✎ Geodesic Equation [The Physics Travel Guide]

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<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.

Geodesics are the "shortest" paths between two points in a flat spacetime and the straightest path between two points in a curved spacetime.  
  
On a sphere the geodesics are "great circles". 
<tabbox Concrete>

**Derivation**

The Lagrangian for a free point particle in a spacetime $Q$ is 
\begin{align*}
 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*}

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*}

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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