equations:geodesic_equation
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equations:geodesic_equation [2018/04/14 11:08] aresmarrero [Intuitive] |
equations:geodesic_equation [2018/12/19 11:01] (current) jakobadmin ↷ Links adapted because of a move operation |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | Geodesics are the straightest path between two points in a curved spacetime. | + | Solutions of the geodesic equations are called geodesics. |
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| + | Geodesics are the "shortest" paths between two points in a flat spacetime and the straightest path between two points in a curved spacetime. | ||
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| + | On a sphere the geodesics are "great circles". | ||
| <tabbox Concrete> | <tabbox Concrete> | ||
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| \end{align*} | \end{align*} | ||
| where the last line follows since $g_{ik}=g_{ki}$. | where the last line follows since $g_{ik}=g_{ki}$. | ||
| - | Now we define the so-called [[advanced_notions:general_relativity:christoffel_symbols|Christoffel symbols]] | + | Now we define the so-called [[advanced_tools:connections:levi_civita_connection|Christoffel symbols]] |
| \[ | \[ | ||
| \Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr) | \Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr) | ||
| Line 66: | Line 69: | ||
| This action is exactly the proper time when $(Q,g)$ is a Lorentzian manifold, or arclength when $(Q,g)$ is a Riemannian manifold. | This action is exactly the proper time when $(Q,g)$ is a Lorentzian manifold, or arclength when $(Q,g)$ is a Riemannian manifold. | ||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| - | | + | Almost every problem in classical mechanics can be regarded as geodesic motion. |
| - | Almost every problem in classical mechanics can be regarded as geodesic motion. | + | In the geometric optics approximation light acts like particles tracing out geodesics, i.e. the shortest paths. |
| Especially in [[models:general_relativity|general relativity]] particles always follow geodesics. To be precise, a free particle in general relativity traces out a geodesic on the Lorentzian manifold, i.e. spacetime. | Especially in [[models:general_relativity|general relativity]] particles always follow geodesics. To be precise, a free particle in general relativity traces out a geodesic on the Lorentzian manifold, i.e. spacetime. | ||
equations/geodesic_equation.1523696901.txt.gz · Last modified: 2018/04/14 09:08 (external edit)
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