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I’m a little bit confused about how to get the four-velocity components from a given metric tensor (or line element). For instance, which are the components of the four-velocity in the Schwarzschild metric? Can anybody help me?

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    $\begingroup$ This is because the metric does not contain any information about four-velocity! The only "connection" that I am aware of, is using the derivatives of the metric (Christoffel symbols), each multiplied by two four-vectors, to calculate geodesic motion. en.wikipedia.org/wiki/Geodesics_in_general_relativity $\endgroup$ – m4r35n357 Nov 11 '18 at 10:51
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The metric and the four velocity are not connected in such a way. For instance, you could consider a plasma around a black hole. You want to use the Schwarzschild metric. In the rest frame of the fluid you can write $u_j=\left(-\sqrt{B},0,0,0\right)$. As the plasma there is only governed by the magnetic field (in this very specific case!). You could as well use another frame and the 4-velocity would look different.

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  • $\begingroup$ But the desplacement vector is related to the metric $\endgroup$ – Joan Nov 11 '18 at 17:06

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