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Didactic image of a heavenly body (star, planet, dense egg)
A star like the Sun is a mass concentration. Around it: the void, or a portion of space that is "almost empty," for it contains very rarefied gas and photons. In 2D, the corresponding didactic image is a blunt (positive) cone:
(18)
You can construct it from two components: a portion of a sphere and a portion of a (positive) cone, glued together. The portion of the sphere is a surface of constant curvature density. The portion of the cone is a flat surface, a surface of zero local curvature density. This last example is a Euclidean surface. The portion of the sphere is a non-Euclidean surface (a Riemannian surface).
This is the 2D didactic image of an object of constant density r, surrounded by void.
How do we join the two elements together to ensure the continuity of the tangent plane? It is simple. Your portion of a cone comes from a cone whose cut corresponded to an angle q. Your portion of the sphere is supposed to be built from elementary mini-positive cones, so that it contains a certain "amount of angular curvature" q. If the two angles are equal, the tangent plane will be continuous.
But how do we measure the amount of curvature contained in a given portion of a sphere?
Total curvature.
We can build a surface by joining elementary positive cones. We can arrange it to obtain a surface of constant curvature density. Then we know that the surface is a portion of a sphere. If we add more and more elementary (positive) cones, this sphere will become complete. It contains a certain amount of angular curvature. All spheres contain the same amount. The total angular curvature of a ping-pong ball and the total angular curvature of the Earth are equal, although they have very different weights.
By the way, the total curvature of an egg is the same, for they have the same topology. In principle, hens lay eggs with spherical topology. Personally, I have never seen an egg with toroidal topology. That would correspond to some strange snake, with no head nor tail, or something like that.
Let us return to ping-pong balls, normal spheres. If this surface has a constant local angular density, it means that the amount of angular curvature (the sum of elementary angles Δq) will be proportional to the area. See figure 19. This area can be bounded by any kind of border. But we can use geodesics of the sphere. Let S be the area of the sphere and s the grey area inside the triangle.
(19)
Above, we saw that the (positive) deviation from the Euclidean sum (180°) for a triangle drawn on a surface depends on the number of cone vertices that were inside. The sum was 180° plus all the angles corresponding to those enclosed vertices.
Conversely, if I measure the deviation from the Euclidean sum, I can measure the amount of curvature contained inside the triangle.
A geodesic of a sphere is called a great circle of the sphere. See figure (20). Meridians and the equator are great circles of the sphere.
(20)
We can cut our sphere into eight equal-area pieces. See figure (21). We get eight triangles whose angles are all 90°. Then the deviation from the Euclidean sum is 90°. Each of these triangles contains an angular curvature equal to 90°. As a conclusion, the total curvature, the total angular curvature of the sphere is 8 x 90° = 720° = 4π.
(21)
Each grey triangle contains π/2.
Do you enjoy curved surfaces, Riemannian surface geometry?
If we return to our blunt cone, we see that the angular curvature is contained inside the circular border, in the constant curvature density area. The cone's flank, or wall, is not a limited surface. You can extend it to infinity if you want. The amount of angular curvature does not depend on the perimeter of the border, nor on the area of the portion of the sphere. The latter can be reduced. See figure (22). Even reduced to a simple point, it will contain the same amount of angular curvature. That is why we say that a conical point was a concentrated curvature point. Conversely, we can build smooth surfaces with a set of conical points.
Matter is made of atoms. Atoms can be considered as point-like objects. They are "concentrated curvature points" in 3D space.
The air you breathe is a constant density medium. It is made of molecules, atoms. It is a set of concentrated curvature points, linked by Euclidean portions of space. You assimilate that to a constant curvature medium.
Next time you breathe, think about it.
(22)
