I was thinking about the geodesics equations and I realised that a particle will not have acceleration if the connection coefficients vanish, which (I think) is to say we are attatching a inertial frame to the particle. But if we attach a non-inertial frame to the particle, it will probably have acceleration. Nothing new to this point as it works this way even in Newtonian mechanics. The problem seems to be this: we might be on a flat space-time but using "curved" coordinates. Then in general the particle will be accelerating. Then General Relativity would dictate that the particle is in a gravitational field. Paradox? Edit: or.... maybe this is not a problem as the equivalence principle says that the space-time is locally flat and the geodesics equation is for analysing a curve only locally?