The most interesting aspects of Special Relativity are its 'paradoxes.' We put 'paradoxes' in quotes because Special Relativity is, in fact, an entirely self-consistent theory that contains no true paradoxes (that is, no paradoxes that cannot be resolved with a little careful thought). The fun part is thinking through the apparent paradoxes to find where the logical error that leads to the inconsistency lies. We have already encountered several of the classic paradoxes of Special Relativity, and in this topic we will uncover several more.
The first section examines the relativistic Doppler effect, which is an interesting and useful extension of the Doppler effect in classical mechanics. It is especially important because relativistic speeds need to be considered for one of the main instances in which the Doppler effect becomes important, the red-shift of light reaching us from far galaxies. The second section deals with the most famous 'paradox' of relativity: the so-called Twin Paradox. The third section extends what we have learned about energy, momentum an d 4-vectors to more interesting problems involving the decay of and collision between particles.
The aim of this SparkNote is to demonstrate that Special Relativity, as unfamiliar and unintuitive as it may seem, does have important applications in areas of physics ranging from the very small (sub-microscopic particle interactions) to the very large (motion of stars and galaxies, cosmology). Moreover, the relationship between the electric and magnetic forces and fields is bound up with Special Relativity; the interaction ultimately produces oscillations of the electromagnetic field that is light itself. The results of Special Relativity are not as abstract as they at first appear to be!