Notion of parallel worldlines in curved geometry

[cianfa72]

TL;DR

About the notion "to be parallel" in the context of curved geometry

The notion of spacetime curvature is just the same as geodesic deviation. Therefore take for instance two bodies at different altitudes from Earth surface. In order to evaluate their geodesic deviation the two worldlines must start parallel in spacetime (actually in tangent spaces at both initial points/events).

We know in curved spacetime the notion of "to be parallel" is path dependent and it is given by the affine connection assigned to the spacetime as metric manifold (i.e. the Levi-Civita connection derivated from the metric tensor $g$ is normally used).

That said, which is the relevant/implied path to say that the two worldlines start parallel ? I believe it is the geodesic path one gets exponentiating the (spacelike) vector orthogonal to the first worldline's 4-velocity at its starting point that intersect the other worldline at its starting point.

 

[Dale]

TL;DR Summary: About the notion "to be parallel" in the context of curved geometry

That said, which is the relevant/implied path to say that the two worldlines start parallel ?

Usually you can only make such a statement when those two worldlines are initially close enough that you can treat them as being in a flat spacetime. Then it is unambiguous.

 

[cianfa72]

Usually you can only make such a statement when those two worldlines are initially close enough that you can treat them as being in a flat spacetime. Then it is unambiguous.

Ok. Does the above definition of parallelism apply also to non geodesic worldlines as well?

Take for instance two concentric circles on the euclidean plane. From any point on the bigger one, draw the orthogonal straight line to the other circle getting a point.The tangents to circles on both points stay always parallel.

 

[Dale]

Does the above definition of parallelism apply also to non geodesic worldlines as well?

I have never seen a definition of parallel that would apply to non-geodesics. There might be one, but I don't know of it.

 

[cianfa72]

Does the definition of "to be parallel" for geodesics also demand that the distance/lenght along the orthogonal geodesics between them at any point stays the same?

See also Clifford parallel.

 

[cianfa72]

I believe the definition of "to be parallel" for geodesics in the context of non-euclidean geometry actually doesn't demand that the (minimum) distance between them must stay constant.