I have a question I wanted to clear up. According to the definition of a "local inertial" frame in GR, you must use a coordinate system that locally looks Cartesian, right? I mean if you had a coordinate system with a basis that wasn't orthogonal, then it would not be considered a local inertial frame, right? I ask because outside of relativity, I would normally say that whether a frame is inertial or not wouldn't depend on the coordinate system you choose. If your frame is in uniform motion, then it's inertial regardless of which way you point your meter stick.