General relativity doesn’t actually talk about ‘extra space’ as if it’s some substance that is coming into being. That’s an analogy, since a lot of things in GR can be analogized by some kind of fluid flow or stretchy material. There are some mathematical parallels, but fundamentally GR only says that lengths and time intervals can change depending on the structure of the universe.
When we say the universe expands, we mean that if you have two non-interacting objects floating in space, with no forces on them from any direction, then over time the distance between them increases. Everything else, like rubber sheets, is a physical analogy – something we can create in the real world that has similar behavior to the space of GR. The only relevant feature of this analogy is that distances can change.
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