advanced_tools:connections:levi_civita_connection
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advanced_tools:connections:levi_civita_connection [2018/04/14 10:59] aresmarrero created |
advanced_tools:connections:levi_civita_connection [2023/04/02 03:28] (current) edi [Concrete] |
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| <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 ====== |
| + | |||
| + | //also known as Christoffel Symbols; see also [[advanced_tools:connections]] // | ||
| <tabbox Intuitive> | <tabbox Intuitive> | ||
| + | The Levi-Civita connection is a mathematical tool that we use to [[advanced_tools:parallel_transport|parallel transport]] vectors around a manifold. | ||
| - | <note tip> | + | Parallel transport is just the simplest way to compare vectors at different points in the manifold. |
| - | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | |
| - | </note> | + | 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/blog/lie-groups-and-their-representations/#metric_connect_curvature|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 21: | 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.1523696388.txt.gz · Last modified: 2018/04/14 08:59 (external edit)
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