I am working through Paul Renteln's book "Manifolds, Tensors, and Forms" (As I am learning General Relativity). I have come across a derivation of a 'parallel transport equation':
$$\frac{d\gamma^i}{dt}\left(\frac{\partial Y^k}{\partial x^i}+\Gamma^k_{ij}Y^j\right)=0,$$
Definition of parallel transport:
(I have only included this so you know what the variables used are referring to)
Let $I\subseteq \mathbb{R}$ be an interval, and let $\gamma:I\rightarrow M$ be a parameterized curve in M with tangent vector field $X=\gamma_*(d/dt)$. Let $Y$ be any other vector field on $M$. We say that $Y$ is parallel transported along $\gamma$ if $\nabla_{X}Y=0.$
The nearly-full derivation is as follows:
Suppose that $X=X^i \partial_i$ and $Y=Y^j \partial_j$. Then:
$$\nabla_X Y=\nabla_X(Y^j\partial_j),$$
$$=X(Y^j)\partial_j+Y^j\nabla_X\partial_j,$$
$$=X(Y^j)\partial_j+Y^j\nabla_X\partial_j,$$
$$X(Y^j)\partial_j+Y^jX^i\nabla_{\partial_i}\partial_j,$$
$$=\left[X(Y^k)+\Gamma^k_{ij}X^i Y^j\right]\partial_k.$$
Hence, our condition for parallel transport $\nabla_X Y=0$, holds provided that:
$$X(Y^k)+\Gamma^k_{ij}X^iY^i=0.$$
Using the notion of integral curves we have:
$$X^i=X(x^i)=\gamma_*\left(\frac{d}{dt}\right)(x^i)=\frac{d}{dt}(x^i \circ \gamma)=\frac{d\gamma^i (t)}{dt},$$
So we write our previous expression as:
$$\frac{d}{dt}Y^k+\Gamma^k_{ij}\frac{d\gamma^i}{dt}Y^j=0,$$
or:
$$\frac{d\gamma^i}{dt}\left(\frac{\partial Y^k}{\partial x^i}+\Gamma^k_{ij}Y^j\right)=0,$$
My Problem
My problem is regarding the first step (perhaps not directly related to parallel transport), when the following choices were made:
$$X=X^i \partial_i,$$
$$Y=Y^j \partial_j,$$
I am completely oblivious as to what this means/why they have been chosen? Would these be different if we were considering perhaps covariant components of a vector field $Y$? I am relatively new to this level of maths (and am a physicist) so am struggling to visualise what this means, I have also been given a question for homework regarding the parallel transport of the covariant component of a vector, so am wondering if I need to perhaps perform a different substitution for $Y$ and re-derive?
Any help is appreciated, Thanks! 🙂
Best Answer
As per @Mark Wildon 's comment, the problem in question was not a choice, but an expression of the vector field $X$ as a linear combination of the standard basis of vector fields, thus for covariant components it would naturally look as such: $$Y=Y_j\partial^j,$$