- #1
Blub
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I am studying GR and I have these two following unresolved questions up until now:
The first one concerns the Levi-Civita connection. There are two conditions which determine the affine connections. The first one is that the connection is torsion-free (which is true for space-time and comes from experiment), the second one is that the parallel transport is an isometry/ the metric is invariant at each point on the manifold. My question is that why can we impose this 2. condition?
My 2. question is about the length of curves on a manifold, which is given by the formula in the pic I attached.
g is the metric tensor and v the tangent vector of gamma at the point gamma of t.
This is maybe a dumb question, but wouldn't this mean by this formula that the length of a curve depends on how fast we move through the curve? Or maybe my understanding of a tangent vector is wrong...
Any help would be greatly appreciated.
The first one concerns the Levi-Civita connection. There are two conditions which determine the affine connections. The first one is that the connection is torsion-free (which is true for space-time and comes from experiment), the second one is that the parallel transport is an isometry/ the metric is invariant at each point on the manifold. My question is that why can we impose this 2. condition?
My 2. question is about the length of curves on a manifold, which is given by the formula in the pic I attached.
g is the metric tensor and v the tangent vector of gamma at the point gamma of t.
This is maybe a dumb question, but wouldn't this mean by this formula that the length of a curve depends on how fast we move through the curve? Or maybe my understanding of a tangent vector is wrong...
Any help would be greatly appreciated.