- #1
Kostik
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- TL;DR Summary
- Dirac is saying that ##A_{\mu;\nu}-A_{\nu;\mu} = A_{\mu,\nu} - A_{\nu,\mu}## only works for covariant vectors. Why?
In Dirac ("GTR") p. 39 he says, "For a covariant vector ##A_\mu##, we have
$$A_{\mu;\nu}-A_{\nu;\mu} = A_{\mu,\nu} - \Gamma^\rho_{\mu\nu} A_\rho - \left( A_{\nu,\mu} - \Gamma^\rho_{\nu\mu}A_\rho \right) = A_{\mu,\nu} - A_{\nu,\mu}.$$ This result may be stated: covariant curl equals ordinary curl. It holds only for a covariant vector. For a contravariant vector we could not form the curl because the suffixes would not balance."
Yet a similar calculation shows that ##A^\mu_{\,\,;\nu}-A^\nu_{\,\,;\mu} = A^\nu_{\,\,,\mu}-A^\mu_{\,\,,\nu}##, which is the parallel result. So, what is he trying to say?
EDIT: Sloppy error - my mistake!
$$A_{\mu;\nu}-A_{\nu;\mu} = A_{\mu,\nu} - \Gamma^\rho_{\mu\nu} A_\rho - \left( A_{\nu,\mu} - \Gamma^\rho_{\nu\mu}A_\rho \right) = A_{\mu,\nu} - A_{\nu,\mu}.$$ This result may be stated: covariant curl equals ordinary curl. It holds only for a covariant vector. For a contravariant vector we could not form the curl because the suffixes would not balance."
Yet a similar calculation shows that ##A^\mu_{\,\,;\nu}-A^\nu_{\,\,;\mu} = A^\nu_{\,\,,\mu}-A^\mu_{\,\,,\nu}##, which is the parallel result. So, what is he trying to say?
EDIT: Sloppy error - my mistake!
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