# How do we assign length to dimensions in string theory?

My current understanding of the variations of string theory includes the sentence "string theory needs (at least) 6 extra spatial dimensions to work, but because our observable universe consists of 3 spatial ones, we could assume they're very tiny".

Now, when we deal with our ordinary 4 dimensions, we don't consider them as finite, you're not gonna run out of space as you move forward in space. (I know this could be paradoxical if the universe actually has an edge, but we don't know that.) So how do we assign length to those "extra" dimensions? I know objects have lengths in their dimensions, but does the fabric of spacetime itself have it? Is this consistent with Einstein's theory of relativity?

one more question popped into my head, sorry

About considering them as tiny, couldn't they be just as infinite as the other three but because everything currently known to us is at most 3D we have no idea of what could be going on in other spatial dimensions? Or are we somehow sure that there exists no more of them?

Ok sorry this is the last one

To the extent of our knowledge, could the universe contain more dimensions that have very little to do with space or time? If yes, do we still have to consider those extra dimensions as spatial?

• The hand-wavy answer that I've heard is that it basically has to be the Planck length, since there is no other length that is "special." Maybe a string theorist could give a more rigorous discussion. – Ben Crowell Oct 24 '18 at 20:32
• Interesting read: arxiv.org/abs/gr-qc/0403053v4 – Avantgarde Oct 24 '18 at 23:31

Let me change the order of your questions a bit:

About considering them as tiny, couldn't they be just as infinite as the other three but because everything currently known to us is at most 3D we have no idea of what could be going on in other spatial dimensions?

The existence of extra dimensions affects the physics that we see in our 3+1 dimensional world, and we have no experimental evidence for such infinite dimensions. However, if the extra dimensions are tiny enough we will not be able to detect them with current experiments.

I should note that there are theories of large extra dimensions, where the Standard Model is located on and restricted to a 1+3 dimensional subspace (a brane more precisely), but even then gravity would still propagate in these large extra dimensions and alter the physics we observe.

So how do we assign length to those "extra" dimensions?

Since large extra dimensions are not observed, one compactifies the extra dimensions. As the simplest example, a single extra dimension could be curled up as a circle, which has a characteristic length given by the radius $$R$$. In most scenarios this length would be of the order of the Planck length. In general to obtain models closer to that of the Standard model, the extra dimensions are compactified on more complicated geometries (Calabi-Yau manifolds).

To the extent of our knowledge, could the universe contain more dimensions that have very little to do with space or time? If yes, do we still have to consider those extra dimensions as spatial?

I don't think this makes much sense. You want to add dimensions, which are then NOT dimensions? So in the end you are asking for the consequence of something you haven't defined. To add to this, the dimensions of string theory are not just something you add by hand as you like. They are required for consistency of the theory.

• About the circular dimensions, I can't really wrap my head around them, because I'm not sure exactly how the fabric of spacetime is associated with dimensions, like, what happens to that circular dimension when it gets close to a black hole? Are their points defined in polar coordinates? (And yes, you're right about the last question, it doesn't make much sense, I don't know what I was thinking.) – Erfan Abedi Oct 25 '18 at 18:17
• @ErfanAbedi I think you might be overthinking the extra dimensions. All spatial dimensions are on completely equal footing initially, and we just choose a compact topology for some and non-compact for 3 spatial dimensions in accordance with what we observe. For a dimension wrapped up as a circle, we can define a coordinate on an interval with the endpoints identified. – Sparticle Oct 25 '18 at 23:16