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.
