## Doppler Shift

Let's take another trip with the twins, but this time John will travel 12 hours away and 12 hours back, as measured by his clock. Every hour he will send a radio signal to Hunter telling him the hour. A radio signal is just another form of electromagnetic radiation; therefore, it also travels at the speed of light. What do we get as John travels away from Hunter? When John's clock reads "1 hour" he sends the first signal. Because he is moving away from Hunter at 60% of the speed of light, the relativistic Doppler Effect causes Hunter to observe John's transmission to be ½ the source value. From our discussion above, ½ the frequency means the time it takes is twice as long, therefore, Hunter receives the John's "1 hour" signal when his clock reads "2 hours". When John sends his "2 hour" signal, Hunter receives it at hour 4 for him. So you can see the relationship developing. For every 1-hour signal by John's watch, the elapsed time for Hunter is 2 hours. When John's clock reads "12 hours" he has sent 12 signals. Hunter, on the other hand, has received 12 signals, but they were all 2 hours apart … thus 24 hours have passed for Hunter. Now John turns around and comes back sending signals every hour in the same manner as before. Since he is approaching Hunter, the Doppler shift now causes Hunter to observe the frequency to be twice the source value. Twice the frequency is the same as ½ the time, so Hunter receives John's "1 hour" signals at 30min intervals. When the 12-hour return trip is over, John has sent 12 signals. Hunter has received 12 signals, but they were separated by 30 minutes, thus 6 hours have passed for Hunter. If we now total up the elapsed time for both twins, we see that 24 hours (12 + 12) have elapsed for John, but 30 hours (24 + 6) have elapsed for Hunter. Thus, Hunter is now older than his identical twin, John. If John had traveled farther and faster, the time dilation would have been even greater. Look at the twins again, but this time let John travel 84 hours out and 84 hours back (by his clock) at 80% the speed of light. The total trip for John will be 168 hours, and the total time elapsed for Hunter will be 280 hours; John was gone for 1 week by his clock, but Hunter waited for 1 week 4 days and 16 hours by his clock. Remember that Hunter will receive John's outgoing signals at half the frequency which means twice the time. Therefore, Hunter receives John's 84 hourly signals every 3 hours for a total of 252 hours (3 is the Relativistic Doppler shift for 80% the speed of light). Likewise, Hunter receives John's return trip 84 hourly signals every 20 minutes for a total of 28 hours (20 minutes is the 1/3 Relativistic Doppler shift for the return). Now you know the total round trip from Hunter's perspective, 252 + 28 = 280 hours or 1 week 4 days and 16 hours. John, on the other hand, traveled 84 hours out and 84 hours back for a total of 168 hours or 1 week.

We'll look closer at the twin paradox in the next section.