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Ah, but then we couldn't see or experience gravity at all!
In differential geometry there's a very important distinction between coordinate distance and actual distance. The globe and GPS coordinates give a good example - one degree of longitude is throwing distance at South Pole Station, but ~111km at the equator, even though it's still one degree. On a curved surface additive coordinates will never describe actual distance exactly. In some cases, like a 2-spherical planet, they're even guaranteed to break down somewhere (like the exact poles).
It was a blunder mentioning rubber - this isn't about bowling balls on a trampoline. I just meant that solid matter has a natural spacing between atoms, and if something continuously pulls it away from that - like expanding space or, I don't know, two conveyor belts going opposite ways - it's going to respond with a constant tension offsetting the effect. Or break.
And annoyingly, that's only possible in more dimensions than we can picture. All a 1+1 dimensional space can do is expand or contract.
Okay, nitpick
but if it's something relativistic space-like momentum can be just as important as energy. The matter half of the Einstein equation(s) treats every component of 4-momentum equally.