I know how they're defined but I never used them. AG On Thursday, September 5, 2024 at 5:26:36 PM UTC-6 Brent Meeker wrote:
> Ever use cylindrical coordinates? > > Brent > > > > On 9/5/2024 2:07 PM, Alan Grayson wrote: > > Since the metric tensor is defined on the* flat *tangent space, will its > matrix representation always be diagonal? TY, AG > > On Thursday, September 5, 2024 at 12:35:03 PM UTC-6 Brent Meeker wrote: > >> 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. >>>>> >>>>> >>>>> -- >>>> You received this message because you are subscribed to the Google >>>> Groups "Everything List" group. >>>> To unsubscribe from this group and stop receiving emails from it, send >>>> an email to [email protected]. >>>> >>>> To view this discussion on the web visit >>>> https://groups.google.com/d/msgid/everything-list/ae301a72-9969-4e60-a132-0c8fe6d6f16cn%40googlegroups.com >>>> >>>> <https://groups.google.com/d/msgid/everything-list/ae301a72-9969-4e60-a132-0c8fe6d6f16cn%40googlegroups.com?utm_medium=email&utm_source=footer> >>>> . >>>> >>>> >>>> -- >> You received this message because you are subscribed to the Google Groups >> "Everything List" group. >> To unsubscribe from this group and stop receiving emails from it, send an >> email to [email protected]. >> >> To view this discussion on the web visit >> https://groups.google.com/d/msgid/everything-list/ddfc2b2c-28de-4338-b2eb-41d69bbcd58an%40googlegroups.com >> >> <https://groups.google.com/d/msgid/everything-list/ddfc2b2c-28de-4338-b2eb-41d69bbcd58an%40googlegroups.com?utm_medium=email&utm_source=footer> >> . >> >> >> -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/17a5202a-0dfd-4cfc-98fd-7e1e20e90f94n%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/17a5202a-0dfd-4cfc-98fd-7e1e20e90f94n%40googlegroups.com?utm_medium=email&utm_source=footer> > . > > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/2a944ec3-bfc7-499e-9928-4be42fcb0037n%40googlegroups.com.
