# Rasing and lowering indicies

1. Jul 12, 2007

### 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?

2. Jul 12, 2007

### nrqed

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.

3. Jul 12, 2007

### S.P.P

Brilliant, thanks very much!

4. Jul 13, 2007

### HallsofIvy

Staff Emeritus
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.