Notion of parallel worldlines in curved geometry

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Discussion Overview

The discussion revolves around the concept of parallel worldlines in the context of curved spacetime geometry. Participants explore the implications of geodesic deviation, the definition of parallelism, and the conditions under which worldlines can be considered parallel, particularly in non-Euclidean geometries.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that the notion of spacetime curvature relates to geodesic deviation and questions the relevant path for determining parallelism between worldlines.
  • Another participant notes that the statement of parallelism is generally applicable when worldlines are sufficiently close to each other, allowing for a flat spacetime approximation.
  • A question is raised about whether the definition of parallelism applies to non-geodesic worldlines, using concentric circles as an example.
  • One participant expresses uncertainty about the existence of a definition of parallelism that applies to non-geodesic worldlines.
  • A further inquiry is made regarding whether the definition of parallelism for geodesics requires that the distance along orthogonal geodesics remains constant.
  • Another participant asserts that in non-Euclidean geometry, the definition of parallelism for geodesics does not necessitate that the minimum distance between them remains constant.

Areas of Agreement / Disagreement

Participants express differing views on the applicability of parallelism to non-geodesic worldlines and the conditions under which worldlines can be considered parallel. The discussion remains unresolved regarding the definitions and implications of parallelism in various contexts.

Contextual Notes

Participants reference specific geometric concepts and conditions, such as the Levi-Civita connection and the treatment of worldlines in flat spacetime, which may depend on the assumptions made about the spacetime structure.

cianfa72
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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.
 
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cianfa72 said:
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.
 
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Dale said:
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.
 
cianfa72 said:
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.
 
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.
 
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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.
 
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