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Question: Starting from an arbitrary spacetime metric, how does one obtain a tangent vector field? (We might need to assume certain geodesic congruences but my understanding is very limited.)

Build up:

My aim is to see that the shear and twist in FLRW metric is zero. For this, I am looking at the Raychaudhuri equation and it involves a certain tensor $B_{\alpha \beta} = u_{\alpha; \beta}$ where $u_{\alpha}$ is a tangent vector field. It is given that $u_\alpha = - \partial_{\alpha} t$, but how do I get it?

If it is possible, please outline the procedure (and reference) to start from any general metric tensor.

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