Relativity Lite

18 | Relativity Lite Then the total distance the rocket travels for the emission and reflection is the sum of these, r vt a b = = + = × + = × 3 2 4 4 3 8 5 8   3  , which is again 3 5 of the distance the light travels in the same time, ct = × + = × 5 2 4 4 5 8 5 5 8    . We found from our time-dilation calculation in figure 2 that if one leg is 3 units long when the hypotenuse is 5 units long, then the other leg has to be 4 units long. This is the case in the two expressions above for the leg and hypotenuse, where the unit of measure is 5 8  . So c τ = × 4 5 8  . But from figure 8, we see that c τ = 2 L . Comparing these two expressions shows that ℓ = 8 5 × 2 4 L = 4 5 L . (Skip to here.) Comparing this to the time-dilation expression we obtained from figure 3, t = 5 4 τ shows that the length ℓ of the rocket measured by Earth is contracted, relative to the proper length L , by the same factor γ = 5 4 , as the time t measured by Earth is dilated relative to proper time τ . That is t = γ τ and ℓ = L γ . This compensation between time dilation and length contraction is necessary for real- ity to be whole. Einstein’s second postulate of relativity was that no experiment you could perform would tell you whether it was your frame of reference that was moving or someone else’s (alternatively stated, motion is relative, or there is no preferred frame of reference). Consider our example of the time dilation of muons created in the Earth’s atmosphere. These muons are free to think they are at rest and it is the Earth that is moving toward them. They see the surface of the Earth traveling toward them at v = 0.9986 c just after they are produced. At this speed, it would take the Earth’s 3 miles of atmosphere 16 microseconds to pass by them before the surface crashes into them. As before, this is about 8 proper half-lives. If there were no relativistic length contraction, most would decay before they saw your head approaching as you stood on the surface of the Earth. But because of relativity, the muons see the Earth’s atmosphere contracted to only 1/6 of a mile. * The time it takes the Earth’s surface to hit them is then 0.8 microseconds, or about half of the muon half-life. † Again, we get the result that roughly 1/4 of the muons decay, leaving 13 muons getting hit by your head each second before their lives are over. Without length contraction, the physical reality of your cancer risk would depend on which frame of reference does the calculation. That would violate Einstein’s principle of relativity . * ℓ = 3 19 miles = 0.158 miles. † t =  .9986 c = 0 158 9986 186000 . . / miles miles second × = 0 158 186 . . × 1 1000000 second = 0.854 microseconds.