Intuitive

Geodesics are the straightest path between two points in a curved spacetime.

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 special relativity but instead of the 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 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 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*}

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.

Why is it interesting?

Almost every problem in classical mechanics can be regarded as geodesic motion.

Especially in 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.