- #1
Burnstryk
Homework Statement
I am trying to derive the following relation using inner products of vectors:
Homework Equations
[tex] g_{\mu\nu} g^{\mu\sigma} = \delta_{\nu}^{\hspace{2mm}\sigma} [/tex]
The Attempt at a Solution
What I have done is take two vectors and find the inner products in different ways with contravariant and covariant components:
[tex] \textbf{v}.\textbf{w} [/tex]
I have obtained the following relations:
[tex] g_{\mu\nu} v^{\mu} w^{\nu} = g^{\mu\nu}v_\mu w_\nu = v_\nu w^\nu = v^\nu w_\nu[/tex]
Using these relations I decided to take a vector with an arbitrary component (sigma) and multiply it by the metric and inverse considering the lowering and operating nature:
[tex] g_{\mu\nu} g^{\mu\sigma} v_{\sigma} = g_{\mu\nu} v^{\mu} = v_\nu = \delta_{\nu}^{\hspace{2mm}\sigma}v_\sigma [/tex]
and hence obtain the original result.
I wanted to see if these arguments and method make sense or if I'm confusing everyone.