The Lorentz transformationRecall that in the previous section, we tried to model space and time in a way that would be consistent with the observed constancy of the speed of light in Nature. The resulting model for space and time measurements has observers measuring time and space differently when they are moving relatively to one another. The two basic differences between what relatively-moving observers measure can be summarized as:
We will now state (without going through the grunge work of proving) that the mathematical model that best encompasses the observed constancy of the speed of light in Nature (pretending for now that gravity doesn't exist) is the one where the spacetime coordinates of two such observers as described above are related through what is now called a Lorentz transformation:
For example, in time dilation: in the blue car's frame of reference, the laser pulse did not travel in the Xb-direction at all, so X' = Xb = 0, leaving T = T' /(1 - U2/c2)1/2. In length contraction: in the red car's frame of reference, the red driver must measure the length of the blue car at a single moment of the red driver's time. This means we have Tr = T = 0 and so c T' = - (U/c) X', leading to X = X' (1 - U2/c2)1/2. The Lorentz transformation is the foundation of relativistic geometry, which we will examine next. |
Wednesday, October 13, 2010
How does the Special Theory of Relativity work?
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