Relativity Lite

To c or Not to c  | 9 If there were no relativistic time dilation, most would decay as they travel through the depth of the atmosphere before crashing through your skull. We will show later that on average, the muons are traveling to the Earth’s surface at v = 0.9986 c . At this speed, it would take them 16 microseconds to travel through the atmosphere from the height they are produced, about 3 miles, * or about 8 half-lives. † It turns out that 18 muons are created each second over an area the width of our bodies. After 8 half-lives, 1 2 8 × 18 = 0.07 muons per second should be left to crash through our bodies. Table 1 shows a progression of halv- ing the prior number, with some rounding. Table 1. The half-life progression for 18 muons at the start. Time Muons left 0 µs 18 2 µs 9 4 µs 4 or so 6 µs 2 8 µs 1 10 µs 0.5 12 µs 0.3 14 µs 0.15 16 µs 0.07 But relativity changes all this. To find the time-dilation factor for this speed, we find that we only need to rotate the square slightly to get the lower right-hand corner of the square to just touch the right-hand edge of the rectangle (see figure 6). * A. W. Wolfendale, Cosmic Rays at Ground Level (Institute of Physics, London, 1973), p. 174–75. † t = d c .9986 = 3 9986 186000 miles miles second . / × = 3 1 86. × 10 1000000 seconds = 16 microseconds.