Relativity Lite

Mixmaster Universe | 23 where our γ 1 loses accuracy. * This becomes apparent as we calculate γ 1 and γ for various velocities: Table 2. Approximate γ 1 and exact values (truncated to five decimal places) for the time-­ dilation factor γ τ = t . v c v c 2 2 1 2 2 2 v c γ 1 2 2 1 1 2 ≅ + v c γ ≡ − 1 1 2 2 v c 0.1 0.01 0.005 1.005 1.00503 0.2 0.04 0.02 1.02 1.02062 0.3 0.09 0.045 1.045 1.04828 0.4 0.16 0.08 1.08 1.09109 0.5 0.25 0.125 1.125 1.15470 0.6 0.36 0.18 1.18 1.25 It seems that γ 1 works pretty well for velocities below half the speed of light. Now would be a good time to go put on some obnoxiously loud rock music to release any tension you may have over going through this derivation. May I suggest Led Zeppelin’s The Ocean (Live at Madison Square Garden, 1973)? When you have done that, come on back. INTO THE MIX Now look closely at figure 8 of chapter 1. The light is reflected from the right-hand mirror at the same time that the light is reflected from the top mirror. These events are simultaneous in the rest frame of the rocket. But in figures 9b and 9c in chapter 1, the light is reflected off the top mirror before it is reflected off the right-hand mirror, not simultaneously! (But the return sequence to the lower left-hand corner is simultaneous for both frames of reference.) The sequence of events in different frames of reference moving relative to each other is not necessarily the same when you live in Einstein’s universe. The time interval between two events I measure on my clock depends not only on the time interval you measure on your clock (time dilation) but also on where you are standing relative to the other event in your frame of reference. In other words, my time depends * Did you ever wonder how your calculator actually finds this beast? It uses a series of approximations: J W { § © ¨ · ¹ ¸ § © ¨ · ¹ ¸ § © ¨ · ¹ ¸ § © ¨ · ¹ ¸ t v c v c v 1 1 1 2 1 2 3 2 2 2 2 1 2 2 2 2 c 2 2 § © ¨ · ¹ ¸  , and so on, which confirms that our “‘slightly more’ than [ f ]” above will depend on the fourth power of the velocity and several of higher powers to get 10-digit accuracy.