Lecture 08 - Parallel Transport

#math

Definition

A vector field $X$ on $M$ is said to be parallely transported along a smooth curve $γ : R \to M$ if:

\nabla_{v_{γ}} X = 0.

Definition

A slightly weaker condition than "parallely transported" is "parallel":

(\nabla_{v_{γ, γ (λ)}} X)_{γ (λ)} = μ (λ) X,

for $μ : R \to R$ .

The difference between the two is imagine Pinocchio moving around but always facing the same direction (lets say north). If Pinocchio doesn't lie then the vector defined by his nose is parallely transported (if he does lie then its only parallel).

Autoparallely tansported curves

Definition

A curve $γ : R \to M$ is called autoparalelly transported if:

\nabla_{v_{γ}} v_{γ} = 0.

Remark: A weaker notion of autoparallel curve (normally in literature we mean the previous definition so one may ignore this):

\nabla_{v_{γ}} v_{γ =} μ v_{γ} .

Autoparallel Equation

Consider the portion of the autoparallel transported curve $γ$ that lies in $U$ , where $(U, x) \in A$ . We would like to express $\nabla_{v_{γ}} v_{γ} = 0$ in terms of chart representatives.

Recall that $v_{γ} = {\dot{γ}}_{(x)}^{m} (\frac{\partial}{\partial x^{m}})$ and $γ_{(x)}^{m} = x^{m} \circ γ$ . So:

\begin{aligned} \nabla_{v_{γ}} v_{γ} & = \nabla_{{\dot{γ}}_{(x)}^{m} (\frac{\partial}{\partial x^{m}})} {\dot{γ}}_{(x)}^{n} (\frac{\partial}{\partial x^{n}}) \\ = {\dot{γ}}_{(x)}^{m} \frac{\partial {\dot{γ}}^{n}}{\partial x^{m}} \frac{\partial}{\partial x^{n}} + {\dot{γ}}^{m} {\dot{γ}}^{n} Γ_{n m}^{q} \frac{\partial}{\partial x^{q}} \\ = {\dot{γ}}_{(x)}^{m} \frac{\partial {\dot{γ}}^{q}}{\partial x^{m}} \frac{\partial}{\partial x^{q}} + {\dot{γ}}^{m} {\dot{γ}}^{n} Γ_{n m}^{q} \frac{\partial}{\partial x^{q}} \\ = {\ddot{γ}}_{(x)}^{m} \frac{\partial}{\partial x^{q}} + {\dot{γ}}^{m} {\dot{γ}}^{n} Γ_{n m}^{q} \frac{\partial}{\partial x^{q}} \end{aligned}

That is in summary:

{\ddot{γ}}_{(x)}^{m} + {\dot{γ}}^{a} {\dot{γ}}^{b} Γ_{a b}^{m} = 0,

chart expression of the condition of the condition that $γ$ be autoparallely transported.

Examples:
a) Euclidean plane, $U = R^{2,} x = {id}_{R^{2}}, Γ_{(x)}^{i}_{j l} = 0 ⟹$ geodesics satisfy ${\ddot{γ}}^{m} = 0$ , that is the second derivative of the curve is zero (the geodesics are straight lines in the traditional sense).
b) Round sphere $(S^{2}, O, A, \nabla_{O})$ . Consider the chart: $x (p) = (θ, φ)$ . Where $Γ^{1}_{22} (x^{- 1} (θ, φ) = - s i n (θ) c o s (θ)$ and $Γ^{2}_{21} = Γ^{2}_{12} = c o t (θ)$ . Let $x^{1} (p) = θ (p)$ and $x^{2} (p) = φ (p)$ .
Then our autoparallel equation:

\begin{array}{r} \ddot{θ} + Γ^{1}_{22} \dot{φ} \dot{φ} = 0 \\ \ddot{φ} + 2 Γ^{2}_{21} \dot{θ} \dot{φ} = 0 \\ ⟺ \\ \ddot{θ} - s i n (θ) c o s (θ) \dot{φ} \dot{φ} = 0 \\ \ddot{φ} + 2 c o t (θ) \dot{θ} \dot{φ} = 0 \end{array} .

A solution to this is $θ (λ) = \frac{π}{2}$ and $φ (λ) = ω λ + φ_{0}$ where $ω \in R$ (which is a point on the equator moving at constant speed). i.e the solutions are great circles on the sphere.

Torsion

Can one use $\nabla$ to define tensors on $(M, O, A, \nabla) ?$

Definition

The torsion of a connection $\nabla$ is the $(1, 2)$ -tensor field

T (ω, X, Y) := ω (\nabla_{X} Y - \nabla_{Y} X - [X, Y]) .

Reminder: $[X, Y] (f) = X (Y f) - Y (X f)$ , where $f \in C^{\infty}$ .

Definition

A $(M, O, A, \nabla)$ is torsion free if $T = 0$ everywhere.
In a chart:
$T^{i}_{a b} := T (d x^{i,}, \frac{\partial}{\partial x^{a}}, \frac{\partial}{\partial x^{b}}) = . . = Γ^{i}_{a b} - Γ^{i}_{b a} = 2 Γ^{i}_{[a b]}$

From now on we only use torsion free connections.

Curvature

Definition

The Riemann curvature of a connection $\nabla$ is the $(1, 3)$ -tensor field,

R (ω, Z, X, Y) := ω (\nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z) .

Algebraic relevance of the Riemann curvature tensor:
$\nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z = R (\cdot, Z, X, Y) + \nabla_{[X, Y]} Z$
in a chart ( $U, x$ ) :
$(\nabla_{a} \nabla_{b} Z)^{m} - (\nabla_{b} \nabla_{a} Z)^{m} = R^{m}_{n a b} Z^{n} + \nabla_{[\frac{\partial}{\partial x^{a}}, \frac{\partial}{\partial x^{b}}]} Z = R^{m}_{n a b} Z^{n} .$
So we cannot swap partial derivatives if the space is not flat! Moreover this computation of the Riemann curvature tensor relates to Walds here.