I have a technical question about the geodesic equation.
Assume we have a frame $(E_{1},E_{2},E_{3},E_{4})$ (not necessarily a coordinate frame). Assume we have a parametrized curve $\gamma(s)\in M$ with the tangent vector $v\in T_{\gamma(s)}M$. Then if I want to write down the geodesic equation $\nabla_{v}v=\nabla_{v^{a}E_{a}}V^{b}E_{b}=0$ in this frame I get a term (among others) which looks like
\begin{equation*} v^{b}E_{b}(v^{a})E_{a} \end{equation*}
How exactly can I understand the $v^{b}E_{b}(v^{a})$ part? The $v^{a}=v^{a}(s)$ and $E_{b}$ acts by taking partial derivatives with respect to coordinates. So how exactly does the frame act on it?
This is how the geodesic equation is usually denoted, but one could argue it's a slight abuse of notation. We know, of course, that the equation should reduce to something like this:
$$\frac{dv}{ds} + \text{connection stuff} = 0$$
That $dv/ds$ is being denoted with a directional derivative instead. Strictly speaking, it doesn't make sense, as that directional derivative would be written with partial derivatives with respect to coordinates.
Often, to make this work, the tangent vector $v$ is extended onto an open set around the curve, so that you have a well-defined vector field that you can take a covariant derivative of. The choice of extension doesn't affect the result, so this point is often tacitly ignored.