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equations:geodesic_equation [2018/04/14 11:09] aresmarrero [Why is it interesting?] |
equations:geodesic_equation [2018/05/05 22:00] jakobadmin [Concrete] |
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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. |
+ | |||
+ | 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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&= m\sqrt{g_{ij}\dot{q}^i\dot{q}^j} | &= m\sqrt{g_{ij}\dot{q}^i\dot{q}^j} | ||
\end{align*} | \end{align*} | ||
- | just like in [[models:special_relativity|special relativity]] but instead of the [[advanced_tools:minkowski_metric|Minkowski metric]] $\eta_{ij}$, we now have a general metric $g_{ij}$. Alternatively we can use | + | just like in [[theories:special_relativity|special relativity]] but instead of the [[advanced_tools:minkowski_metric|Minkowski metric]] $\eta_{ij}$, we now have a general metric $g_{ij}$. Alternatively we can use |
\begin{align*} | \begin{align*} | ||
L(q,\dot{q}) &= \tfrac{1}{2}m g(q)(\dot{q},\dot{q}) \\ | L(q,\dot{q}) &= \tfrac{1}{2}m g(q)(\dot{q},\dot{q}) \\ | ||
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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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In the geometric optics approximation light acts like particles tracing out geodesics, i.e. the shortest paths. | 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 [[theories: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. |