Relativity Lite

To c or Not to c  | 15 all, the downside of using a convenient, though incorrect, model to make predictions is revealed well in the story of the Ptolemaic vs. Copernican models of the Solar System. A SHORT TALE So everything you believe about time has just gotten blown out the window. About now, I would expect you to be wondering if space gets messed with too. It does. Unfortunately, the pictures below show this only with the aid of a series of length compar- isons. In my experience, those who are a bit wigged out by math may put up with a single such relation here and there, but a series of about eight length comparisons that build up to the answer may well lead to frustration. So let me simply give you the result here and you can just skim over the notes below as if they were written in Portuguese, from which a recognizable word or two might pop out if you know a bit of Spanish or French. If you would like to bypass the compari- sons entirely, read the following paragraph and then just jump to the words Skip to here . The length ℓ of the rocket measured by Earth is contracted by the same factor, relative to the proper length L , as the time t measured by Earth is dilated relative to proper time τ . Since we keep on finding this “factor,” we had better give it a name. It is always represented by the Greek letter for g , gamma, which is written γ . That is, t = γ τ , and ℓ = L γ . Here are the details. Consider our spaceship at rest, with an additional pair of mirrors set at the same distance horizontally as the original ones were set vertically (see figure 8). Then if a light wave is emitted from the lower left-hand corner of the set of mirrors, it will travel outward in concentric rings, strike the top and right-hand mirrors simultaneously, and be reflected back to the emission point. The total distance traveled in this proper-time interval is 2 L = c τ . Figure 8. A spaceship with vertical and horizontal light clocks.