I thought you could write general relativity in a coordinate free manner that specifically doesn't rely on strong coordinate choices Indeed. But even before yo

I thought you could write general relativity in a coordinate free manner that specifically doesn't rely on strong coordinate choices Indeed. But even before you start messing with the formalism, the metric alone cannot be used to define a speed as 4-velocities are normalized. You need to decompose your tangent space into timelike and spacelike subspaces so you can have a dx and a dt.