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
I'd ask for some of whatever you're smoking that makes you say all of this in response to a map, but I need to be able to drive a car today and I don't want a DUI.
What makes you think I smoked anything?
Why couldn't it be... edibles? :-P
And don't get me started on the topologies and symmetries of fields in quantum mechanics!
But seriously, I enjoy letting the mind take me for a nice long ride, using an artifact to trace connections to other things throughout history, and in this case it's the math of maps, their topologies.
Random topics on Lemmy sometimes lighting up my mind like a Christmas tree.
Also trying a little to get into the mindset of the era when some of these things were being cooked up and explored, shift perspective as best I can, from where and when I am. It's incomplete, but it's still fun. I probably got this type of "pinball narrative" from James Burke and his old history shows Connections and The Day The Earth Changed.
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)
Would that be the Lorenz Transformations?
It's not what I was thinking of, but that is an often used coordinate transform too.
Schwarzschild coordinates, which you use to derive the solution for a basic gravity well around a non-rotating spherically symmetric body, take the form of spherical coordinates plus time, and so inherit coordinate singularities at the poles of each "layer". I don't know which ones they're referring to exactly, but apparently some systems commonly used for studying space asymptotically far away from a black hole create singularities across the whole event horizon, which is actually a pretty normal patch of space. In this case it's not inevitable, though; Kruskal–Szekeres coordinates allow the entire black hole to be represented smoothly, obviously excluding the actual physical singularity. In other cases (like a normal 2D sphere) every coordinate system must have at least one coordinate singularity.
This isn't my specialty, but every Wikipedia article on differential geometry is loaded with information which objects translate between coordinate systems safely and how, so it's fair to say it's a big deal.