User Tools

Site Tools


Sidebar


Add a new page:

advanced_tools:connections:levi_civita_connection

$ \Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)$

Levi-Civita Connection

also known as Christoffel Symbols; see also Connections

Intuitive

The Levi-Civita connection is a mathematical tool that we use to parallel transport vectors around a manifold.

Parallel transport is just the simplest way to compare vectors at different points in the manifold.

Parallel is necessary, for example, to define the covariant derivative.

Concrete

Christoffel symbols $\Gamma^i_{jk}$ are a particular type of connection that a Lorentzian manifold has (called the Levi-Civita connection).

Abstract

The motto in this section is: the higher the level of abstraction, the better.

Why is it interesting?

The Christoffel symbols appear in the most important equations of general relativity: the Einstein equation and the geodesic equation.

advanced_tools/connections/levi_civita_connection.txt · Last modified: 2018/04/14 11:51 by theodorekorovin