equations:geodesic_equation
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equations:geodesic_equation [2018/04/14 10:56] aresmarrero [Concrete] |
equations:geodesic_equation [2018/05/05 22:00] jakobadmin [Concrete] |
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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> | + | |
| | | ||
| + | On a sphere the geodesics are "great circles". | ||
| <tabbox Concrete> | <tabbox Concrete> | ||
| Line 17: | Line 17: | ||
| &= 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}) \\ | ||
| Line 41: | Line 41: | ||
| \begin{align*} | \begin{align*} | ||
| \frac{d}{dt}g_{ij}\dot{q}^j &= \tfrac{1}{2}\partial_{i}g_{jk}\dot{q}^j\dot{q}^k \\ | \frac{d}{dt}g_{ij}\dot{q}^j &= \tfrac{1}{2}\partial_{i}g_{jk}\dot{q}^j\dot{q}^k \\ | ||
| - | \hspace{-3ex}\therefore\quad | + | \hspace{-3ex}\rightarrow\quad |
| \partial_{k}g_{ij}\dot{q}^k\dot{q}^j + g_{ij}\ddot{q}^j | \partial_{k}g_{ij}\dot{q}^k\dot{q}^j + g_{ij}\ddot{q}^j | ||
| &= \tfrac{1}{2}\partial_{i}g_{jk}\dot{q}^j\dot{q}^k \\ | &= \tfrac{1}{2}\partial_{i}g_{jk}\dot{q}^j\dot{q}^k \\ | ||
| - | \hspace{-3ex}\therefore\quad | + | \hspace{-3ex}\rightarrow\quad |
| g_{ij}\ddot{q}^j &= \bigl(\tfrac{1}{2}\partial_{i}g_{jk}-\partial_{k}g_{ij}\bigr)\dot{q}^j\dot{q}^k \\ | g_{ij}\ddot{q}^j &= \bigl(\tfrac{1}{2}\partial_{i}g_{jk}-\partial_{k}g_{ij}\bigr)\dot{q}^j\dot{q}^k \\ | ||
| &= \tfrac{1}{2}\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)\dot{q}^j\dot{q}^k | &= \tfrac{1}{2}\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)\dot{q}^j\dot{q}^k | ||
| \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 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 57: | Line 57: | ||
| \begin{align*} | \begin{align*} | ||
| \ddot{q}_i = g_{ij}\ddot{q}^j &= -\Gamma_{ijk}\dot{q}^j\dot{q}^k \\ | \ddot{q}_i = g_{ij}\ddot{q}^j &= -\Gamma_{ijk}\dot{q}^j\dot{q}^k \\ | ||
| - | \hspace{-3ex}\therefore\quad | + | \hspace{-3ex}\rightarrow\quad |
| \ddot{q}^i &= -\Gamma^i_{jk}\dot{q}^j\dot{q}^k. | \ddot{q}^i &= -\Gamma^i_{jk}\dot{q}^j\dot{q}^k. | ||
| \end{align*} | \end{align*} | ||
| Line 63: | Line 63: | ||
| <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?> | ||
| - | | ||
| - | |||
| 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. |
| + | |||
| + | 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. | ||
| + | |||
| </tabbox> | </tabbox> | ||
equations/geodesic_equation.txt · Last modified: 2018/12/19 11:01 by jakobadmin
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