Proposition: Coordinate forcing and compact-domain miss

On $R_{\lambda }$,

Moreover,

The forcing is smooth, flatly zero for $t\le 0$, and past compact relative to compact observation sets.

For $\lambda >0$ it is not in ${C_c}^{\infty }({M_T}^{\circ };\mathbb{C})$. Both its real and imaginary time components are nonzero spatially noncompact functions.

Proof. By (eq:coefficients) and (eq:P2),

All coefficients depend only on time, proving (eq:Fco). Equations (eq:q) and (eq:d1) give ${S_{\lambda }}^{0i}=0$, hence ${V_{\lambda }}^{0i}=0$.

Smoothness and flat past vanishing follow from Lemma lem:R. Lemma lem:past proves past compactness.

On the nonempty interval $(\epsilon ,T)$,

The derivative is not identically zero, so ${F_{\mathrm{co}}}^0$ is not identically zero. Likewise its real and imaginary derivatives are $-2{\lambda }^2\cos (2\lambda t)$ and $-2{\lambda }^2\sin (2\lambda t)$. Neither vanishes identically on that interval. Thus both real and imaginary components are nonzero functions independent of $\mathbf{x}$, and neither has compact spatial support.