Lesson 3 - Length Contraction

Derivation

page1 This lesson is about length contraction. I'm adding two people, called $A_1$ and $A_2$ . In their reference frame, they are 2 units away from each other.

page2 Now I can change the velocity and see that they move closer to each other. If I pick a velocity of 0.87, then the scaling factor $\gamma = \frac{1}{\sqrt{1- v^2/c^2}}$ is about 2. That means that in our current reference frame, the distance between them is about $\frac{1}{2}$ as big as in their reference frame.

page3 It is difficult to derive length contraction just from drawing pictures. So I won't try. There are some nice references about the this in the Lorentz transformation section on the discussion page under the help menu.

page4 As with time dilation, it seems like there is a paradox. If $A$ thinks $B$ is shorter, and $B$ thinks $A$ is shorter, which is right?

page5 The classic way to talk about this paradox is by trying to drive a train into a barn. Let's add a train in our current reference frame. Now we'll adjust our velocity to be the same as $A_1$ and $A_2$ . Notice that the train is shorter now.

page6 I'm going to adjust the time so that the train has moved past the two people, and then I'm going to add a barn. The barn is positioned so that the train is completetly inside it.

page7 But when I adjust the velocity so that I'm looking at the train's reference, the train no longer fits in the barn. What happened?

page8 Let's go back to the barn's reference frame. I'm going to add two events: a red one corresponds to the left hand side of the train is inside the barn, and a blue event is that the right hand side is inside the barn. In the barn's reference frame, these events happen at the same time.

page9 But in the train's reference frame, the two events do not happen at the same time. As we move the time forward and back, we can see that they mean the same thing: the red event means the left hand side of the train is inside the barn, and the blue event is the right hand side of the train is inside the barn. However, by the time the left hand side is inside, the right hand side has already passed beyond the edge of the barn, causing expensive damage. The paradox is resolved by saying that the two different reference frames do not agree on which events are simultaneous.