Griffiths 12.26, 12.27, 12.29, 12.30, 12.34
1. Consider two points traveling at the speed of light and separated by a distance , in an inertial frame . We can write and . In this problem you will find out how these points are described in a reference frame traveling to the right at a speed of with respect to . The easiest way to do this is to use the Lorentz transformation which gives in terms of .
Why should anyone be interested? If you interpret points and as the nodes of an electro-magnetic plane wave, then the difference is the wavelength. This is therefore a calculation of the Doppler shift for light.
2. You are a TV hosts for the tenth annual ``Nerdathon''. Two contestants push their answer buttons almost simultaneously. (The question concerns the color of Einstein's socks when traveling at half the speed of light, see previous problem for the answer). Because you are happy, you are skipping between the contestants at an approximate speed of . According to the audience, the interval between the two buzzes is spacelike. In other words the audience says that the difference in position between the contestants divided by the time between the buzzes is greater than the velocity of light.
3. During the Nerdathon you somehow got lost. You find yourself standing on some railroad track between two high speed trains that are going towards you with equal and opposite velocities. The engineer on one train says his speed is relative to the other train. What is the train's speed relative to you?
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