*Here’s a short guest post by my old friend Anton. As usual, please feel free to discuss the paradox through the comments box!*

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I thought of a physics paradox the other day and Peter has kindly granted me a guest post here about it, as follows. Consider a homogeneous isotropic closed universe as described by general relativity. Let it contain a uniform density of a single species of electrically charged particle, so that this universe has a net charge. The charged particle density is sufficiently low, however, that the perturbation from the regular uncharged metric is negligible. Since this universe is homogeneous and isotropic the electric field in it is everywhere zero. BUT if I consider a conceptual 3-dimensional sphere, small enough for space-time curvature to be neglected, then it contains a finite amount of electric charge, and therefore by Gauss’ theorem a nonzero electric field points out of it at every point on its surface. This contradicts the zero-field conclusion based on the metric.

Here are three responses (one my own) and my further responses to these, in brackets:

- In a closed universe it is not clear what is the outside and what is the inside of the sphere, so Gauss’ law is not trustworthy (tell this to a local observer!);
- the electric field lines due to the charges inside this (or any) conceptual sphere wrap round the universe an infinite number of times (this doesn’t negate Gauss’ theorem!);
- the curved rest of the Universe actually adds a field that cancels out the field in your sphere (neither does this negate Gauss’ theorem!)