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From: Hendrik van Hees <hees@itp.uni-frankfurt.de>
Newsgroups: sci.physics.research
Subject: Re: The momentum - a cotangent vector?
Date: Fri, 09 Aug 2024 13:53:39 PDT
Organization: Goethe University Frankfurt (ITP)
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Approved: Jonathan Thornburg [remove -color to reply]" <dr.j.thornburg@gmail-pink.com (sci.physics.research)
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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/