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Total curvature.
** ...We can construct a sphere by joining together small posicones. However, during this process, this surface with constant curvature (or curvature density or local curvature) will close up. Therefore, it contains a certain amount of curvature—but what is it?
...If I draw a geodesic triangle on a sphere, it will enclose a certain number of mini posicones, a certain "amount of curvature," which is an angle. This will simply be proportional to the area of the triangle, or more precisely to the ratio between the area s of the triangle and the area S of the sphere.
...But earlier, we saw that when drawing a geodesic triangle on a surface made of joined posicones, the deviation from the Euclidean sum of angles equals the sum of the curvatures concentrated at each cone vertex contained within our triangle. Thus, it suffices to measure the sum of the angles a, b, g of the triangle above, constructed from three geodesic arcs of the sphere, to obtain a measure of the angular curvature contained within this triangle. The geodesics of the sphere are its "great circles."
...Let us cut our sphere into eight equal parts. We will obtain eight triangles formed by geodesic arcs, each with three right angles.
...Each of these triangles therefore contains a curvature equal to π/2. Since there are eight of them, the total curvature of the sphere is therefore 4π.
...This small observation serves to show that we can derive geometric results using extremely simple reasoning.
...Returning to the topic of the blunted cone, we see that the side of the object depends on the amount of curvature "contained inside"—this curvature may be point-like (conical point) or distributed over a spherical cap. We can make the cap shrink toward a point by homothetically reducing it (in such a way that it always contains the same "amount of curvature").
Paths.
...In General Relativity, the key idea is simple: treat the trajectories of objects, particles, photons, or matter as geodesics. Of course, these are geodesics of a four-dimensional hypersurface. Thus, we again have only didactic images.
If we take our blunted cone, we can draw geodesics on it and project them onto a plane.
...All particles follow geodesics of the hypersurface: particles of matter, as well as photons and neutrinos. This is why we amused ourselves by depicting a geodesic that passes completely through the object. A neutrino can pass through the Sun without any problem.
...But what is this plane onto which we project these geodesics? It is the way we represent space. Our "mental universe" is completely Euclidean, and our thinking is "flat." When we see a comet brushing past the Sun, it never occurs to us that it is actually going "straight ahead," meaning it follows a geodesic of the hypersurface. Our perception of the world is Figure 24', where a celestial body "attracts" objects passing nearby.
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