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Originally Posted by Tempus
As I said, I have yet to read the paper to see how the GaussīLaw and the inverse N-1 law are ruled out for small space dimensions. Other references on that would be appreciated. Has it to do with those dimensions being closed?
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Yes, it is due to the extra dimensions being compact. Take a look at
this link: the gravitational potential scales as 1/r^(d+1) for r << L and as 1/r for r >> L, for d compact dimensions of size L.
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So string theories in general state that gravity leaks through the extra dimensions. Since the leakage from other branes into ours in an invisible elf, the way to prove that would be to measure the actual leakage from our brane out to the bulk. I donīt know if it has been tried in laboratory whether it has been looked for in planetary orbits, but from your words I would imply that no such shortage in the gravity field expected has been found whithin the current error margins. Am I right?
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The difference from a non-Newtonian force becomes highly suppressed at larger distances, with corrections on the order of (L/r)^2. If there was a "large" extra dimension of size 1 micrometer, then the variation of the force between the Earth and Sun (150 million km apart) would be about one part in 10^34, well beyond what we are capable of measuring (and may ever be capable of measuring). To search for extra dimensions, you want to look at variations of Newton's Law at
small distances. However, gravity is pretty weak and it is difficult to measure it on such small scales since electromagnetic forces have such a huge effect on objects small enough to put only 1 mm or less apart (and Earth's gravity is a huge background that must be dealt with). That is why we currently only have upper limits of around mm's, even though we can see (using photons) much smaller distances (photons can be constrained to a brane and do not have to behave the same way as gravity).
I put "large" extra dimensions in quotes above because, in extra dimension papers and discussions, the "large" is relative to the Planck scale and is not associated with how we view the traditional 3 non-compact dimensions (which are obviously large). That is, an extra dimension can be 1/1000 the size of a nucleus, far smaller than the experimental limits, and would still be considered "large" because it is much larger than the Planck scale.
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Then coming back to the effect of other branes into our brane, I wonder how that small scale effect adds up to the candidates as observed phenomena (galaxy rotation, clusters) which happen to be extralarge scale without affecting the moderate large scale (planetary orbits and lab measurements).
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Because the variation gets smaller with larger distances, as noted above, the non-Newtonian effect is negligible on planetary scales, much less galactic scales.
If you have other branes near our brane that have massive objects, that is a different story. I suppose you can have gravitational interactions between particles on two different nearby branes (this has nothing to do with non-Newtonian forces), but I don't know enough about branes to say much about this.