In article <8uk1ta$afm$1@nnrp1.deja.com>,

arcsign@my-deja.com wrote:

Indeed, not only is =91the ratio of the circumference to the
diameter of a circle' not part of the definition of pi, but for any
particle or object with mass in GR, that ratio is also not the
CODATA Standard International or historical =91magnitude' of pi.

=
Isn't that what we've been telling you? That the ratio of the
circumference of a circle to its diameter in non-Euclidean geometry is
not pi?
=

~~~~~~~~~~~~webnotes~~~~~~~~~~~~~~~~
1. Weinberg was recently quoted on s.p.research as saying:
"..and the geometric interpretation of the theory of gravitation
has dwindled to a mere analogy, which lingers in our language
in terms like "metric", "affine connection", and "curvature", but
is not otherwise very useful."
=
2. Feynman said, quite explicity, that "the geometric
interpretation is not really necessary or essential to physics".

=
When they talk about a non-geometric interpretation, they speak of a
field theory on ordinary flat Minkowskian spacetime, not the curved
spacetime of general relativity. This _only_ works if spacetime has
a trivial topology; it can't possibly reproduce full GR. On the other
hand, we don't know whether spacetime has trivial topology or not.
=

3. Chris Hillman said, "And the geometric "interpretation", as
you put, of gtr, is an -essential part- of that theory,..".

=
What you apparently don't understand is that, Weinberg and Feynman
notwithstanding, if you actually go and measure the ratio of the
circumference of a circle to its diameter, you will not get pi -- exact=

what you expect in a non-Euclidean geometry. This is true *even if*
-- as Weinberg and Feynman mention -- you drop the curved-spacetime
interpretation and treat GR as a theory with Euclidean (or rather
Minkowskian) spacetime geometry.
=
Hence: although you can write down GR (in appropriate topology) withou=

referring to curved spacetime geometry, if you make measurements in
spacetime you still get exactly the results you would get if spacetime
were curved, so you might as well just say that it IS curved.
=

If, in ANY space, a measured =91circumference' is divided by
a measured =91diameter' and that ratio is NOT pi, then that
circumference is not a circle, within experimental limits.

=
It can be a circle; you are just wrongly restricting "circles" to be
those closed curves with a ratio of pi.
=

That ratio divided by pi is the =91..variable coefficient of pi'.

=
Well, whatever you want to call it.. but it's not pi, and the curves
_are_ still circles.
=

Here, the definition of roundness coefficient is particularly
stated such so as NOT to exclude non-Euclidean geometry,

=
There's no need to introduce a "roundness coefficient" so as to not
exclude non-Euclidean geometry; if you want to include non-Euclidean
geometry you simply use the standard definition of a circle, which I've=
defined for you, and which has nothing to do with ratios.
=

Except in GR,

=
or except in any other non-Euclidean geometry, since ...
=

..always measure =91diameter' straight through
the centerline axis and in the flat plane of the circumference,
..NEVER along any kind of arc or curve, ..even if that
circumference describes a hole in a curved surface.

=
... cannot be done in a non-Euclidean geometry, so your prescription
is invalid in any geometry but Euclidean.

<> >><> >><> >><> >><>

In article <8uk1ta$afm$1@nnrp1.deja.com>,

arcsign@my-deja.com wrote:
~~~~~~~~~~~~webnotes~~~~~~~~~~~~~~~~
1. Weinberg was recently quoted on s.p.research as
saying: "..and the geometric interpretation of the theory
of gravitation has dwindled to a mere analogy, which
lingers in our language in terms like "metric", "affine
connection", and "curvature", but is not otherwise very
useful."

2. Feynman said, quite explicity, that "the geometric
interpretation is not really necessary or essential to
physics".

[4.]Nathan Urban: ..we don't know whether spacetime
has trivial topology or not.

3. Chris Hillman: "And the geometric "interpretation"..
..of gtr, is an -essential part- of that theory,..".

````SNiP````..if you actually go and measure the ratio of the
circumference of a circle to its diameter, you will not get pi
-- exactly what you expect in a non-Euclidean geometry.

This is true *even if* -- as Weinberg and Feynman
mention -- you drop the curved-spacetime
interpretation and treat GR as a theory with Euclidean
(or rather Minkowskian) spacetime geometry.

`````SNiP```````
..if you make measurements in spacetime you still get
exactly the results you would get if spacetime were
curved, so you might as well just say that it IS curved.

If, in ANY space, a measured =91circumference' is divided
by a measured =91diameter' and that ratio is NOT pi, then
that circumference is not a circle, within experimental
limits.

It can be a circle; you are just wrongly restricting
"circles" to be those closed curves with a ratio of pi.

Well, whatever you want to call it.. but it's not pi,
and the curves _are_ still circles.

Here, the definition of roundness coefficient is
particularly stated such so as NOT to exclude
non-Euclidean geometry,..

if you want to include non-Euclidean geometry you
simply..