Definition of parallel transport

Question

The definition of parallel transport is $t^iD_i u^j=0$, where $vec{t}$ is the tangent vector to the curve and $vec{u}$ is the vector being parallel transported along the curve. In flat space, using above definition, we have $t^i partial_i u^j=0$, and usually parallel transport means $partial_i u^j=0$, which suggests that every components in flat coordinates are constant. However, from $t^i partial_i u^j=0$, it seems that as long as the change of the vector $vec{u}$ is perpendicular to the tangent vector $vec{t}$, it is also a solution. Wouldn't this lead to ambiguity about the definition of parallel transport?

benrg · Answer

There's no sum over $j$ in $t^i partial_i u^j=0$. It's four equations (in 4D) which express the constancy of every component of $u$ along the curve. $partial_i u^j=0$ is sixteen equations which express the constancy of every component of $u$ in every direction.

Answered by benrg on February 20, 2021

Javier · Answer

$t^i partial_i u^j$ is the directional derivative of $u^i$ in the direction of $t^i$. It being zero is the condition for parallel transport along a curve: you don't need all partial derivatives of $u^i$ to be zero, because you don't care about what happens if you don't move in the direction of $t^i$.
The component of the change of $u^i$ in the direction of $t^i$, however, is $t_j partial_i u^j$. Note that it's the $u$ index that's contracted with $t$, not the derivative. This is because $t$ is not telling you in which direction to take the derivative (to have a directional derivative we would need an additional $t^i$), but in which direction you're projecting the result of the derivative. So as you noted, you can have a vector field which is not parallel transported but for which this is still zero,.

Definition of parallel transport

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