How Does the Metric Affect Index Position in Tensor Contractions?

[S.P.P]

In GR the metric can raise or lower indices, but which index will it raise in this case:

$\delta^{ij} \partial_i \xi_j$

is it,

$\partial^j \xi_j$

or,

$\partial_i \xi^i$

Or are these equal?

 

[nrqed]

In GR the metric can raise or lower indices, but which index will it raise in this case:

$\delta^{ij} \partial_i \xi_j$

is it,

$\partial^j \xi_j$

or,

$\partial_i \xi^i$

Or are these equal?

Didn't you actually mean $g^{ij} \partial_i \xi_j$??
In that case, yes, it is equal to both the expressions you gave, which are equal to one another.

 

[S.P.P]

Brilliant, thanks very much!

 

[HallsofIvy]

Note that int both
$\partial^j \xi_j$
and
$\partial_i \xi^i$
you will be doing a further contract. What you are really saying is that the order in which the contractions are done does not matter.
$\delta^{ij} \partial_i \xi_j$
is a scalar quantity.