Short version: Is a geodesic a parametrised curve or only the image of a parametrised curve?
Long version: I am wondering if the notion of geodesic describes the concept of just a path within spacetime, or describes a parametrised curve such that the choice of parameter is important whether a curve is called geodeosic or not.
Consider e.g. a curve $\gamma : s \longmapsto \gamma(s)$ that satisfies
$$\nabla_{\dot{\gamma}}\dot{\gamma} = 0 \tag{1}$$
This is the geodesic equation and so gamma is obviously a geodesic by definition (with the definition being that it must satisfy equation (1)). Consider now instead the curve $\alpha : s \longmapsto \alpha(s)$ satisfying
$$\nabla_{\dot{\alpha}} \dot{\alpha} = f \dot{\alpha}$$
for some scalar field $f$. Then a well-known fact states that $\alpha$ can be reparametrized so that it satisfies equation (1). Since the image of the curve $\alpha$ does not change but only its parametrisation, I wonder (Q1) if it is appropriate to call $\alpha$ a geodesic and if it is (Q2) common to call it a geodesic.
