Theorem: External witness with zero free term

For any nonzero real TT wave $h$ in the plane, there are real compact $f,g$ and a normalized scalar $v\in {C_c}^{\infty }({\mathbb{R}}^3)$ such that

Proof. Choose $x_{\ast }=(t_{\ast },{\mathbf{x}}_{\ast })$ and ${\mathbf{p}}_{\ast }$ such that $h^{ij}(x_{\ast })p_{\ast i}{p}_{\ast j}\ne 0$. Choose a nonnegative nonzero smooth momentum bump $\beta$ near ${\mathbf{p}}_{\ast }$ on which that quadratic expression has one sign, and set

For

the number

is real and nonzero. Its real part retains that sign on a small neighborhood.

Take a nonnegative nonzero real test $\rho$ supported there, set $f=P\rho$, and choose compact real $g=1$ near $\mathrm{supp}\rho$. Then $\varphi (f)=0$, $u_f=-\rho$, and $q_{g,h,f}=-{\mathsf{Q}}_h\rho$. Hence

This also proves that the mass-shell vector of $q_{g,h,f}$ is nonzero.