Question: Finding 4-vector velocity in different metric form for comoving and independent observer

Hello guys,

Assume we have metric ds^2=a(r,t)*dt^2-b(r,t)*dr^2-r^2*dtheta^2-r^2*sin^2(theta)*dphi^2 wherein a(r,t) and b(r,t) are two arbitrary metric functions. how we can find 4-vector velocity for comoving and independent observers?

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