By Silvan S. Schweber
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We are done! Just substitute the expressions for b and r found in step 3 into the equation from step 4, and then use the result from step 2 that s/a = v/ v c to reproduce the famous result first obtained by Einstein. This is Einstein’s beautiful formula for the addition of velocities. u= 48 u’+v u’v 1+ 2 c Having done all this geometric labor, let us not forget to contemplate the result by checking whether it satisfies the qualitative statements made in the previous section. • If we substitute the special values v = 1⁄2 c and u’ = 1⁄2 c, which were used in the previous example, we see that the graph did not betray us: we obtain u = 4/5 c as we did before by direct visual inspection.
Note that the future and past light cones are identical in all inertial frames. The cones are universal: they are attached to an event, not to a specific observer. However, any point P located outside the light cones of (say) 1 may, dependent on the particular velocity of an observer passing through 1, lie in the future, the past or the present for that observer. But that ambiguity in the time ordering is innocuous, because there is no signal which can travel between point 1 and point P. There can be no causal relation between events at points 1 and P.
In the figure we have depicted a moving light source that flashes with frequency fs . As you can see, the observer at rest receives the light signals with a different frequency, and the question is what frequency fo she will measure. The frequency is just the number of pulses per second. So from the figure we see that fs = 4 / w’0 and that fo = 4/w1. This means the ratio of the observed to the emitted frequency is fo / fs = w’0 / w 1. The time dilation formula tells us that w 0 = gw’ w’0 . From the figure we can read off that w 1 – w 0 = bw 0 , because this distance equals the length of the horizontal arrow, which is the distance traveled by the source in time w0 at velocity bc.
An Introduction to Relativistic Quantum Field Theory by Silvan S. Schweber