# Four-vectors and metric tensor

I think it's safe to say that if $x^\mu=(x^0,x^1,x^2,x^3)$, then $x_\mu=(x^0,-x^1,-x^2,-x^3)$. But I don't really understand why one follows from the other. Could someone explain?

Also, I've been churning my head around the difference between $\eta_{\mu\nu}$ and and $\eta^{\mu\nu}$. What's the difference anyway?

I understand that $\eta_{\mu\nu}$ stands for the generic matrix element of the metric tensor $\eta = \left(\begin{array}{cccc}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{array}\right)$. That I understand. But I don't understand the utility of the upstairs notation.

• – Jim Feb 13 '15 at 18:50