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## Section 3.6: Exploring the Twin Paradox Using the Doppler Effect

Relative speed v/c = (0.4 < v/c < 0.9)

One twin, Pink, remains on Earth while her twin brother, Green, leaves in a rocket ship to a planet. When he reaches the planet, he turns around at that planet and returns to Earth 20 years later (as measured by Pink, on Earth).  Even though Pink is 20 years older, her brother is younger. To measure this, each twin sends out a flash of light on his/her birthday. In his or her reference frame, then, these flashes occur with a frequency of one. What does it look like from the other reference frame?  First look at it from Pink's view on Earth. The time clock in the left corner shows the years as measured on Earth (20 years total), while the counter counts the flashes that Pink receives from Green. Restart.

1. What is the frequency of the pulses that she receives from Green on his outbound trip?
2. What is the frequency of the pulses that she receives from Green on his inbound trip?

Verify that your answers are consistent with the equation for the relativistic Doppler shift, Eq. (3.9), by using the substitution that 1/λ = ν/c.

Now, from Green's point of view, the situation is a bit different. In order to make this trip, he has to switch reference frames so in the animation, we show the view of the Pink (and Earth) from the two frames: White and Black that Green jumps between. In the first reference frame, Pink (and Earth) are moving away from Green, while in the second reference frame, Pink (and Earth) is moving toward him. The pulse counter only counts pulses in the reference frame that Green is currently in.

1. Verify that the total number of pulses Green receives is equal to Pink's change in age when Green returns to Earth. Notice that Green's clock (in the upper left hand corner) shows him to be younger.
2. What is the frequency of the pulses that Green receives while in the White reference frame?  While in the Black reference frame?
3. Show that these equations, too, are consistent with the Doppler shift equation, Eq. (3.9).

The Doppler shift only depends on the relative speeds between the two objects. Both Pink and Green measure the same frequency of light from the other person, but Pink and Green receive a different number of signals (Green receives half as many on his trip out as on his return trip) which accounts for the age difference.