this post was submitted on 29 Jul 2024
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All projections of a sphere onto a flat plane introduce distortion. But there are lots of different projections though.
See this relevant XKCD:
The way I understand it, when you plot out the topologies of spacetime via Relativity, the same kind of thing can happen, the maths gets all weird and funky on you, can blow up into infinities at certain points, and we call them singularities.
It still kinda blows my mind that these things popped up in the math first (by Karl Schwarzschild in the trenches of WWI, 1915) and it wasn't until exactly half a century later their existence was physically detected for the first time (Cygnus X-1, in 1965).
Then you use a different mathematical tool to plot out spacetime, and you get white holes, Einstein-Rosen bridges and parallel universes.
Different maths for 3D (or 4D) topological maps of spacetime show us different phenomena, and that relevant XKCD serves as a perfect analogy of how many ways there can be of approaching the same topologies.
Or why the hell not try mapping things out in ten dimensions! Come up with M-theory for a multiverse!
just say wormholes you NERD
Black holes are actual singularities (by existing physics), though, not mere coordinate singularities like the poles. Coordinate singularities show up in GR too, but you can get rid of them by changing your coordinate system - a very common operation, apparently.
You're right the math is similar; you can learn a lot of differential geometry concepts just by looking at coordinates of, or projections onto a sphere. Curvature gets exponentially more complex as you step into 3 and then 4 dimensions, but the same mathematical objects apply.
(Interestingly, differential geometry in dimensions 5 and up is the same as in 4, and topology actually gets easier)