Re: The momentum - a cotangent vector?

It's only that vectors or covectors and in general any tensor of any
rank do not transform at all under coordinate transformations (i.e.,
diffeomorphisms). What transforms are the basis vectors and the
corresponding dual base and correspondingly the components of the
tensors wrt. these bases. I tried to summarize this briefly for vectors
and covectors in my posting.

Note that in physics you usually have more structure in your manifolds.
E.g., in GR you assume a pseudo-Riemannian manifold, i.e., a manifold
with a fundamental form (of Lorentz signature (1,3) or (3,1) depending
on your sign convention) with the torsion-free compatible affine
connection. Then you can also canonically (i.e., independent of the use
of bases and cobases) identify vectors and covectors, as is usually done
by physicists.

On 09/08/2024 06:15, Mikko wrote:

On 2024-08-07 11:37:02 +0000, the moderator said:
 

I think Stefan is using "tangent vector" and "cotangent vector"
in the sense of differential geometry and tensor calculus.  In
this usage, these phrases describe how a vector (a.k.a a rank-1
tensor) transforms under a change of coordintes: a tangent vector
(a.k.a a "contravariant vector") is a vector which transforms the
same way a coordinate position $x^i$ does, while a cotangent vector
(a.k.a a "covariant vector") is a vector which transforms the same
way a partial derivative operator $\partial / \partial x^i$ does.

 
Thank you. That makes sense.
 
--=20
Mikko

--
Hendrik van Hees
Goethe University (Institute for Theoretical Physics)
D-60438 Frankfurt am Main
http://itp.uni-frankfurt.de/~hees/