00:01
All right, so let's say we have a vector w that is composed of a interproduct between a vector a and a covariant derivative of a, sorry, covariant derivative alpha with respect to, or on ub, basically.
00:20
Or sorry, this is v, not u.
00:22
Then what we want to show is there's an expression for the contravariant vector.
00:28
So what we can do is take the inner product of vb with the metric tensor.
00:36
So let's let v beta, which is our covariant vector be g alpha beta times v beta, where that's the covariant or contravariant vector.
00:50
All right.
00:54
And remember that the covariant derivative of the metric tensor is zero.
01:01
It doesn't even matter if i pick the index alpha.
01:04
We could pick the index kappa or whatever you want.
01:06
It's going to be zero by definition.
01:08
So let's in fact not use the letter beta here...