|
The first thing I noticed was that when I was on the moving train, the tunnel no longer appeared to be perpendicular to the track. This is because light going through the tunnel appears to be coming towards me at an angle. This effect is called aberration of light, and is in fact observed in stars.
A good analogy for aberration is vertically-falling rain. If you walk forwards, the rain will appear to be falling on you at an angle. However, unlike rain, the speed of light always remains the same.
Consider the photon that begins at the far end of the tunnel and ends at the camera. We are only considering my perspective on the train.
Let t=the time required for the photon's journey.
During this period, the tunnel travels backwards 0.5ct.
Total distance traveled by the photon = sqrt((30 km)^2 + (0.5ct)^2)
Therefore, ct = sqrt((30 km)^2 + (0.5ct)^2)
c^2*t^2 = 900 km^2 + 0.25*c^2*t^2
c^2*t^2 = 1200 km^2
ct = sqrt(1200) km = 34.64 km
cos(angle) = 30 km / 34.64 km
angle = 30 degrees
In conclusion, the tunnel appears to be pointing 30 degrees from perpendicular, and appears to be about 34.64 km long.
In general, the tunnel will actually appear to be hyperbolic in shape, depending on exactly when I took the photo. Here, our assumptions of negligible width and a perfect photo straight down the tunnel cause the hyperbola to degenerate into straight line. |
