general relativity - Is it possible to express "free"-ness of a time-like world line without referring to "tangent space" (but only directly to causal relations )?

I don't know much about tangent spaces, or tangent vectors, "as such"; nor about affine parametrization (which seems to be closely related to the notion of tangent vectors, as far as I understand for instance MTW, Box 10.2, section B).

Is it possible to explain/express whether some particular identifiable point (cmp. MTW, Box. 13.1) had been "free" (or had "moved freely"; or was "in free fall"; or was represented by a "geodesic"; etc.) without explicitly using the notion of tangent vector or affine parametrization?

I'd be especially looking for such a description being given explicitly and exclusively in terms of particular identifiable points and coincidence events in which they took part (or also, which identifiable points didn't take part in some particular coincidence event of other participants); or (equivalently, as far as I understand) in terms of whether coindicence events under consideration are time-like or space-like related to each other (or neither, i.e. light-like).

(I have already put forth some related attempts here or there. But perhaps this rephrased question helps to focus the effort ...)