Why does gravity forbid local observables?

Question

I heard in a conference that gravity forbids to construct local gauge invariants like $\mathrm{Tr}\left\{−\frac{1}{4} F_{μν}^{a}F_{a}^{μν}\right\}$ and only allows non-local gauge invariant quantities like Wilson Loops: $\mathrm{Tr}\mathcal{P}\exp\left[\oint_{\gamma} A^{a}dx_{a}\right]$. Could someone explain me where does it come from?

Answer 1

General coordinate invariance lets you arbitrarily set the values of the metric and it's first derivative at any one point-- Fermi coordinates . Since you can do this, constructions like the maxwell term you describe above will be necessarily coordinate-dependent, and thus, not local observables.