How do we parallel transport a vector?

[pellman]

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

$$\nabla_V X=0$$

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

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

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

 

[The_Duck]

Rewrite your equation using the chain rule as

$$\frac{d X^\alpha}{d \tau} = -{\Gamma^\alpha}_{\mu\nu} \frac{d x^\mu}{d\tau} X^\nu(\tau)$$

This tells you how the components of the vector $X$ change if you parallel transport it along an infinitesimal increment in the affine parameter $\tau$. You have to integrate up these changes to get the change when you change $\tau$ by a finite amount. Unfortunately the right hand side depends on $X$ so if you actually write down the integral you get an equation that only determines $X^\alpha(\tau)$ implicitly. It would be straightforward to do the integral numerically, though.

 

[WannabeNewton]

Given a manifold $M$, a derivative operator $\nabla$ on $M$, a smooth curve $\gamma: I\subseteq \mathbb{R} \rightarrow M$ with tangent $V$, and a tensor $T_0$ at some point $\gamma(s_0)$, there exists a unique tensor field $T$ on $\gamma$ such that $\nabla_{V}T = 0$ and $T(\gamma(s_0)) = T_0$ i.e. $T_0$ is parallel transported along $\gamma$ with respect to $\nabla$. This statement can be proven by choosing a set of local coordinates and invoking the uniqueness and existence theorem for ODEs.

To actually find $T$, you just solve the ODE initial-value problem you get from the parallel transport condition (the initial value being $T(\gamma(s_0)) = T_0$) by choosing a set of local coordinates. So for example in the case of a vector $X$, you have $\frac{dX^{\mu}}{ds} + \Gamma ^{\mu}_{\nu\gamma}V^{\nu}X^{\gamma} = 0$. This is an ODE that you can (in principle) solve.

 

[Bill_K]

pellman, Please note that the way The Duck writes it:
$$\frac{d X^\alpha}{d \tau} = -{\Gamma^\alpha}_{\mu\nu} \frac{d x^\mu}{d\tau} X^\nu(\tau)$$
is the ONLY way to write the condition. What you wrote:
$$\frac{dx^\mu}{d\tau}\left(\partial_\mu X^\alpha + {\Gamma^\alpha}_{\mu\nu}X^\nu\right)=0$$
is a logical impossibility. Inside the parenthesis you have written ∂_μX^α as if X^α was a field, a function of four variables, so that you could take its four-dimensional gradient. It's only a function of one variable τ, and only defined along a single worldline.

You have to integrate up these changes to get the change when you change $\tau$ by a finite amount. Unfortunately the right hand side depends on $X$ so if you actually right down the integral you get an equation that only determines $X^\alpha(\tau)$ implicitly. It would be straightforward to do the integral numerically, though.

In other words, it's not an integral, it's a differential equation.

 

[sardar]

Parallel transport is a concept in differential geometry that is used to move a vector from one point to another along a given curve while maintaining its direction and magnitude. This process is important in understanding the behavior of vectors in curved spaces, such as in general relativity.

To parallel transport a vector along a curve c(τ), we use the condition that the covariant derivative of the vector field X along the tangent vector V is equal to zero at each point along the curve. This means that the vector field X remains unchanged as it is moved along the curve.

In order to calculate the parallel transported vector X^\alpha(\tau) for a given vector X^\alpha(\tau_0) at a specific point c(\tau_0), we can use the covariant derivative formula in components as shown in the content. This formula takes into account the coordinates of the curve as well as the Christoffel symbols, which represent the curvature of the space.

The integral form of the parallel transport can then be written as an integral along the curve c(τ), with the starting point at c(\tau_0) and the end point at c(\tau). The specific form of the integral will depend on the specific curve and space being considered.

In summary, to parallel transport a vector along a given curve, we use the condition that its covariant derivative is zero and calculate the integral along the curve using the coordinates and Christoffel symbols. This process allows us to understand the behavior of vectors in curved spaces and is an important tool in the study of differential geometry and general relativity.