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I'm interested in a frame bundle over spacetimes with different topologies. In the trivial case of Minkowskian space ($\mathbb{R}^{3,1}$), a frame (or tangent space) at one point is going to be related to a frame (or tangent space) at another point via an $SO(3,1)$ rotation. Hence we get a spin connection valued in that group. In fact we can always reduce the structure group locally to this for a Lorentzian manifold (ie spacetime). I'm more interested in global structures on spacetimes.

Let's consider a spatial slice in a closed Friedman-Lemaitre-Robertson-Walker type spacetime, which has a spatial $S^{3}$ topology. A frame (or the tangent space) at an arbitrary point on the space is related to a frame (or the tangent space) at another point via an $SO(4)$ rotation, and any spatial triad (or tangent spaces) could be thus related.

Considering a full spacetime then with topology $S^{3}\times\mathbb{R}$ , wouldn't the structure group for a parallelization necessarily be $SO(4,1)$? In other words wouldn't the covariant derivative in an oriented orthonormal frame (the parallelization) necessarily have an $SO(4,1)$ spin connection? Which itself would reduce to the standard $SO(3,1)$ spin connection in the neighborhood of a point? If you want to formulate your answer in terms of fiber bundles that's totally fine! (:

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Are you trying to embed the de Sitter group $SO(4,1)$ into your theory one way or another? If so, you may refer to the MacDowell–Mansouri version of gravity here, where the expanded spin connection on the de Sitter group $SO(4,1)$ encompasses both the original $SO(3,1)$ spin connection and the coset $SO(4,1)/SO(3,1)$ tetrad.

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  • $\begingroup$ +1 thank you! I'm actually not going quite that route, I'm actually just wondering if different topologies in general affect the spin connection when it's over a finite area. For example a closed-cyclic FLRW universe would have something like an $S^{3}xS^{1}$ topology which then by my reasoning would have an $SO(4,2)$ connection which upon reduction locally to the Lorentz group would have to bear traces of that former symmetry (since the Lorentz group locally would be made of broken generators from the initial group. I am trying to stay within General Relativity though, if possible. $\endgroup$
    – R. Rankin
    Feb 8, 2021 at 2:22
  • $\begingroup$ I was thinking if those groups describe the spacetime by relating the tangent space at each point on the manifold then the connection is a position determined transformation valued in that group as viewed from the embedding space? $\endgroup$
    – R. Rankin
    Feb 8, 2021 at 2:24
  • $\begingroup$ "would have something like an $S^3 x S^1$ topology which then by my reasoning would have an $SO(4,2)$ connection", I am not sure about the $S^1$ part, since you may prefer not going down the closed-time-loops rabbit hole. As for $SO(4,2)$ conformal gravity, there are some attempts, see e.g. : journals.aps.org/prd/abstract/10.1103/PhysRevD.51.1674 $\endgroup$
    – MadMax
    Feb 8, 2021 at 17:17
  • $\begingroup$ Thanks! I agree about the closed timelike loops! In reality, it's really an approximation, by relating two points of full contraction to one another and saying they're the same. Since that state is alleged to be in thermodynamic equilibrium, they should be relatable up to fluctuations. A spatial anology would be a waveguide made of several hollow metal spheres aranged linearly connected by tiny holes where they touch. Away from the hole, one might approximate the normal modes as though it were a sphere, though at high frequencies the size of the hole to come into play. Does that sound Legit? $\endgroup$
    – R. Rankin
    Feb 8, 2021 at 21:17
  • $\begingroup$ I suppose my question is if the mathematics is correct, that these are the groups that naturally correspond to the connection in these different topologies, if you can address this I'd happily accept your answer. $\endgroup$
    – R. Rankin
    Feb 8, 2021 at 23:08

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