The Twin Paradox

What is the paradox?

page1 Lets look at a pair of twins, traveling apart at a speed of $0.5c$ . Each one thinks the other's clock is moving slower. Let's add some people, with their clocks turned on. I'll add Alice in our initial reference frame. Then I'll change my velocity and add Bob. I'll also add an event $e_1$ that represents the beginning of their separation.

page2 Since they start at the same location, they both agree that their clocks read 0 at that time. But when I move the time forward, their clocks do not increase at the same rate.

page3 What will happen if Bob turns around and comes back? When Bob's clock reads 2, I will add an event, $e_2$ , that shows where he wants to turn around. When Alice and Bob get back together again, will their clocks match? The answer is no -- let's see why. Instead of turning Bob around, I am going to add a third twin, Carl. Carl is a clone of Bob, but is traveling in the other direction.

page4 First I change the velocity to $-0.5c$ and then I add Carl to my drawing.

page5 To make him a clone, I want the clocks of Bob and Carl to both read 2 at this event. So I select Carl, and move his starting point 2 units before the turn around event. Look at the animation canvas: Both Bob and Carl have clocks reading 2.

page6 Now let's look at the event where Carl and Alice meet again. I'll label it $e_3$ . If I move time forward in Carl's reference frame, his clock reads 4. That makes sense: it took Bob 2 units to travel to the turn around spot, so it should take Carl 2 units to travel back.

page7 Let's switch to Alice's field of view by adjusting our velocity to 0. Then I can adjust the time until I see the event where Alice and Carl meet.

page8 Alice's clock reads somewhere between $4.4$ and $4.8$ . So when Carl returns, he is younger than Alice.

page9 Does that mean Bob or Carl's clocks changed speed? No — in their reference frame, their clocks advanced at a constant rate. However, let's go back to Bob's reference frame. You can see that he thinks Alice's clock is moving slower than his own. However, Carl is moving even faster than Alice, relative to Bob. So Bob thinks that Carl's clock is even slower than Alice's. And, in fact, Bob thinks that the time between $e_2$ and $e_3$ is longer than the time between $e_1$ and $e_2$ , so there is more time for Alice's clock to catch up.