The MTW method has one limitation though - it assumes that all dimensions are of equal size. If that’s not the case, then the result obtained by this procedure may become scale-dependent.

We all know about compactified dimensions (ref String Theory). I’m wondering though - is the opposite possible? What I mean is - could one configure a spacetime manifold such that one of its dimensions becomes detectable only at large scales, but is hidden at smaller scales?

I can’t think of a way to do that, but would like to hear others’ opinions on this.

On 9/2/2023 at 10:46 PM, Genady said:

Imagine that you are given a smooth space of unknown geometry. What kind of constructions would you use to figure out the number of dimensions of this space?

 

I think this subject has links to your other current question about tensors so the two threads should be read together.

Consequently also the references I make to in each have material relevent to the other thread.

 

Here is a good maths text about the requirements of dimension theory,  for a variety of spaces,

The index of this book is particularly unusual as it contains potted definitions for many important terms.

 

 

A second book I am going to place here, although it also contains good material about Hilbert Spaces it it more relevant to spaces in general than visualising tensors.

 

Like in the related thread, thanks a lot for all the suggestions and thoughts.