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Recognitions:
 Quote by Ben Niehoff Another thing to add: There are additional ways to make gauge-invariant scalars besides merely making contractions like $$R_{abcd} X^a Y^b Z^c W^d$$ at a point. One can also make nonlocal measurements, by parallel-transporting a vector around a given path, and finally comparing it with its original image (you can imagine carrying this process out using two observers, each carrying a copy of a vector, travelling two different paths, and then comparing their vectors). In Yang-Mills theory, such a scalar measurement is called a Wilson loop. In GR, we call it holonomy. It is this kind of measurement that gives us the Aharonov-Bohm effect: A Wilson loop going around a perfect solenoid. An analogous process can happen in geometry: Consider a path going around the base of a cone. Everywhere along the path, the geometry is locally flat. But there will be a nontrivial holonomy around this loop, due to the curvature concentrated at the tip of the cone. (There is no need to have a curvature singularity; you can imagine smoothing out the tip of the cone.) So there are other ways to make measurements. But ultimately, you end up taking the dot product between two vectors.