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Curvature (positive).
...When we had drawn our triangle,
made up of geodesic lines, on a plane, the sum
of its vertex angles was equal to p.
A plane... is a flat, "non-curved", Euclidean surface. The sum
of the angles of this triangle is therefore the Euclidean sum. In the previous
experiment we saw that if a triangle did not contain the
apex of our cone, the sum remained Euclidean. On the other hand, when the
triangle contains the apex S, then this sum presents an excess
q, whatever
the triangle, as long as it contains this point. We will say that the apex of the
cone is a point of concentrated curvature.
...We can now move on to other
experiments. After having made two cones, with
cuts q1 and
q2, we can
glue these two surface elements to one another.
...A simpler way of proceeding
consists in making two cuts in a sheet of Bristol board
and making the following surface:
You will then be able to draw on this surface as many geodesic triangles
as you wish:
-
Enclosing neither S1 nor S2: Sum of the angles: p
-
Enclosing only S1: Sum of the angles p
- q1
- Enclosing only S2: Sum of the angles: p
- q2
- Enclosing both points S1 and S2: Sum of the angles
: p + q1+
q2
...It is easy to imagine that one could make
a large number of mini posicones with a small angle Dq
and that one could glue them one on top of another. One could
even manage for there to be a constant density of curvature
per unit area, by treating this curvature as
the sum of the Dq values associated
with each apex of these mini posicones.
...By making these mini posicones smaller
and smaller (as well as the elementary angle Dq
associated with them) one can use this to construct
a portion of surface with constant density of curvature.
The sphere is a surface with constant density of curvature.
More simply, we will say with constant local curvature.
An egg is a curved surface, with variable density of curvature. More simply, we will say with variable local curvature.
...General Relativity
consists in identifying density r
and local curvature. Of course, General Relativity
deals not with two-dimensional surfaces, nor even with
three, but with four-dimensional hypersurfaces. One must therefore not
ask too much of the foregoing, and we should consider
these figures only as didactic images, intended to fix
ideas. But they are not as bad as all that.
2d didactic image of a celestial body.
A celestial body, like the sun, is a concentration of matter, surrounded, if not
by vacuum, at least by a quasi-vacuum (therefore by a region with very
low curvature). In two dimensions the didactic image will be that of a blunt
cone.
...A blunt cone is made
using two elements: a spherical cap,
with constant curvature (or with "constant density of curvature")
and a truncated cone. This truncated cone is "flat", its density
of curvature is zero. It is a Euclidean surface. It is the didactic image
2d of a celestial body of constant density r.
...One might incidentally wonder how
to perfectly join a truncated cone and a spherical cap,
so that the tangent plane is continuous.
...It is simple. The truncated cone is made
from a cone, which implies a cut of an angle
q . The spherical cap
contains a certain "quantity of curvature", which is also an angle.
It is the sum of all the angles of the mini posicones that constitute it.
These two angles must be equal.
But how can one evaluate the quantity of curvature contained in a given
spherical cap? .
...
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