Lemma: Compact detection

Let

be a nonzero compactly supported distribution with support in M_+^\circ^2. On the unrestricted smooth off-shell configuration space, ${\mathcal{C}}_{f,g}\ne 0$ as an actual functional.

Proof. There exist compactly supported smooth $a(x),b(y)$ such that

To justify this reduction, choose a compact product rectangle containing the relevant supports. A compactly supported smooth test on the rectangle can be periodized inside a larger rectangle and expanded in its smooth Fourier series. Multiplication by separate compact cutoffs in the two variables gives approximating finite sums of tensor-product tests, converging with every derivative on a fixed compact set. If a distribution vanished on all tensor-product tests, continuity would therefore make it vanish on every test.

Choose a smooth scalar configuration satisfying

and choose $h_{00}=b$, with the other metric components zero. Equation (eq:commutator) detects $T_{f,g}$. If real configurations are required, decomposing $a,b$ into real and imaginary parts gives a real tensor-product detector. Compactness is required of the kernel, not of $\varphi$.