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Relativistic snapshot (Posted on 2006-05-26)

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?

Submitted by Tristan

Rating: 3.3333 (3 votes)

Solution: (Hide)

This problem requires distinction between what is observed and what is calculated from these observations. If you were to film the tunnel as it passes by, you would be able to calculate that the tunnel is 100 km, exactly as given, and as predicted by the Theory of Special Relativity. The calculations that lead to this require the knowledge that light is not instantaneous. If light originates 100 km away, it takes about a third of a thousandth of a second to reach you. During this time, the light source may have moved.
The tunnel travels .5 c (where c is the speed of light), so the velocity of light relative to the velocity of the tunnel from my frame of reference should be .5 c if I'm looking forwards, and 1.5 c if I'm looking backwards.
If I'm looking backwards, the time required for light to reach from one end to the other is 100 km / (1.5 c). During this time, the closer end will have traveled 100/3 km away from me. Therefore, the tunnel would look about 66.7 km if I were looking out the back window.
If I'm looking forwards, the time required for light to reach from one end to the other is 100 km / (.5 c), during which the closer end would have traveled 100 km towards me. Therefore, the tunnel would look twice as long, 200 km, if I were looking out the front window.
200 km, front window

Comments: ( You must be logged in to post comments.)

Subject	Author	Date
Puzzle Thoughts	K Sengupta	2023-05-06 00:02:05
re: false!	Tristan	2007-02-28 03:06:35
false!	matt	2006-08-06 12:39:31
dfntly nt th sltn	Leming	2006-06-12 16:53:23
re(5): Solution	Eric	2006-06-07 00:21:52
re(4): Solution	Ken Haley	2006-06-06 01:56:59
re(3): Solution	Tristan	2006-06-04 09:29:42
re(2): Solution	Ken Haley	2006-05-30 02:05:05
re: Solution	tomarken	2006-05-29 07:41:03
Solution	Ken Haley	2006-05-29 01:41:32
re: A Question	Tristan	2006-05-26 16:41:19
A Question	tomarken	2006-05-26 16:19:54

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