> You also seem to not realize the difference between a metric and a line
> element. A metric is a tensor, which is of course completely independent
> of coordinates (technically it is a tensor field on the manifold, but
> the words "tensor" and especially "field on the manifold" are often
> omitted for brevity). A line element gives an expression for
> infinitesimal distances expressed in terms of the coordinate
> differentials of a specific coordinate system. The line element depends
> on the metric tensor, and the coefficients of the coordinate
> differentials are the components of the metric tensor projected onto
> those coordinates.

If you have a different set of coordinate system, you need another set
of metric.  For example, the rectangular coordinate and polar
coordinate.

> Thus the Schw. metric can be projected onto many different coordinate
> systems, yielding many different line elements all representing the same
> metric.

Thus, you are wrong here.  You need to go back to read more carefully
of my thread-opening post listed as follows.