Date of Award
Kaluza-Klein theory is a fascinating piece of physics history and, while not something actively researched today, is a precursor to string theory and is the first example of a unification theory by adding an extra dimension. The origins date back to shortly after general relativity was first published. Theodore Kaluza, while looking at the Einstein equations in 5 dimensions instead of four, realized the equations became noticeably simpler if he imposed something on it called the cylinder condition. This is equivalent to saying the 5th dimension is a closed loop, rather than an infinitely long straight line. Using this assumption he derived the equations of motion along the 5D paths and saw that, in the absence of forces, became approximately the same as those in GR that were under the in uence of electromagnetic forces. This result sparked an interest in the unification of the fundamental forces of nature by adding extra dimensions, especially ones with cylinder conditions of small radius. Most of the rest of these theories were quantum in nature. But this only talks of Kaluza's contributions to the theory and leaves nothing of Klein's. While we will only be focusing on the classical versions of this theory, Oskar Klein was interested in how this might translate to quantum mechanics. The results were interesting and provided a massive stepping stone to similar theories of unification but has some incorrect predictions. While this theory is not actively considered as a candidate for our laws of physics as they stand, it is still an important piece of physics history and understanding it can help to provide an gateway to understanding the theories that have branched from it. In the following I intend to make some more advanced topics in theo- retical physics more accessible mathematically and more intuitive. The end goal is to provide an intuitive and sufficient for the purposes of physics at this level, but not complete, understanding of the mathematics that makes up Kaluza- Klein theory and why the theory unifies electromagnetism and gravity. We will start with a special relativity review and delve more into the geometry thereof than you would have in a special relativity course. Then we will move onto a dierential geometry introduction and work to build an intuitive understanding of the mathematics involved until we have enough to produce the Einstein Field Equations. Then we will progress onto the derivation of the field equations of Kaluza-Klein and show why they have approximately the same predictions as the Einstein equation.
Statter, Samantha, "A Beginner’s Guide to Kaluza-Klein: The Basics of Spacetime, Curvature, and the Classical Unification of Electromagnetic and Gravitational Fields" (2020). Senior Theses. 1470.
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