Re:The Geometric Tool That Solved Einstein's Relativity Problem
by cfgauss2718 on Monday August 12, 2024 @07:44PM
(#41230626)
Here here! Functions do not depend on your choice of coordinates, only the components of tensors do! I think this is why it’s important to keep covariance and contravariance in mind. While tensor(fields) do not depend on coordinates intrinsically, the way we represent them when doing calculations most certainly does, and this is usefully characterized by co/contravariance.
Re:The Geometric Tool That Solved Einstein's Relativity Problem
by Koshkin on Tuesday August 13, 2024 @05:41PM
(#41240148)
"Here here!" is indeed topological. "Hear, hear!", on the other hand, is conversational.
Re:The Geometric Tool That Solved Einstein's Relativity Problem
by cfgauss2718 on Thursday August 15, 2024 @10:45AM
(#41256756)
I love when grammatical mistakes become unintentional puns
Re:The Geometric Tool That Solved Einstein's Relativity Problem
by prof-dr-ir on Tuesday August 13, 2024 @11:07AM
(#41236209)
In the end tensor fields are sections of a bundle.
I insist that calling them "just functions" is simplistic. In fact, I'd say that the complexity of your elaborations kind of proves my point.
Note that I deliberately use "simplistic" and not "wrong", since a section is a function of sorts.
This is the essence of notions like scalar, vector, tensor, that they do not depend on the chosen coordinate system.
Only their numeric representations associated with a chosen coordinate system do depend on that system.
If you compute some arbitrary functions of the numeric components of a tensor in a certain coordinate system, in most cases the array of numbers that composes the result will not be a tensor, precisely because the result will really be different in any other coordinate system, while a tensor must be invariant.
All physical laws are formulated only using various kinds of tensors, including vectors and scalars, precisely because they must be invariant at the choice of the coordinate system.