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 waht!

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.