Geodesic equation (free particle)

Question

How to find a coordinate system whose geodesic equation does not have the "Christoffel symbol" term? (i.e. free particle - generalized Newton's second law.)

Answer 1

Suppose you're in a coordinate system where the Christoffels don't vanish at some point.

To choose a coordinate system where the Christoffel symbols vanish at a given point $p$, you must apply a Christoffel symbol change of variables:

$$0={\bar\Gamma}^k{}_{ij} = \frac{\partial x^p}{\partial y^i}\, \frac{\partial x^q}{\partial y^j}\, \Gamma^r{}_{pq}\, \frac{\partial y^k}{\partial x^r} + \frac{\partial y^k}{\partial x^m}\, \frac{\partial^2 x^m}{\partial y^i \partial y^j}$$

For simplicity, maybe $\frac{\partial x^a}{\partial y^b}=\delta^a_b$ (evaluated at point $p$ and point p only, so this says nothing about the second derivatives), in which case the equation becomes:

$$0= \Gamma^k_{ij} + \frac{\partial^2 x^k}{\partial y^i \partial y^j}$$

if $x^k=y^k+C^k_{i j} y^i y^j$ and $p$ is the origin, this tells you immediately that if you choose $C^k_{ij}=-\Gamma^k_{ij}$ then you're in a frame where all of the Christoffels vanish.