• Choose a category
• All
• Classic-Fiction

By Root 2126 0

use the Pythagorean theorem to calculate that the length of each of the diagonal paths in Figure 2.3 is (vt/2)2 + h2, where h is the distance between the two mirrors of a light clock (taken to be six inches in the text). The two diagonal paths, taken together, therefore have length 2(vt/2)2 + h2 . Since the speed of light has a constant value, conventionally called c, it takes light 2(vt/2)2 + h2/c seconds to complete the double diagonal journey. And so, we have the equality t = 2(vt/2)2 + h2/c, which can be solved for t, yielding t = 2h/c2-v2. To avoid confusion, let's write this as tmoving = 2h/c2-v2, where the subscript indicates that this is the time we measure for one tick to occur on the moving clock. On the other hand, the time for one tick on our stationary clock is tstationary = 2h/c and as a little algebra reveals, tmoving = tstationary/1-v2/c2, directly showing that one tick on the moving clock takes longer than one tick on the stationary clock. This means that between chosen events, fewer total ticks will take place on the moving clock than on the stationary, ensuring that less time has elapsed for the observer in motion.

4. In case you would be more convinced by an experiment carried out in a less esoteric setting than a particle accelerator, consider the following. During October 1971, J. C. Hafele, then of Washington University in St. Louis, and Richard Keating of the United States Naval Observatory flew cesium-beam atomic clocks on commercial airliners for some 40 hours. After taking into account a number of subtle features having to do with gravitational effects (to be discussed in the next chapter), special relativity claims that the total elapsed time on the moving atomic clocks should be less than the elapsed time on stationary earthbound counterparts by a few hundred billionths of a second. This is just what Hafele and Keating found: Time really does slow down for a clock in motion.

5. Although Figure 2.4 correctly illustrates the shrinking of an object along its direction of motion, the image does not illustrate what we would actually see if an object were somehow to blaze by at nearly light speed (assuming our eyesight or photographic equipment were sharp enough to see anything at all!). To see something, our eyes—or our camera—must receive light that has reflected off the object's surface. But since the reflected light travels to us from various locations on the object, the light we see at any moment traveled to us along paths of different lengths. This results in a kind of relativistic visual illusion in which the object will appear both foreshortened and rotated.

6. For the mathematically inclined reader, we note that from the spacetime position 4-vector x = (ct, x1, x2, x3) = (ct, ) we can produce the velocity 4-vector u = dx/dt, where t is the proper time defined by dt2 = dt2-c-2(dx21 + dx22 + dx23). Then, the "speed through spacetime" is the magnitude of the 4-vector u, ((c2dt2-dx2)/(dt2-c-2dx2)), which is identically the speed of light, c. Now, we can rearrange the equation c2(dt/dt)2-(d/d)2 = c2, to be c2(d/dt)2 + (d/dt)2 = c2. This shows that an increase in an object's speed through space, (d/dt)2 must be accompanied by a decrease in dt/dt, the latter being the object's speed through time (the rate at which time elapses on its own clock, dt, as compared with that on our stationary clock, dt).

Chapter 3

1. Isaac Newton, Sir Isaac Newton's Mathematical Principle of Natural Philosophy and His System of the World, trans. A. Motte and Florian Cajori (Berkeley: University of California Press, 1962), Vol. I, p. 634.

2. A bit more precisely, Einstein realized that the equivalence principle holds so long as your observations are confined to a small enough region of space—that is, so long as your "compartment" is small enough. The reason is the following. Gravitational fields can vary in strength (and in direction) from place to place. But we are imagining that your whole compartment accelerates as a single unit and therefore your acceleration simulates a single, uniform gravitational force field.