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?

I realize this response is far too late, but here it is.  Spoilers ahead!

"First off Einstein stated that no possible speed could ever exceed the speed of light;c."

Agreed.  However, it is possible for one object to be moving past me at the velocity of +.75c and another to be moving past at -.75c.  From my perspective, the distance between these two objects is increasing at the rate of 1.5c.

In the solution, I erred in calling this the "relative velocity" between the objects, for in relativity, the relative velocity between two objects is not equal to the difference of their velocities.  The equation is:

let vr = relative velocity;
v1 = first velocity;
v2 = second velocity;
vr = (v1 - v2)/(1-v1*v2/c^2)

Rather than calling it the "relative velocity", I should have called it the rate at which the distance between the two objects (the tunnel end and the photon) is increasing, from my perspective.

However, this is the only correction I need to make.  The rest of the solution follows.  The tunnel indeed does appear longer.

This lengthening effect is an illusion caused by the finite speed of light.  The shortening effect caused by Special Relativity is real.  Together, the tunnel appears longer, though it is actually shorter.

Posted by Tristan on 2007-02-28 03:06:35