Carmen and Dan are in their physics lab room. They each select a meter stick, lay the two side by side, and agree that the meter sticks are exactly the same length. Then, for an extra-credit project, they go outside and run past each other, in opposite directions, at a relative speed $v=0.9c$. FIGURE 36.26 shows their experiment and a portion of their conversation.

FIGURE 36.26 Carmen and Dan each measure the length of the other’s meter stick as they move relative to each other.

Now, Dan’s meter stick can’t be both longer and shorter than Carmen’s meter stick. Is this another paradox? No! Relativity allows us to compare the same events as they’re measured in two different reference frames. This did lead to a real paradox when Priya rolled past Ryan on the train. There the signal light on the box turns green (a single event) or it doesn’t, and Priya and Ryan have to agree about it. But the events by which Dan measures the length (in Dan’s frame) of Carmen’s meter stick are not the same events as those by which Carmen measures the length (in Carmen’s frame) of Dan’s meter stick.

There’s no conflict between their measurements. In Dan’s reference frame, Carmen’s meter stick has been length contracted and is less than 1 m in length. In Carmen’s reference frame, Dan’s meter stick has been length contracted and is less than 1 m in length. If this weren’t the case, if both agreed that one of the meter sticks was shorter than the other, then we could tell which reference frame was “really” moving and which was “really” at rest. But the principle of relativity doesn’t allow us to make that distinction. Each is moving relative to the other, so each should make the same measurement for the length of the other’s meter stick.