Would, for example, a train traveling from south to north experience this force? If so, why? I would think it does not since its in contact with the earth and so it does not experience the difference in speed when going from point to the other.

In Sean Carroll's GR book, the differential $p$-form is defined as a $(0,p)$ tensor that is completely antisymmetric, which I would think is something like $$\frac{1}{2!}(t_{ab}-t_{ba})\textbf {e}^a \otimes \textbf{e}^b.$$ In Zee's GR book, part IX.7, he defined the $p$-form in a similar way, whi...