### Modern Physics Problem 10.1.3: Twin Paradox

#### Description

Students can be easily confused by the effect of transit time in the twin paradox. Unlike Minkowski diagrams, this stimulation explicitly shows what the earth-bound twin sees by adding and subtracting the transit time to the display.

The animation has 3 moving dots. The black dot corresponds to the Earthbound twin. The leading green dot gives the simultaneous (according to Earth) position of the traveling twin. The trailing  red dot, which will change to blue, indicates what the Earthbound twin actually sees. (The color change represents the Doppler shift as the earthbound twin looks at his brother.)  It lags the other red dot on the outbound leg, as the image of the traveling twin received by the Earthbound twin is late due to the light travel time. On the return leg, this dot turns blue and rapidly catches up to the green dot.

The clock on the left shows proper time for the Earthbound observer. It moves at a constant rate, with one full rotation corresponding to one grid spacing on the time axis of the diagram.

The middle clock corresponds to the leading green dot.  This is the time in the twin's frame of reference and is time-dilated according to special theory of relativity.

The clock on the right shows what the Earthbound observer would actually see through a telescope focused on his traveling twin's clock. This time takes into account both the relativistic Doppler effect and the light travel time. If the earthbound twin were to subtract the light travel time from these readings, he would obtain the values shown in the second clock.  Note that it is slow up until almost the very end, when it speeds up as the blue-shifted signals from the traveling twin begin to arrive back at Earth.

To the left of each clock is a time bar, indicating the total number of rotations of the clock, i.e., the age. The leftmost bar shows the age of the Earthbound twin, the middle bar gives the simultaneous (according to Earth) age of the traveling twin, and the rightmost bar gives the age of the traveling twin as viewed by the Earthbound twin through his telescope.

#### Question

• How fast is the moving twin traveling? Assume time is measured in years and distance is measured in light years.
• What is the age difference between the twins when they reunite?
• How can you convert the readings from the third clock to the second clock?
• Some dots follow world lines in the diagram, others do not. Which dots follow world lines?

#### Reference

See Giancoli-PA 26-4.