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--- equations:geodesic_equation [2018/04/14 10:47]
aresmarrero
+++ equations:geodesic_equation [2018/05/05 22:00]
jakobadmin [Concrete]
@@ Line 1: / Line 1: @@
-<WRAP lag> \ddot{q}_i =  -\Gamma_{ijk}\dot{q}^j\dot{q}^k</WRAP>
+<WRAP lag> $\ddot{q}_i =  -\Gamma_{ijk}\dot{q}^j\dot{q}^k$</WRAP>
 ====== Geodesic Equation ======
 <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>
-<note tip>
+**Derivation**
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas.
-</note>
-<tabbox Abstract>
-<note tip>
+The Lagrangian for a free point particle in a spacetime $Q$ is
-The motto in this section is: //the higher the level of abstraction, the better//.
+\begin{align*}
-</note>
+ L(q,\dot{q}) &= m\sqrt{g(q)(\dot{q},\dot{q})} \\
+ &= m\sqrt{g_{ij}\dot{q}^i\dot{q}^j}
+\end{align*}
+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*}
+ L(q,\dot{q}) &= \tfrac{1}{2}m g(q)(\dot{q},\dot{q}) \\
+ &= \tfrac{1}{2}m g_{ij}\dot{q}^i\dot{q}^j
+\end{align*}
-<tabbox Why is it interesting?>
+We now want, as usual, to find the equations of motion.  Using the [[equations:euler_lagrange_equations|Euler-Lagrange equations]] we get
+\begin{align*}
+ p_i = \frac{\partial L}{\partial\dot{q}^i} &= mg_{ij}\dot{q}^j \\
+ F_i = \frac{\partial L}{\partial q^i}
+ &= \frac{\partial}{\partial q^i}\Bigl(\tfrac{1}{2}mg_{jk}(q)\dot{q}^j\dot{q}^k\Bigr)\\
+ &=\tfrac{1}{2}m\partial{i}g_{jk}\dot{q}^j\dot{q}^k,
+ \quad(\text{where } \partial_i=\frac{\partial}{\partial q^i}).
+\end{align*}
+So the Euler--Lagrange equations say
+\[
+ \frac{d}{dt}mg_{ij}\dot{q}^j = \tfrac{1}{2}m\partial_{i}g_{jk}\dot{q}^j\dot{q}^k.
+\]
+An important observation is that the mass factors away. Therefore, the motion is independent of the mass!
+We can rewrite the geodesic equation as follows
+\begin{align*}
+ \frac{d}{dt}g_{ij}\dot{q}^j &= \tfrac{1}{2}\partial_{i}g_{jk}\dot{q}^j\dot{q}^k \\
+ \hspace{-3ex}\rightarrow\quad
+ \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 \\
+ \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 \\
+ &= \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*}
+where the last line follows since $g_{ik}=g_{ki}$.
+Now we define the so-called [[advanced_tools:connections:levi_civita_connection|Christoffel symbols]]
+\[
+ \Gamma_{ijk} \equiv -\bigl(\partial_{i}g_{jk}-\partial_{k}g_{ij}-\partial_{j}g_{ki}\bigr)
+\]
+Using this definition, we can write the geodesic equation as
+\begin{align*}
+ \ddot{q}_i = g_{ij}\ddot{q}^j &= -\Gamma_{ijk}\dot{q}^j\dot{q}^k \\
+ \hspace{-3ex}\rightarrow\quad
+ \ddot{q}^i &= -\Gamma^i_{jk}\dot{q}^j\dot{q}^k.
+\end{align*}
+<tabbox Abstract>
+Geodesics are paths $q:[t_0,t_1]\rightarrow Q$ that are critical points of the action
+\[
+ S(q) = \int_{t_0}^{t_1}\sqrt{g_{ij} \dot{q}^i\dot{q}^j}\,dt
+\]
+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?>
 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 [[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>

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