If you change coordinate systems the vectors and tensors transform in
such a way that the physics is unchanged. That's the defining property
of vectors and tensors and why they are not just the arrays used to
represent them.
Brent
On 9/4/2024 10:42 PM, Alan Grayson wrote:
How is the foregoing consistent with the statement that tensors are
independent of coordinate systems? TY, AG
On Wednesday, September 4, 2024 at 10:39:33 PM UTC-6 Alan Grayson wrote:
No; you never posted any page numbers in your reference. Had you
done that, I would have immediately studied your specific
reference. But I was about to post something like what you wrote;
namely, that *solving* for the metric tensor *MEANS* solving for
the 16 components defining its function, given a stress-energy
tensor and a coordinate system. The form of the components must
depend on the coordinates. And given a coordinate system, the
components will vary depending on location in spacetime, and this
is what's meant by the "metric tensor field". I thought the
"field" refers to a *unique* *real numbe*r at each point in
spacetime, but it must refer to the components of the matrix
representing the tensor. Wiki's definition seems misleading since
it states that the metric tensor is a bilinear function of two
vectors on the tangent plane. Those vectors are its arguments, but
sort-of misleading. Is there anything in the foregoing that I got
wrong? AG
On Wednesday, September 4, 2024 at 8:09:07 PM UTC-6 Brent Meeker
wrote:
The metric field is the set of metric tensors, one at each
point. It's not some vector lengths.
When are you gonna read "Relativity DeMystifie". I told you
the page numbers and you can look up more in the index.
Brent
On 9/4/2024 5:59 PM, Alan Grayson wrote:
But my point is therefore that the metric tensor field is
ambiguous !
On Wednesday, September 4, 2024 at 6:40:38 PM UTC-6 Brent
Meeker wrote:
It explains this "...and yields *different *real values
for most different pairs."
Brent
On 9/4/2024 1:03 PM, Alan Grayson wrote:
and yields *different *real values for most different pairs.
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