# Rolled-up dimensions vs surfaces in a higher dimensional space

All the accounts of dimensions higher than 4 seem to talk about them being 'rolled up'. Is this different to being confined to a 4D surface that exists in a higher dimensional space?

• What's your precise mathematical definition of "confined to a 4D surface in a higher dimensional space?" Commented Mar 12, 2021 at 23:52
• @RamiroHum-Sah: Confined could be like a moebius or klein bottle in 3D, something we can't make sense of viewed from the lower dimensions of such a surface, that 'prevents' leaving the surface without recognising, or having the energy, to leave the 'surface'. The holographic principle pictures us on a 4D surface in 5D space as I understand it. Is that contrary to the 5th D being 'rolled up'? Commented Mar 13, 2021 at 0:23

A "4D surface that exists in a higher dimensional space" is called an embedding in differential geometry. Almost all the work in differential geometry is dedicated to making statements independent of any embedding. Hence, physically, it shouldn't matter if 4D space (i.e. Einstein's description of the gravitational universe) is embedded in a higher dimensional space if we cannot experience the additional dimensions of this embedding.

Likewise, rolled-up dimensions in string theory could be understood by a corresponding embedding, but as long as this does not add any physically measurable extra information, the embedding is unnecessary. Moreover, embeddings will never be unique. You can always add another hypothetical dimension of which the mathematical surface description is independent.

The actual point about rolled-up dimensions is that you can get back to a starting point by moving in one direction only, and especially after an extremely short distance. In 4D space, however, it is not possible to move in all the same direction and finally return to where you started. At least as far as we know today.

• Our own 4Ds, are thought to bring us back to the same point we started if we travel far enough..? Commented Mar 13, 2021 at 0:13
• No, not from our current knowledge. I have seen a TV documentary where the possibility of a periodic universe has been discussed, but the conclusion was that if the universe were periodic (i.e. compactified or rolled-up in a large scale) the cosmic background radiation should show certain resonances that have never been observed. Commented Mar 13, 2021 at 0:17
• I think it depends on positive vs negative space curvature, and currently we think it's flat & accelerating towards saddle shaped + big rip long term? In this context though, we could imagine positive curvature in relation to the 'surface', as bringing a journey away from the surface so rapidly back to the same point, the extra dimensional space isn't (readily) percieved Commented Mar 13, 2021 at 0:29

Yes these are conceptually different. The 'rolled up' dimensions are generally compactified (i.e. very small). Hence, they're not immediately in conflict with observation - we only experience 4 spacetime dimensions of course. For models with branes, typically there are certain boundary conditions that seperate the bulk space from the branes. I should also add that in this case, usually these other dimensions aren't compact.

These two Wikipedia pages give a brief description of the differences: https://en.m.wikipedia.org/wiki/Randall%E2%80%93Sundrum_model https://en.m.wikipedia.org/wiki/Compactification_(physics)

• Ooh thanks for that! Have been meaning to follow Lisa Randall's work more closely, she seems one of the most innovative interesting thinkers on gravity Commented Mar 13, 2021 at 0:30
• @CriglCragl yeah she's a great physicsts in general, & the RS models dating back around 20 years are worth studying. Actually realised one of the original RS papers is called "An alternative to compactification" which gives away the fact they're different! Commented Mar 13, 2021 at 0:52