When we try to find the geodesic of a partical at rest, in the second term of the geodesic equation we use dt/dtau = 1. Shouldn’t it be i (for imaginary number), since the time component of the 4-velocity vector is imaginary?
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1$\begingroup$ Imaginary time isn't really taught in modern GR. See, for instance, this Physics.SE Q&A or this Physics.SE Q&A $\endgroup$– Kyle KanosCommented Sep 2, 2022 at 21:15
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$\begingroup$ So, when we use imaginary time, vt.vt would result in a negative term. But if we use real time, vt.vt would result in a positive term. Won't that result in a wrong direction of acceleration? $\endgroup$– Nayeem1Commented Sep 2, 2022 at 21:24
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$\begingroup$ The signs are handled by the metric tensor. $\endgroup$– Kyle KanosCommented Sep 2, 2022 at 21:27
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$\begingroup$ So proper signs are handled by the Crystoffel symbols (which are derived from the metric tensor), so we don't need to worry about them in velocity components? $\endgroup$– Nayeem1Commented Sep 2, 2022 at 21:35
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$\begingroup$ Yes, that is the case, AFAIK. $\endgroup$– Kyle KanosCommented Sep 2, 2022 at 21:41
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