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Covariant and Contravariant
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J. B. Wood  
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  More options Dec 1 2008, 9:26 pm
Newsgroups: sci.math
From: w...@itd.nrl.navy.mil (J. B. Wood)
Date: Mon, 01 Dec 2008 11:26:19 -0500
Local: Mon, Dec 1 2008 9:26 pm
Subject: Covariant and Contravariant
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Hello, all, and while I understand the math behind and use of the above in
vector analysis, I can't figure out why vector components are labeled as
such.  
The fact that a vector in space can be represented in space in two ways
(via the metric tensor) as either with "contravariant" components and one
set of basis vectors or with "covariant" components with reciprocal basis
vectors is clear.  But what are the vector components varying "with" or
"against"?  It appears these terms are arbitrary.  Thanks for your time
and comment.  Sincerely,

John Wood (Code 5550)        e-mail: w...@itd.nrl.navy.mil                    
Naval Research Laboratory
4555 Overlook Avenue, SW
Washington, DC 20375-5337


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Mariano Suárez-Alvarez  
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  More options Dec 1 2008, 9:36 pm
Newsgroups: sci.math
From: Mariano Suárez-Alvarez <mariano.suarezalva...@gmail.com>
Date: Mon, 1 Dec 2008 08:36:17 -0800 (PST)
Local: Mon, Dec 1 2008 9:36 pm
Subject: Re: Covariant and Contravariant
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On Dec 1, 2:26 pm, w...@itd.nrl.navy.mil (J. B. Wood) wrote:

> Hello, all, and while I understand the math behind and use of the above in
> vector analysis, I can't figure out why vector components are labeled as
> such.  
> The fact that a vector in space can be represented in space in two ways
> (via the metric tensor) as either with "contravariant" components and one
> set of basis vectors or with "covariant" components with reciprocal basis
> vectors is clear.  But what are the vector components varying "with" or
> "against"?  It appears these terms are arbitrary.  Thanks for your time
> and comment.  Sincerely,

The vary "with" and "against" changes of coordinates: their
transformation formulas either involve the coefficients of
the change-of-basis matrix, or the coefficients of the
inverse matrix to the change-of basis matrix.

It is of course arbitrary which one is "with" and which
one is "against", but there *is* a difference.

-- m


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Ken S. Tucker  
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  More options Dec 2 2008, 1:24 am
Newsgroups: sci.math
From: "Ken S. Tucker" <dynam...@vianet.on.ca>
Date: Mon, 1 Dec 2008 12:24:08 -0800 (PST)
Local: Tues, Dec 2 2008 1:24 am
Subject: Re: Covariant and Contravariant
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Hi Mr. Wood

On Dec 1, 8:26 am, w...@itd.nrl.navy.mil (J. B. Wood) wrote:

> Hello, all, and while I understand the math behind and use of the above in
> vector analysis, I can't figure out why vector components are labeled as
> such.
> The fact that a vector in space can be represented in space in two ways
> (via the metric tensor) as either with "contravariant" components and one
> set of basis vectors or with "covariant" components with reciprocal basis
> vectors is clear.  But what are the vector components varying "with" or
> "against"?  It appears these terms are arbitrary.  Thanks for your time
> and comment.  Sincerely,

> John Wood (Code 5550)        e-mail: w...@itd.nrl.navy.mil
> Naval Research Laboratory
> 4555 Overlook Avenue, SW
> Washington, DC 20375-5337

In physics, covariant is demo'd by Planck's "h",

h = erg*sec * (a scalar ~ 6.6x10^27)
h=h'  , meaning h is invariant,
h == erg'*sec'
therefore
erg*sec = erg'*sec'
so
erg' = (sec/sec') * erg

OTOH, the speed of light "c" is also an invariant,
c = c', but transforms contravariantly,
c = meter/sec * (a scalar ~ 3*10^8)
c == meter'/sec'
meter' = (sec'/sec) * meter.

Take note the erg uses (sec/sec') to transform,
but the meter uses (sec'/sec), which is a very
important difference, that became evident in
the 1983 definition of the meter,

http://physics.trak4.com/modern-spacetime.pdf

and then furthermore at this link,

http://physics.trak4.com/

is Modern SpaceTime (MST) articles in briefs,
proving the redefinition of the meter is in
accord with General Relativity.
Regards
Ken S. Tucker


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