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.