This is just the start: when you do *quantum* gravity in this context you get into lots of mindblowing math. At this stage I'm mainly just learning that math and applying it to this problem. But that's a fun way to learn math. Quantum polyhedra! 👍
More later, but now I need breakfast.
(9/n)
This is good because in 2d space, unlike in 3d space, the equations of general relativity say that spacetime is *flat* in a vacuum. So if we have massive point particles at rest in empty 2d space, space will be flat - except at the particles, which give cone points!
(4/n)
In a crazy way, a cone is actually *flat* except at the cone point. Of course the cone's surface bends, but it's 'intrinsically' flat. This means that if you were a tiny ant living on the cone, studying geometry, you'd see angles of triangles add up to 180°, just like on a flat table!
(2/n)
In general relativity, when space is 2-dimensional, a particle creates a singularity that's a cone point. And here's the cool part: the angle deficit of that cone is proportional to the *mass* of that particle!
So there's an upper bound on mass, since the angle deficit can't be > 360°.
(6/n)
Even better, *any* metric on the 2-sphere that's flat except at finitely many cone points with positive angle deficits comes from some convex polyhedron! This is called Alexandrov's theorem.
So general relativity when space is a 2d sphere has a lot to do with polyhedra!
(8/n)
To see this, just notice that you can take a flat piece of paper, cut a slice out of it, and glue together the two edges of the slice to get a cone. If you'd drawn a triangle on the flat piece of paper now you have a triangle on the cone, and no angles have changed! They still sum to 180°.
(3/n)
In general relativity, a point mass just sitting there is inevitably a black hole. But that's when you're living in 3d space (and one dimension of time). In 2d space, a point mass creates a less drastic singularity: a 'cone point'. I've been studying these cone points lately, just for fun.
(1/n)
Now suppose space is not only 2-dimensional but topologically a sphere. It's flat except at cone points, which are particles. Can you visualize this?
Yes you can! Any convex polyhedron gives an example! Its geometry is intrinsically flat except at the vertices, which are cone points.
(7/n)
A cone can be sharp and pointy or flat and squat. It's described by its 'angle deficit' - the angle of the slice you cut out to create that cone. The bigger this angle, the more pointy your cone. When it hits 360° you get an infinitely pointy cone.
(5/n)
