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## Summary:

- Parallel transport vs Lie dragging when 'transporting/dragging' a curve on a metric manifold along a killing vector field

## Main Question or Discussion Point

Hi,

I would like to ask for a clarification about the difference between parallel transport vs Lie dragging in the following scenario.

Take a vector field ##V## defined on spacetime manifold and a curve ##C## on it. The manifold is endowed with the metric connection (I'm aware of it does exist just one unique torsion-free connection compatible with the metric tensor ##g_{\mu\nu} ##: the connection such that ##\nabla g=0##).

The curve ##C## itself defines a vector field thus we can Lie drag or parallel transport it along the flow of vector field ##V## (##V## congruence curves).

From my understanding for any ##V## we have ##\mathcal L_V V =0## meaning that Lie dragging the vector field ##V## itself from point P to point Q along its congruence curve results in a vector tangent to the congruence curve at Q.

Said that take now the vector field ##V## to be a killing vector field. In that case I believe the Lie dragging along its flow preserves the inner product between the ##C## tangent vector at P and vector ##V(P)## (basically the vector field ##V##

Consider now the parallel transport along ##V## congruence curve according to the metric connection. For any such curve it preserves the inner product between parallel transported vectors nevertheless the 'parallel transported' tangent vector to the congruence curve at P could be different from tangent vector to the congruence curve at Q (namely when the curve is

To fix that we can use the Fermi-Walker transport.

Does it make sense ?

I would like to ask for a clarification about the difference between parallel transport vs Lie dragging in the following scenario.

Take a vector field ##V## defined on spacetime manifold and a curve ##C## on it. The manifold is endowed with the metric connection (I'm aware of it does exist just one unique torsion-free connection compatible with the metric tensor ##g_{\mu\nu} ##: the connection such that ##\nabla g=0##).

The curve ##C## itself defines a vector field thus we can Lie drag or parallel transport it along the flow of vector field ##V## (##V## congruence curves).

From my understanding for any ##V## we have ##\mathcal L_V V =0## meaning that Lie dragging the vector field ##V## itself from point P to point Q along its congruence curve results in a vector tangent to the congruence curve at Q.

Said that take now the vector field ##V## to be a killing vector field. In that case I believe the Lie dragging along its flow preserves the inner product between the ##C## tangent vector at P and vector ##V(P)## (basically the vector field ##V##

*evaluated*at P) when both 'dragged' at Q.Consider now the parallel transport along ##V## congruence curve according to the metric connection. For any such curve it preserves the inner product between parallel transported vectors nevertheless the 'parallel transported' tangent vector to the congruence curve at P could be different from tangent vector to the congruence curve at Q (namely when the curve is

*not*a geodesic according ##g_{\mu\nu} ##).To fix that we can use the Fermi-Walker transport.

Does it make sense ?

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