Tensor Indices Switch with Infinitesimals and Space-Time Derivatives

[waht]

Wondering if this is valid to do, if I start with the expression

\delta\omega^{u}_{ \singlespacing v} x^v \partial_u

where \delta\omega is an infinitesimal, and \partial a space-time derivative,

is it still valid to drop and raise the u to obtain

\delta\omega_{u v} x^v \partial^u

without involving the metric tensor?

 

[tiny-tim]

Wondering if this is valid to do, if I start with the expression

\delta\omega^{u}_{ \singlespacing v} x^v \partial_u

where \delta\omega is an infinitesimal, and \partial a space-time derivative,

is it still valid to drop and raise the u to obtain

\delta\omega_{u v} x^v \partial^u

without involving the metric tensor?

Hi what!

Yes, it's just a (double) dot-product:

\delta\omega^{u}_{ \singlespacing v} x^v \partial_u

=\ \delta\omega_{w\singlespacing v}g^u_w x^v \partial_u

=\ \delta\omega_{w\singlespacing v} x^v \partial_w

 

[waht]

Thanks Tim, that cleared it up.