# In the adapted coordinate system of a static 1+1D manifold do geodesic's tangent vector component depend on time?

I was trying to derive the tangent vector of an arbitrary geodesic on a static bidimensional manifold with metric $$ds^2 = F(r)dt^2 - 1/F(r)dr^2,$$ where $F$ is the square modulus of the timelike Killing vector field, and where the coordinate system is chosen in order to be adapted to the metric.

In this derivation I was asking myself if time derivatives of the tangent vector component can be put equal to zero.