general relativity - What are world lines as opposed to arbitrary curves in spacetime?

In GR the spacetime manifold is equipped with a metric which makes it a Lorentzian manifold. It is the metric that is doing the separation of space and time (so that we end up with three dimensions of space and one of time for each spacetime event covered by the metric). Is that correct?

If so: I am trying to understand what kind of curve in spacetime stands as a world line. I know of world lines as timelike curves. But since the metric is needed to tell apart null, spacelike and timelike curves, defining a world line this way does not seem straightforward. Is there a metric-independent way to define what a world line is?

For example which geodesics are world lines and which are not?

Answer

Yes the metric does the separation of space and time. Without the metric you just have a 4d manifold with 4d points called events and a topology (a sense of small regions of spacetime). You can even have curves and they can have tangents.

But once you have the metric then a 4d curve can have tangents that can be identified as spacelike or timelike or lightlike. And now you can ask whether your manifold is Lorentzian or not.

It is the metric that is doing the separation of space and time (so that we end up with three dimensions of space and one of time for each spacetime event covered by the metric). Is that correct?

Yes.

Is there a metric-independent way to define what a world line is?

Absolutely not. Consider the manifold $$M=\{(a,b,c,d):a,b,c,d\in\mathbb R\},$$ and the curve $t\mapsto (t,0,0,0)$.

Either of the two metrics $$\mathrm ds^2=\mathrm da^2-\mathrm db^2-\mathrm dc^2-\mathrm dd^2,$$ and $$\mathrm ds^2=\mathrm db^2-\mathrm da^2-\mathrm dc^2-\mathrm dd^2,$$ make it a Lorentzian manifold. But the curve $t\mapsto (t,0,0,0)$ is a worldline (has all tangents be timelike) with the first metric and the curve $t\mapsto (t,0,0,0)$ is not a worldline (has nontimelike tangents) in the second metric.

Whether or not something is a worldline depends on the metric. That's life.

For example which geodesics are world lines and which are not?

The curves (differentiable maps from parameters into the manifold) that have a timelike tangent at every 4d point (event) along the curve are the worldlines. The other ones are not worldlines.

But even being a geodesic depends on the metric. So if you had two metrics on the same manifold they could disagree on whether a curve is a geodesic.

That said, when someone hands you a Lorentzian manifold they hand you a manifold and a metric. So you always have a metric.