✎ Levi-Civita Connection [The Physics Travel Guide]

You've loaded an old revision of the document! If you save it, you will create a new version with this data.

Creating directory /var/www/html/physicstravelguide.com/public_html/data/tmp/captcha/2025-04-18 failed

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

====== Levi-Civita Connection ======

//also known as Christoffel Symbols; see also [[advanced_tools:connections]] //

<tabbox Intuitive> 
The Levi-Civita connection is a mathematical tool that we use to [[advanced_tools:parallel_transport|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.
<tabbox Concrete> 
Christoffel symbols $\Gamma^i_{jk}$ are a particular type of connection that a Lorentzian manifold has (called  the Levi-Civita connection).

----

**Examples**

The diagram below shows three concrete examples for connections (Christoffel symbols) on simple 2-dimensional manifolds. For a more detailed explanation see [[https://esackinger.wordpress.com/|Fun with Symmetry]].

<note tip>
The motto in this section is: //the higher the level of abstraction, the better//.
</note>

<tabbox Why is it interesting?> 
The Christoffel symbols appear in the most important equations of general relativity: the [[equations:einstein_equation|Einstein equation]] and the [[equations:geodesic_equation|geodesic equation]].

</tabbox>

Please solve the following equation to prove you're human. 236 +7 = Please keep this field empty:

Edit summary

Note: By editing this page you agree to license your content under the following license: CC Attribution-Share Alike 4.0 International

User Tools

Site Tools

Page Tools