Curvature is one of the most fascinating mathematical properties. It is gradual, an incremental change on an infinitesimal level. Locally, all curves are flat and Euclidean. This means that on the smallest levels, the calculations of Euclid on a flat plane holds true. However, once one escapes the realm of the infinitesimal and reaches the world of curvature, Euclid’s plane bends. This idea, of curvature, is central to two breakthroughs in humanity’s eternal quest for understanding: the discovery of calculus and the discovery of the laws of relativity. Calculus was able to describe curvature as a property based on the sum of infinitesimally small pieces of length dx. Einstein’s work on relativity and on the field equations come from this idea. The curvature of space can be throught of as a metric built on a gradient, or a gradual change in the bending of space. Although I am currently still trying to wrap my mind around the mathematics behind the field equations, I can still comprehend the utter beauty of curvature.
Curvature is synonymous with “change”. Change is composed of infinitely many stills, just like how a movie is composed of many frames. Of course, a movie doesn’t have infinite frames, but if it theoretically did have infinite frames I imagine it would be the smoothest, most “natural” film of all time. Could reality be seen as being composed of an infinite number of still frames, like the theoretical “smooth movie”? Or maybe, at some point, one cannot cut another layer of reality. Maybe reality is not composed of infinitely thin layers (or frames) and “degrades”, producing imperfections on the smallest (quantum) level. From our perspective, everything behaves with curvature because things change. But maybe our reality is built on approximations, essentially rounding off the quantum imperfections. These are all speculations, but these thought games are a nice way to think.