Averaging of unit vectors in Euclidean space over $S_1$ is a well-known result. If we consider unit vector $\vec{n}=(\cos(\phi),\sin(\phi))$ such as $\vec{n}^2=1$, then the following average: \begin{align} \langle n_i n_j \rangle=\frac{1}{2}\delta_{ij}, \end{align} where \begin{align} \langle f(\phi) \rangle=\frac{1}{2\pi}\int_0^{2\pi} d\phi f(\phi). \end{align} But what about two dimensional Minkowski spacetime? Then unit vector coorsponds to $n_t^2-n_x^2=1$. It is natural to expect, that: \begin{align} \langle n_\mu n_\nu \rangle=\frac{1}{2}g_{\mu\nu}, \end{align} but how to show this?
Thank you in advance.