I've been very impressed to learn about kaluza-klein theory and compactification strategies. I would like to read more about this but in the meantime i'm curious about 2 different points. I have the feeling that there are no precise answers to these questions at the current moment but nonetheless i feel obligued to ask:
- Does a compactification strategy has to impose a fixed thickness in the compactification or this arises as a dynamic consequence of the evolution equations? and if i start with a small thickness, does the dynamic in all cases keeps the thickness in a bounded value? are there results that addresses this?
EDIT answer appears to be No, it is not assumed to be fixed. There is an effective scalar field that precisely describes this.
- Given that the argument is that vibrations in the compactified dimensions determine the internal states, any significant gradient ratio in the compactification length would have to be probably measurable as either spontaneous decay or spontaneous stimulation events of particles. My question is, would experimental effects of any thickness gradients would be as difficult to measure as expected effects from a fixed but non-zero compactified dimension?
EDIT a better rephrase of this question is that; since energy of non-gravitational fields like electromagnetism, etc. is stored in vibrations along the compactified dimensions, it would seem that even a small gradient in the radion field (even 0.1%) should be measurable as a effective diffraction index (since a 0.1% variation in the length of the scale would affect each non-gravitational mode by that amount)
I guess even another way to frame the question is: is the low-energy phenomena we are able to see currently, invariant/non-dependent under small changes of scale (from point to point in space-time) of the compactification? isn't scale of compactification itself a vibrating degree of freedom?
So bottom line: since we haven't see any low-energy consequences from small gradients in the radion field, can't we infer that the radion field, is, by whatever reasons (dynamics, broken symmetry, etc.) effectively fixed? can't we currently estimate a bound in the radion gradient given the negative results mentioned above it would have?
I hope i've made my questions clear and interesting enough.