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Relativistic snapshot (Posted on 2006-05-26) Difficulty: 3 of 5
The Theory of Relativity is not required to solve this problem.

The Lightway Express boasts half the speed of light. According to the advertisements, this relativistic speed literally shortens long trips. This is true. At one point, the train goes through a tunnel of about 111.8 km, but from the train's point of view, it is exactly 100 km long.

As a curious tourist, I resolved to experience relativistic speeds, and furthermore, bring home memories in photo form. So while I was riding the Lightway Express, I pointed my camera out a window, and took a picture of the entire 100 km tunnel. Later, when I examined my excellently timed photo, I was disappointed to find that the picture showed a tunnel that was much longer than 100 km.

How long is the tunnel in my photo, and why is it longer than I expected? Was I looking out the front or the back window of the train?

See The Solution Submitted by Tristan    
Rating: 3.3333 (3 votes)

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Solution Solution | Comment 3 of 12 |

Assume the tourist snapped the picture from the rear of the train, pointing backwards, at the instant the rear of the train exits the tunnel.  At that instant, the light from the tunnel's exit reaches the camera since it's right there.  At a later point in time, the light from the the other end of the tunnel (its "entry") reaches the camera. At this point in time, the train has traveled 100m while the light from the entry has traveled 200m (in order to satisfy the condition that the train is moving at 1/2 c).  So, in the picture, the tunnel's entrance is 200m away when the camera records it.  So the tunnel appears to be 200m long.

If the tourist were looking forward and snapped the picture just as the train entered the tunnel, he'd get the opposite effect, and the tunnel would appear to be 66.67 m long.

You get the same result (66.7m or 200m) when the picture is snapped not just at the tunnel entry or exit, but anywhere outside the tunnel (looking forward or back, respectively).

An interesting followup question would be, assuming our tourist has a camera that simultaneously snaps a pair of pictures both forward and backwards, at what point in the tunnel would he have to be in order for the two apparent distances in the pictures to add up to exactly 100m? 


  Posted by Ken Haley on 2006-05-29 01:41:32
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