What Special Relativity Taught Me About the Incredibly Diverse Phenomena in the World (And my Explanation of the Twin Paradox)

The genius of special relativity is the derivation of far-reaching consequences from a small group of postulates. In a sense, Einstein's proposal was simply that Maxwell's equations hold in all reference frames, which in turn means that the speed of light in a vacuum is measured to be the same value c in all reference frames. From this you can derive Lorentz transformations for the standard Galilean transformation. From this you naturally arrive at the idea of a 4-vector, and from there all sorts of profound physical discoveries follow - the equivalence of time with space dimensions, the famous formula E=mc^2, the twin paradox, and so forth.

This semester I took a course (PHYS2100) where the first three weeks where focused on introducing special relativity in a mathematically heavy way. It is important to approach SR from two angles: conceptually, so as to properly understand the uniquely relativistic way of thinking of physics problems, and mathematically, to understand the consequences of the theory beyond the surface level. With this in mind I have found that the best introduction to special relativity is the course offered by Leonard Susskind as part of the "Theoretical Minimum." I should be clear, however, before someone takes this comment as a slam on Dr Plakhotnik who gave the SR lectures for my course, that I never went to any of his lectures, I looked only at his lecture notes.

You can reduce the postulates of special relativity from the usually-quoted two to only one by saying "all laws of physics (including Maxwell's equations) are equally valid in all inertial reference frames." As I said, what is exciting about special relativity is how diverse the consequences of such a simple assertion are, how much that idea changes how we have to view our world. It is not at all obvious from that statement that we can no longer operate in a Euclidean geometry, that we live consequently in Minikowskii space. But it is a consequence. It's not clear at all that this implies that two twins separated, one staying on earth and the other going for a very fast trip around the solar system, should have aged by different amounts upon the twin's return. But it does follow. All sorts of amazing phenomena in the world really do arise from simple postulates.

The beauty of physics is perhaps entirely contained in the elegance of explanation that physics has. There does not need to be a plethora of different mechanisms and explanations for all the different occurrences in the world, complex phenomena result from simple rules. The mystery of emergence in science is exactly the emergence of complexity from simplicity. That means that, most likely, physics will never give a model for some complicated object like the human brain - and yet! That does not at all imply that the intricate operations of the human brain are not somehow (unbeknownst to us) reducible to simply physical laws. I think biology is complicated physics, but it will almost certainly never be studied or taught as such.

Plakhotnik writes at the end of his notes that everyone that teaches or studies SR should have an opinion on the twin paradox (that weird age difference between the twins that I noted above) so I will end with mine. Both the mathematical equations and the experimental evidence demonstrate that it is correct, but why? How can one interpret such an occurrence? His answer is more rigorous but for the same reason probably loses most people. Here is an answer which helps think of it more intuitively (even though, when the rigour is re-introduced the answer is the same): given that the time and space coordinates in space-time are equivalent, the fact that the travelling twin moves more through space means that she moves less through time. They start at the same spot and end up together again at the end, so if you include time then they must have traveled the same "space-time distance", one more through space, the other more through time.