I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same):
$$g^{\mu\nu}g_{\mu\nu}=\delta_{\nu}^{\nu}=\delta_{0}^{0}+\delta_{1}^{1}+\delta_{2}^{2}+\delta_{3}^{3}=1+1+1+1=4.$$ But I've also seen (where the upstairs and downstairs indices are different):
$$g^{\mu\nu}g_{\nu\lambda}=\delta_{\lambda}^{\mu}=\left(\begin{array}{cccc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array}\right).$$
How can there be two different answers to (what appears to me to be) the same operation, ie multiplying the metric tensor by its inverse? Apologies if I've got this completely wrong.
Answer
In terms of your ordinary matrix multiplication, you have, for the case of a 4x4 matrix $M = g_{ab}$:
$M\cdot M^{-1} = I$, which is the same thing as $g_{ab} g^{bc} = \delta_{a}{}^{c}$
and
$Tr\left(M\cdot M^{-1}\right) = 4$, which is the same thing as $g_{ab}g^{ab} = \delta_{a}{}^{a} = 4$
No comments:
Post a Comment