"One consequence of the transformation rules for tensors is that, if all components of a tensor vanish in one coordinate system, they vanish in all of them. The Christoffel symbols then are clearly not tensors. For many modern relativists, this disqualifies them as candidates for the mathematical representation of the gravitational field.
Instead, as mentioned in Section 2, the non-vanishing of the curvature tensor is used as a coordinate-independent criterion for the presence of a gravitational field. To the end of his life, however, Einstein preferred to use the Christoffel symbols instea…