Summation notation and general relativity derivatives

[Maniac_XOX]

Do you know what a dummy index is?

yeah any index summed over..

 

[PeroK]

yeah any index summed over..

And if you sum over an index, then the symbol you use for the index is irrelevant. Do you understand what I wrote in post #26?

$$ A_{\nu} \partial_\alpha A^{\nu} = A_{\mu} \partial_\alpha A^{\mu} = A_0\partial_\alpha A^0 + A_1\partial_\alpha A^1 + A_2\partial_\alpha A^2 + A_3\partial_\alpha A^3$$

 

[Maniac_XOX]

And if you sum over an index, then the symbol you use for the index is irrelevant. Do you understand what I wrote in post #26?

yeah i understand post #26 is obvious but isn't it wrong to sum over those indices with a symbol that is already on another term? or do they all mean the same? so even $\mu^2 A_\alpha$ i could make it $\mu^2 A_\nu = \mu^2 A_\mu$ since it's defined in terms of partial derivatives of A?

 

[PeroK]

yeah i understand post #26 is obvious but isn't it wrong to sum over those indices with a symbol that is already on another term? or do they all mean the same? so even on $F_{\alpha\beta}$ i could make it ##F_\nu\mu# since it's defined in terms of partial derivatives of A?

You'll need to sort your latex. An index must never appear three times. It either appears once (in which case it is a single, but unspecified index); or, it appears twice, in which case it is a dummy index that may at any stage be replaced by any other symbol.

 

[PeroK]

yeah i understand post #26 is obvious but isn't it wrong to sum over those indices with a symbol that is already on another term? or do they all mean the same? so even $\mu^2 A_\alpha$ i could make it $\mu^2 A_\nu = \mu^2 A_\mu$ since it's defined in terms of partial derivatives of A?

The only thing you can't use for a dummy index is a symbol you're already using for a free index. That should be obvious.

Okay, so you've also got $\mu$ as a parameter in the equation of motion.

 

[Maniac_XOX]

right, that sorted it, cheers
is there any way that $A_\mu \partial_\alpha A^\mu - A_\alpha \partial^\rho A^\rho$ can be simplified as well or are the indices too different in this case. It seems to me that it can't be simplified

 

[PeroK]

right, that sorted it, cheers
is there any way that $A_\mu \partial_\alpha A^\mu - A_\alpha \partial^\rho A^\rho$ can be simplified as well or are the indices too different in this case. It seems to me that it can't be simplified

Do you mean? $$A_\mu \partial_\alpha A^\mu - A_\alpha \partial_\rho A^\rho$$

You can standardise the dummy indices:$$A_\mu \partial_\alpha A^\mu - A_\alpha \partial_\mu A^\mu$$ But, as one summation is over partial derivatives and one summation is over the components of $A$, there's nothing meaningful to be done.

 

[Maniac_XOX]

Do you mean? $$A_\mu \partial_\alpha A^\mu - A_\alpha \partial_\rho A^\rho$$

You can standardise the dummy indices:$$A_\mu \partial_\alpha A^\mu - A_\alpha \partial_\mu A^\mu$$ But, as one summation is over partial derivatives and one summation is over the components of $A$, there's nothing meaningful to be done.

right, that still helped tbh, learned a lot since this whole indices thing has been seriously new to me and trying to learn how to use general relativity has been like trying to start in a new language lol, cheers for the help

 

[Maniac_XOX]

Do you mean? $$A_\mu \partial_\alpha A^\mu - A_\alpha \partial_\rho A^\rho$$

You can standardise the dummy indices:$$A_\mu \partial_\alpha A^\mu - A_\alpha \partial_\mu A^\mu$$ But, as one summation is over partial derivatives and one summation is over the components of $A$, there's nothing meaningful to be done.

Would it make sense to use Lorenz gauge condition $\partial_\mu A^\mu=0$ and turn the equations of motion into $$\Box A_\alpha + \mu^2 A_\alpha = 2\beta A_\mu \partial _\alpha A^\mu + \frac {4\pi}{c} J_\alpha$$ and then use the $\partial_\alpha$ as gauge covariant derivative expanding it into $\partial_\alpha - i q A_\alpha -i q B_\alpha$ ?? Or is that only for $D_\alpha$? I've seen it being used in both notations on wikipedia