## Exercise 1: The Twin "Paradox"

In this Exploration we will be considering different aspects of the so-called twin paradox.  Restart.  At t = 0 years the traveling twin (represented by the green circle) heads out on her journey and then returns at t = 10 years (position given in lightyears).  In the top panel the spacetime diagram for the stationary frame is shown.

Select View from Earth-Bound Twin and play.

1. How fast (distance/time) is the moving twin traveling relative to the stationary twin (measured in c)?
2. What are the slopes of the red and green worldlines?  What are the reciprocals?  Are the results consistent with your expectations from part (a)?

Select Pulses from Earth-Bound Twin and play.  The two twins agree to send each other a light pulse once a year on the anniversary of the traveling twin's departure.  Also shown is the stationary observer's clock.

1. What is the frequency of the stationary twin's light pulse?
2. According to the traveling twin, how much time elapses during the outbound and inbound trips? (Hint: use the time dilation formula!)
3. What light pulse frequency does the moving twin observe during her outbound trip?  During her inbound trip?  What average frequency does she observe?
4. Compare the emission and average reception frequencies.  Explain the discrepancy.

One of the most important concepts in special relativity is the idea of the spacetime interval.  The spacetime interval Δs is defined by the equation (Δs)2 = (Δx)2 - (Δ ct)2 which is the Pythagorean theorem (or metric) of spacetime.

Select View from Earth-Bound Twin and play.

1. Calculate the spacetime interval for the stationary observer during the animation.  Compare this quantity to the aging of the stationary twin.
2. Calculate the spacetime interval for the traveling twin's outbound trip (as measured by the stationary observer).
3. Calculate the spacetime interval for the traveling twin's inbound trip (as measured by the stationary observer).
4. Compare the sum of (h) and (i) to the aging of the traveling twin.
5. Why do the spacetime intervals match the aging of the two twins?  Think about how far each twin sees herself travel.

Original Physlet-based material authored by Mario Belloni and Wolfgang Christian and appears in the paper,
"Teaching Special Relativity Using Physlets," The Physics Teacher, May 2004.
Narrative of the laboratory exercises authored by Tim Gfroerer.
© 2007 by Mario Belloni, Wolfgang Christian, and Tim Gfroerer.