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Is there anything mathematically significant about studying a theory in $n$ dimensions or $m+1$ dimensions if $n=m+1$? For instance in the context of general relativity I hear people talk about the theory in 4 dimensions of (3+1) dimensions commonly and I am not sure about what the difference is. As far as mathematical structure is concerned, is it fair to say that the manifold background is diffeomorphic or otherwise structurally identical to a product manifold when the dimensions are written in split notation?

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When people say $3+1$ dimensions, or $n+1$ dimensions, what they mean is that the metric has $n$ space dimensions and 1 time dimension. What this means in GR is that the space is locally diffeomorphic to $n$-dimensional Minkowski space, with metric $\text{diag}(1,-1, \ldots -1)$ (or $\text{diag}(-1,1, \ldots 1)$, depending on convention).

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When people say 3 + 1 dimensions they really want to enphasize that the manifold has been foliated with a set of three dimensional spacelike surfaces parameterized by time. This is often used when you want to give initial data in numerical relativity.

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  • $\begingroup$ A spacetime isn't necessarily foliable into spacelike hypersurfaces but the term $3+1$ is still used $\endgroup$ – Slereah Feb 27 at 7:31
  • $\begingroup$ @slereah can you give a specific example please? $\endgroup$ – magma Feb 27 at 17:59

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