A very interesting example of the slowing of time with motion is furnished by (muons), which are particles that disintegrate spontaneously after an average lifetime of 2. X 10-6 sec. They come to the earth in cosmic rays, and can also be produced artificially in the laboratory. Some of the disintegrate in midair, but the remainder disintegrate only after they encounte a piece of material and stop. It is clear that in its short lifetime a muon cannot travel, even at the speed of light, much more than 600 meters. But although the muons are created at the top of the atmosphere, some 10 kilometers up, yet they are actually found in a laboratory down here, in cosmic rays. How can that be? he answer is that different muons move at various speeds, some of which are very close to the speed of light. While from their own point of view they live only about 2 usec, from our point of view they live considerably longer-enough longer that they may reach the earth, The factor by which the time is increased has already been given as 1/vI-u2 he average life has been measur for muons of different velocities, and the values agree closely with the formula We do not know why the meson disintegrates or what its machinery is, but we do know its behavior satisfies the principle of relativity. That is the utility of the principle of relativity--it permits us to make predictions, even about things that otherwise we do not know much about. For example, before we have any idea at all about what makes the meson disintegrate, we can still predict that when it is moving at nine-tenths of the speed of light, the apparent length of time that t lasts is(2.2×10-6 92/102 sec; and our prediction works-that is the good thing about it. 15-5 The Lorentz contraction Now let us return to the Lorentz transformation (15.3)and try to get a better understanding of the relationship between the (x, y, z, t and the (x', y, z, r,) coordinate systems, which we shall call the S and S systems, or Joe and Moe systems, respectively. We have already noted that the first equation is based on the Lorentz suggestion of contraction along the x-direction; how can we prove that a contraction takes place? In the Michelson-Morley experiment, we now appre- ciate that the transverse arm BC cannot change length, by the principle of relativity order for the experiment to give a null result, the longitudinal arm BE must shorter, by the squ VI-u2/c2. What does this in terms of measurements made by Joe and Moe? Suppose that Moe, moving with the S system in the x-direction, is measuring the x-coordinate of some point with a meter stick. He lays the stick down x'times, so he thinks the distance is x meters. From the viewpoint of Joe in the S system, however, Moe is using a Then if the S system has travelled a distance ut away from the S system, the s observer would say that the same point, measured in his coordinates, is at a distance=x'vI-u2/ c2+ut, or which is the first equation of the lorentz transformation 15-6 Simultaneity =presse an analogous way, because of the difference in time scales, the denominator ion of the lore most interesting term in that equation is the ux/e in the numerator, because at is quite new and unexpected. Now what does that mean? If we look at the situation carefully we see that events that occur at two separated places at the time, as seen by Moe in S, do not happen at the same time as viewed by Joe in S. If one event occurs at point x, at time fo and the other event at x2 and to(the same