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

Question

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 ...)

Answer 1

Since I'm in a car right now I can't double check my answer, but I'll try my best not to lie to you.

To begin with I am a little unclear as to what you are asking. In most circumstances (i.e if the region under consideration is geodesically complete, which in physics is the case if there is no singularity present) any two points can be connected by a geodesic, and to any one point and direction with magnitude (tangent vector) there is a geodesic that passes through that point with the same direction and magnitude (velocity).

Now I believe that you are asking whether the motion of some body is that of a freely falling body, and specifically whether or not there is some description of the answer that does not specifically mention tangent vectors or affinely parametrized curves. The best I can do is that the body is in free fall if and only if the proper time between any two points on its world line is (locally) maximized. That is to say if any slightly different path that it would have followed would have produced a smaller proper time interval between the two points. The local part (slight variations) is there because there may be more than one geodesic connecting the same two points.

Of course, mathematically the proper time is defined through affinely parametrized curves, but at least it's a description of the physics behind the answer without explicit reference to the mathematics. I believe most of us can have a sort of intuitive understanding of proper time from introductory special relativity.

Answer 2

A spacetime is a manifold with two different, and technically independant, structures : a metric, which describes the distances between two points, and a connection, which describes how to transport a vector "straight". In general relativity, there is a formula to link the two, but at their core, the two notions are independant.

A free particle is one that follows a geodesic, hence that notion depends on the connection. The causal structure, on the other hand, depends on the metric. Even worse, it's a notion that loses informations, since the causal structure can be the same for different metrics (basically it loses the notion of distances and only keeps the information on whether or not the two points are timelike or null separated). So you can't even exploit the general relativity link between metric and connection (a geodesic will minimize the length of the curve as given by the metric), since we have also lost the notion of distances.

So I'd say that no, the causal structure by itself isn't enough to determine if a curve is a geodesic. To convince yourself of it, take a geodesic curve, apply a Weyl transformation to the metric ($g(x) \rightarrow e^{\omega(x)} g(x)$), as those preserve the causal structure, and see how the geodesic equation changes. The curve will most likely cease to be a geodesic.