Parallel Transport

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tags: - colorclass/differential geometry ---Parallel transport is a fundamental concept in differential geometry, providing a method for moving vectors along curves on a manifold in a way that preserves their “direction” relative to the manifold’s geometry. It is closely associated with the notion of a connection, which offers a precise mathematical framework to describe how vectors are “transported” along paths or curves on the manifold.

Mathematical Formulation

Given a smooth manifold $M$ equipped with a connection (which can be thought of as a rule specifying how to differentiate vector fields), parallel transport along a curve $\gamma :[a,b]\to M$ is defined by the condition that the vector being transported has a covariant derivative of zero along $\gamma$. If $V(t)$ is a vector field along $\gamma (t)$, then $V$ is said to be parallel along $\gamma$ if

${\nabla }_{\dot{\gamma }(t)}V=0$

for all $t\in [a,b]$, where $\dot{\gamma }(t)$ is the tangent vector to the curve $\gamma$ at $t$, and $\nabla$ denotes the covariant derivative associated with the connection on $M$.

Properties and Significance

- Path Dependence: Unlike in Euclidean Space, parallel transport on a curved manifold is generally path-dependent. That is, the result of transporting a vector from one point to another may depend on the path taken between the points. This path dependence is a manifestation of the manifold’s curvature.

- Geodesics: Geodesics, the generalization of straight lines to curved manifolds, can be characterized as curves along which the tangent vector is parallel transported along itself. This means that if $\gamma (t)$ is a geodesic, then ${\nabla }_{\dot{\gamma }(t)}\dot{\gamma }=0$.

- Curvature: The failure of parallel transport to preserve the orientation of vectors after being transported around a closed loop is directly related to the curvature of the manifold. The Riemann curvature tensor, a fundamental object in differential geometry, quantifies this failure and provides a measure of the manifold’s intrinsic curvature.

- Applications in Physics: Parallel transport plays a crucial role in general relativity, where the connection (specifically the Levi-Civita connection) is determined by the spacetime metric, and parallel transport reflects the influence of gravity on the motion of particles and light. In gauge theories of particle physics, parallel transport is related to the concept of gauge Covariant Derivatives, which describe how fields change under local gauge transformations.

Example: Parallel Transport on a Sphere

A classic example of parallel transport is moving a vector along the surface of a sphere. If a vector is parallel transported along the equator, then moved up to the north pole along a meridian, then transported back along another meridian to the equator, and finally moved along the equator back to its starting point, the vector will have rotated with respect to its original orientation. This illustrates the path dependence of parallel transport and is a manifestation of the sphere’s curvature.

Parallel transport thus serves as a key concept in understanding the geometry of curved spaces and the dynamics of fields and particles within those spaces, bridging the realms of pure mathematics and theoretical physics.