cianfa72
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- 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.
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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