Intricate geometries on spacetime’s rubber sheet

posted this question on Quora:

What unusual geometries on the Reimannian ‘rubber sheet‘ of spacetime exist according to general relativity

Hello Astrophysicists, a very specific question for you.

It seems that the Reimannian ‘rubber sheet’ model of spacetime is useful in describing phenomena that we observe. Some examples below.

  1. vanilla gravity: curved spacetime given a mass
  2. blackholes: essentially a vortex geometry using the spacetime ‘sheet’
  3. wormholes: a touching or folding of spacetime such that two points of the sheet touch one another

My question is, what other exotic geometries may arise and are allowed by the fabric.

For eg. I can see a ‘polyp’ or scrunched ball geometry similar to a sock wrapped in a bedsheet during the wash, that would be inaccesible by most other points on the sheet, but still made from the same ‘sheet’.

This would be essentially a untouchable region made from the same fabric of space time.

Is this possible and has this been theorized? Curious to hear more.

Thank you