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-   -   Covariant Derivation (https://www.mersenneforum.org/showthread.php?t=8325)

Damian 2007-06-04 21:22

Covariant Derivation
 
Can someone explain me in simple terms what is the covariant derivation?
For example if we take as a Manifold the unit circle in R^2, what would be its covariant derivation?

m_f_h 2007-06-04 23:56

[quote=Damian;107635]Can someone explain me in simple terms what is the covariant derivation?
For example if we take as a Manifold the unit circle in R^2, what would be its covariant derivation?[/quote]
the explicit expr of the corvar deriv depends on the object : you have to add a term with the gauge potential (connection) for each index.
e.g. for R_ab^cd
DR_ab^cd = dR_ab^cd+R_eb^cd phi_a^e +R_ae^cd phi_a^e +R_ab^ed phi_e^c+R_ab^ce phi_e^d

sorry I may have not 100% mainstream conventions ,
also that expression might simplify (to zero of course, but suppose R was sth else than d phi + phi phi),
i dont remember well : I did that in an earlier life...

Damian 2007-06-05 18:20

I don't get it :sad:

ewmayer 2007-06-05 20:45

The proper term is "covariant differentiation."

Any decent text or webpage on differential geometry should have an adequate description.

Damian 2007-06-06 15:41

[QUOTE=ewmayer;107729]The proper term is "covariant differentiation."

Any decent text or webpage on differential geometry should have an adequate description.[/QUOTE]

I know, I wanted a less formal definition, maybe with some numerical example, to make it less abstract and more easy to understand it.

m_f_h 2007-06-07 17:04

[quote=Damian;107791]I know, I wanted a less formal definition, maybe with some numerical example, to make it less abstract and more easy to understand it.[/quote]
I suggest you the very nice book by Nakahara: "Geometry, Topology and Physics" (if I recall correctly). It has lots of explicit examples etc. on this and related subjects (even if there are typos in several formulae, but usually just signs (+/-) or so.)

PS: it seems there is some explicit calculation on
[URL="http://en.wikipedia.org/wiki/Connection_%28mathematics"]http://en.wikipedia.org/wiki/Connection_(mathematics[/URL])
PPS: well, not much... I think you have to plug in those into the formulae on the page "covariant derivative"

In fact, there are different notions of covariant derivatives. In general, "covariant" is w.r.t. some local ("gauge") transformation. In general relativity, there are 2 such transformations to be considered : local Lorentz transformations (SU(2) or SO(3,1) acting on "Lorenz" indices), and local coordinate transformations (diffeomorphisms ; acting on "Einstein indices"). "of course", both are linked...


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