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## Main Question or Discussion Point

Given a curve c(τ) with tangent vector V, a vector field X is parallel transported along c if

[tex]\nabla_V X=0[/tex]

at each point along c. Let [itex]x^\mu(\tau)[/itex] denote the coordinates of the curve c. In components the parallel transport condition is

[tex]\frac{dx^\mu}{d\tau}\left(\partial_\mu X^\alpha + {\Gamma^\alpha}_{\mu\nu}X^\nu\right)=0[/tex]

If we are given a vector [itex]X^\alpha(\tau_0)[/itex] of the tangent space at [itex]c(\tau_0)[/itex], how do we obtain the parallel transported vector [itex]X^\alpha(\tau)[/itex] for finite [itex]\tau-\tau_0[/itex]? Clearly it will be an integral taken along c but what is the form of that integral?

[tex]\nabla_V X=0[/tex]

at each point along c. Let [itex]x^\mu(\tau)[/itex] denote the coordinates of the curve c. In components the parallel transport condition is

[tex]\frac{dx^\mu}{d\tau}\left(\partial_\mu X^\alpha + {\Gamma^\alpha}_{\mu\nu}X^\nu\right)=0[/tex]

If we are given a vector [itex]X^\alpha(\tau_0)[/itex] of the tangent space at [itex]c(\tau_0)[/itex], how do we obtain the parallel transported vector [itex]X^\alpha(\tau)[/itex] for finite [itex]\tau-\tau_0[/itex]? Clearly it will be an integral taken along c but what is the form of that integral?