylinder (assuming that he doesn’t go all the way around, but just makes local measurements) to discover that his space is curved. In our technical sense, then, we consider that his space is not curved. What we want to talk about is more precisely called intrinsic curvature; that is, a curvature which can be found by measurements only in a local region. (A cylinder has no intrinsic curvature.) This was the sense intended by Einstein when he said that our space is curved. But we as yet only have defined a curved space in two dimensions; we must go onward to see what the idea might mean in three dimensions. 