equations:geodesic_equation
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equations:geodesic_equation [2018/04/14 10:59] aresmarrero [Concrete] |
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> | ||
| + | Solutions of the geodesic equations are called geodesics. | ||
| - | <note tip> | + | Geodesics are the "shortest" paths between two points in a flat spacetime and the straightest path between two points in a curved spacetime. |
| - | 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> | + | |
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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) | ||
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| <tabbox Abstract> | <tabbox Abstract> | ||
| - | <note tip> | + | Geodesics are paths $q:[t_0,t_1]\rightarrow Q$ that are critical points of the action |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | \[ |
| - | </note> | + | S(q) = \int_{t_0}^{t_1}\sqrt{g_{ij} \dot{q}^i\dot{q}^j}\,dt |
| + | \] | ||
| + | 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?> | ||
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| Almost every problem in classical mechanics can be regarded as geodesic motion. | Almost every problem in classical mechanics can be regarded as geodesic motion. | ||
| - | Especially in [[models:general_relativity|general relativity]] particles always follow geodesics. Geodesics are the straightest path between two points in a curved spacetime. | + | In the geometric optics approximation light acts like particles tracing out geodesics, i.e. the shortest paths. |
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| + | 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. | ||
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| </tabbox> | </tabbox> | ||
equations/geodesic_equation.1523696397.txt.gz · Last modified: 2018/04/14 08:59 (external edit)
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