Consider a closed causal curve $\gamma(t)$, $\gamma(0) = \gamma(1)$, with $\gamma'$ causal everywhere. If we take this curve to be contractible (that is, homotopic to a single point), can it be contained entirely within a single coordinate patch to a simply connected subset of $\Bbb R^n$? Basically is there a coordinate patch $$\phi : U \to \Bbb R^n$$
such that $\gamma \in U$?
I know this is not true for all curves, the simplest example being $\Bbb R^2 \setminus \{0\}$, with a curve $\gamma = \theta$ (a circle around the point), then taking the concatenation $\gamma * \gamma^{-1}$, which is contractible to a point but can't be contained within a simply connected single chart of $\Bbb R^2$. This example won't affect the treatment of causal curves, as even for a piecewise causal curve, this would involve a reversal of time orientation at the corner.
There are slightly smoother examples one can construct, for instance
I think that example as well would involve some bad business with the time orientation near the intersection point, though I'm not 100% sure.
What would be a proof that a contractible closed causal curve is part of a single simply connected chart, if true, then?
