I have a question regarding the Coulomb interaction in spaces with non-trivial topology.(adsbygoogle = window.adsbygoogle || []).push({});

Suppose we have D large spatial dimensions (D>2). Then the Coulomb potential is V_{C}(r) ~ 1/r^{D-2}. Usually one shows in three dimensions that the Coulomb potential V_{C}(r) is nothing else but the Fourier transform of G(k) ~ 1/k² which is a Greens function for a massless particle; with a mass-term there would be an additional factor exp^{-mr}.

Now suppose that we do not have flat space but that space is compactified. The simplest example is a 3-torus with size L. Then one charge located at r=0 "feels" another charge located at r as if it were located at r, r + Le_{i}, r + 2Le_{i}, ... this is equivalent to say that the potential must be periodic on the 3-torus, i.e. it must respect the condition

V(r) = V(r + Le_{i})

One can discretize the theory by using Fourier modes on the 3-torus T^{3}= S^{1}* S^{1}* S^{1}. Then the Coulomb potential V(r) can be derived from G(k) via a discrete Fourier series where one sums over 1/k² where k respects the periodicity of the 3-torus.

So far so good.

What happens if

1)the topology of space (spacetime) becomes more complex?There are e.g. speculations that our universe could have the topology of a dodecahedral space which could explain suppressions of CMB multipole moments.

2)the geometry of space becomes dynamic?In GR the geometry of space is not fixed; typically space(time) will expand.

I have no idea how a mode decomposition in a dodecahedral space would look like. I have no idea how this could affect the Coulomb interaction at early times (for a small universe). I have no idea how the topology of an expanding universe would restrict the mode decomposition.

Are there any hints how all this could affect such a simple law as the Coulomb interaction?

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# Coulomb interaction and non-trivial topology

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