- #1

gerald V

- 67

- 3

In electrodynamics in a curved background, the relation ##F^{\mu \nu} = A^{\mu , \nu} - A^{\mu , \nu}## stays in terms of ordinary derivatives. So, in particular ##F_{,\mu \nu}^{\mu \nu}## holds in terms of ordinary derivatives. One can define ##F_{,\mu}^{\mu \nu} = - \hat{j}^\nu##, which is ordinarily conserved.

In contrast, the field equation ##F_{;\nu}^{\mu \nu} = \frac{1}{\sqrt{- \det g}} (\sqrt{- \det g} F^{\mu \nu})_{,\nu} = -j^\nu## and the conservation law for the current ##j_{;\nu}^\nu = \frac{1}{\sqrt{- \det g}} (\sqrt{- \det g} j^\nu)_{,\nu} = 0## are in terms of covariant derivatives.

Starting from the identiy ##(\sqrt{- \det g} \; F^{\alpha \beta})_{, \beta} = (\sqrt{- \det g})_{, \beta} F^{\alpha \beta} + \sqrt{- \det g} \; F^{\alpha \beta}_{, \beta}## and using the field equation, one gets ##- \sqrt{- \det g} \; j^\alpha = (\sqrt{- \det g})_{, \beta} F^{\alpha \beta} + \sqrt{- \det g} \;( -\hat{j}^\alpha)## or ##j^\alpha = - (\ln \sqrt{- \det g})_{, \beta} F^{\alpha \beta} + \hat{j}^\alpha = - \Gamma_{\mu \beta}^\mu F^{\alpha \beta} + \hat{j}^\alpha##, where the current ##j## is covariantly conserved, while the current ##\hat{j}## is ordinarily conserved.

Questions:

- Is my analysis correct?

- If so, what does it mean that there co-exist a covariantly as well as an ordinarily conserved current? Is ##\hat{j}## observable?

- Is there anything comparable for gravitation, that means the co-existence of a covariantly as well as an ordinarily conserved energy-momentum Tensor?

Thank you very much in advance.