[Physics] Are the Christoffel symbols all zero in gravity-free space

Question

I was looking at the geodesic equation,
$$\ddot{x}^\mu + \Gamma^\mu{}_{\nu\rho} \dot{x}^\nu \dot{x}^\rho = 0,
$$
and thinking about how to identify gravity-free spaces by looking at the Christoffel symbols $\Gamma^\mu{}_{\nu\rho}$. For example, if they were zero, under certain limits I could check what kind of expressions/signs metric derivatives take. Can this be used to show that the space is gravity-free?

Answer 1

No, not necessarily.

Flat space (which is what I assume you mean when you say gravity-free space) is special because it's possible to choose a global coordinate system in which all of the Christoffel symbols vanish. However, it's easy to make a choice of coordinate system (for example, spherical coordinates) for which the Christoffel symbols are generically non-zero, even in the absence of spacetime curvature.