On May 15, 5:04 pm, JanPB wrote:
On May 14, 10:08 pm, Koobee Wublee wrote:

Mathematically, you are just wrong. For example, describing flat
spacetime using the linear rectangular (Euclidean) and using the
spherically symmetric polar coordinate systems require you to supply
different metric for each choice of coordinate system. shrug

No, it's the same metric in both cases, e.g. in the plane the
following are equal:

dx^2 + dy^2

and:

dr^2 + r^2 dtheta^2

...where x = r cos(theta) and y = r sin(theta) (polar coordinates).

The above are two different coordinate decompositions of the same
metric. To see that they are equal, just evaluate them on an arbitrary
vector and see you get the same value in both cases.

You are making the same mistake again. What you refer to is the
geometry itself not the metric. The equations above represent the
same geometry, yes. They are equivalent. However, the coordinates
are different, and the metrics are different. The metric cannot
adequately describe the geometry despite your voodoo conjectures of
dot products, and the coordinates itself cannot adequately describe
the geometry. It takes both well specified coordinate systems and the
metrics to fully describe the geometries. shrug

We cannot go on without you understand my point of view, and I have
understood yours and pointed the errors in your logic. If you are not
malicious as Eric Gisse is, you need to understand my point of view.
shrug