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By Root 2080 0

the ride with an arrow that indicates the momentary direction of motion at each point. As Slim begins to measure the circumference, we immediately see from our bird's-eye perspective that he is going to get a different answer than we did. As he lays the ruler out along the circumference, we notice that the ruler's length is shortened. This is nothing but the Lorentz contraction discussed in Chapter 2, in which the length of an object appears shortened along the direction of its motion. A shorter ruler means that he will have to lay it out—head to tail—more times to traverse the whole circumference. Since he still considers the ruler to be one foot long (since there is no relative motion between Slim and his ruler, he perceives it as having its usual length of one foot), this means that Slim will measure a longer circumference than did we. (If this seems paradoxical, you might find endnote 5 helpful.)

What about the radius? Well, Jim also uses the head-to-tail method to find the length of a radial strut, and from our bird's-eye view we see that he is going to find the same answer as we did. The reason is that the ruler is not pointing along the instantaneous direction of the motion of the ride (as it is when measuring the circumference). Instead, it is pointed at a ninety-degree angle to the motion, and therefore it is not contracted along its length. Jim will therefore find exactly the same radial length as we did.

Figure 3.1 Slim's ruler is contracted, since it lies along the direction of the ride's motion. But Jim's ruler lies along a radial strut, perpendicular to the direction of the ride's motion, and therefore its length is not contracted.

But now, when Slim and Jim calculate the ratio of the circumference of the ride to its radius they will get a number that is larger than our answer of two times pi, since the circumference is longer but the radius is the same. This is weird. How in the world can something in the shape of a circle violate the ancient Greek realization that for any circle this ratio is exactly two times pi?

Here is Einstein's explanation. The ancient Greek result holds true for circles drawn on a flat surface. But just as the warped or curved mirrors in an amusement park fun-house distort the normal spatial relationships of your reflection, if a circle is drawn on a warped or curved surface, its usual spatial relationships will also be distorted: the ratio of its circumference to its radius will generally not be two times pi.

For instance, Figure 3.2 compares three circles whose radii are identical. Notice, however, that their circumferences are not the same. The circumference of the circle in (b), drawn on the curved surface of a sphere, is less than the circumference of the circle drawn on the flat surface in (a), even though they have the same radius. The curved nature of the sphere's surface causes the radial lines of the circle to converge toward each other slightly, resulting in a small decrease in the circle's circumference. The circumference of the circle in (c), again drawn on a curved surface—a saddle shape—is greater than that drawn on a flat surface; the curved nature of the saddle's surface causes the radial lines of the circle to splay outward from each other slightly, resulting in a small increase in the circle's circumference. These observations imply that the ratio of the circumference to the radius of the circle in (b) will be less than two times pi, while the same ratio in (c) will be greater than two times pi. But this deviation from two times pi, especially the larger value found in (c), is just what we found for the spinning Tornado ride. This led Einstein to propose an idea—the curving of space—as an explanation for the violation of "ordinary," Euclidean geometry. The flat geometry of the Greeks, taught to schoolchildren for thousands of years, simply does not apply to someone on the spinning ride. Rather, its curved space generalization as schematically drawn in part (c) of Figure 3.2 takes its place.5

Figure 3.2 A circle drawn on a sphere (b) has a shorter circumference