advanced_tools:connections:levi_civita_connection
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
advanced_tools:connections:levi_civita_connection [2018/04/14 11:01] aresmarrero [Intuitive] |
advanced_tools:connections:levi_civita_connection [2025/03/04 00:55] (current) edi [Concrete] |
||
|---|---|---|---|
| Line 1: | Line 1: | ||
| <WRAP lag>$ \Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)$</WRAP> | <WRAP lag>$ \Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)$</WRAP> | ||
| - | ====== Christoffel Symbols ====== | + | ====== Levi-Civita Connection ====== |
| - | <tabbox Intuitive> | + | //also known as Christoffel Symbols; see also [[advanced_tools:connections]] // |
| - | Christoffel symbols $\Gamma^i_{jk}$ are a particular type of connection that a Lorentzian manifold has (called the Levi-Civita connection). | + | |
| - | A connection is a tool that we use to parallel transport tangent vectors around the manifold. | + | |
| + | <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 transport is just the simplest way to compare vectors at different points in the manifold. | ||
| - | Parallel is necssecary, for example, to define the covariant derivative. | + | Parallel is necessary, for example, to define the covariant derivative. |
| <tabbox Concrete> | <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/appendices/#riemannian_geometry|Fun with Symmetry]]. | ||
| + | |||
| + | {{:advanced_tools:metric_connect_curvature.jpg?nolink}} | ||
| - | <note tip> | ||
| - | In this section things should be explained by analogy and with pictures and, if necessary, some formulas. | ||
| - | </note> | ||
| - | |||
| <tabbox Abstract> | <tabbox Abstract> | ||
| Line 23: | Line 29: | ||
| </note> | </note> | ||
| - | <tabbox Why is it interesting?> | + | <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]]. | ||
advanced_tools/connections/levi_civita_connection.1523696493.txt.gz · Last modified: 2018/04/14 09:01 (external edit)
Page Tools
Except where otherwise noted, content on this wiki is licensed under the following license: CC Attribution-Share Alike 4.0 International
