Is there any known parallel vector field in a Schwarzschild spacetime? Or any method to identify parallel vector fields in any spacetime, given the metric $g$?
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2$\begingroup$ Does parallel mean $\nabla_Y X=0$ for every vector field $Y$? $\endgroup$– Valter MorettiCommented Aug 13, 2020 at 17:29
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$\begingroup$ Yes, so I calculated $\nabla_X \partial t$, for example taking $X=\partial t,\partial r,\partial \theta, \partial \phi$ etc. $\endgroup$– Avik DeCommented Aug 13, 2020 at 18:13
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1$\begingroup$ I suspect that there are not such parallel fields since there are too many Killing Fields... $\endgroup$– Valter MorettiCommented Aug 13, 2020 at 18:42
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1$\begingroup$ I mean in the Schwarzshild spacetime: moving one of these vectors with the flow of the Killing Fields you should obtain four linearly independent parallel vector fields and I think it implies that the manifold is locally flat (and we know that this is false)... $\endgroup$– Valter MorettiCommented Aug 13, 2020 at 18:55
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$\begingroup$ Thank you for the comment. $\endgroup$– Avik DeCommented Aug 14, 2020 at 13:57
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1 Answer
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The relevant concept is holonomy group of a spacetime. Holonomy group of a Schwarzschild spacetime coincides with full Lorentz group, so there are no parallel vector fields. See the paper by Hall & Lonie for more general study of holonomy and Rothman et al for Schwarzschild spacetime specifically.
Also see this answer for more context.
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$\begingroup$ Thank you, I will try to understand the articles. This is very helpful. $\endgroup$– Avik DeCommented Aug 14, 2020 at 13:58