Because gravity will be observed to decay with distance cubed for distances on the scale of the extra dimension, and distance squared beyond that; and we have not found a scale where we see gravity decay faster than distance squared (but it gets harder and harder to measure at small scale, so the error bars grow).
IIRC experimental gravity data rules out any compactified dimension bigger than 50μm, but a question I keep coming back to is "surely the pictures of atomic bonds taken by electron microscopes rules compactified dimensions larger than 1Å?"
interesting question. my (somewhat naive) thought about it is that bonds are maintained by the EM force, which is so strong that it swamps out any contribution from gravity.
If a compactified spatial dimension exists in our universe, and was big enough to fit an atom, why couldn't we see two atoms that seem like they're in the same 3-dimensional coordinates?
Sometimes compactified dimensions are analogised to a straw: seen from a distance it seems one dimensional, up close (an ant's perspective) it's got one long dimension and one short dimension.
I don't know how far to take the analogy. It sounds like surely photons with wavelengths smaller than the compactified dimension would be likely to take a spiral path, looping around compact dimension n times for every m units of 3-space travelled, which would seem like they were mysteriously slow if you weren't expecting the compact dimension to exist.
I vaguely remember the idea of wavelength-dependent speed of light is a thing that's been ruled out by tests with supernova data, but not to what wavelength or sigma.
I think you're describing a completely different geometry than I'm describing.
An ℝ²-brane such as flatland existing in a ℝ³ bulk is different to an ℝ²⨯S¹.
If the S¹ part* is present in our universe to the degree that it can explain anything about gravity, it should also have an impact on everything else in the universe larger than the radius of the S¹ dimension's circumference.
* well, S^n ⨯ T^m, the version of string theory I hear most about has n+m = 6, but there are others, and this thread is a toy model where n=1, m=0
Edit: Apparently the U+1D54A character is stripped, so put a plain ASCII "S" back in.
Yes but you would sure as heck bump into it if it was big.
Like literally in the middle of your sitting room. Isn’t it a known meme horror thing - monster slices from another dimension splicing across into ours as they move through their planes .
Basically it doesn’t happen but the dimensions do exist so they must be small.