Compact representatives on a finite TT plane

Choose independent real waves $v_1=e,v_2=r$ satisfying

with Schwartz Cauchy data and a common open Fourier gap around zero momentum. Put ${\mathcal{V}}_2={\mathrm{span}}_{\mathbb{R}}\{e,r\}$.

Such waves are obtained by applying the smooth TT projector away from momentum zero to compact smooth Fourier data and then performing wave evolution. A nonzero Fourier-gap datum is not compactly supported in position: compact support would make its Fourier transform entire, and an open gap would force it to vanish.

For a spatially Schwartz descriptor, $K\xi$ cannot be a nonzero spatial-TT tensor. Indeed, trace and divergence give ${\partial }_i{\xi }_i=0$ and $\Delta {\xi }_j=0$, hence ${\xi }_j=0$; the temporal-spatial component and decay then give ${\xi }_0=0$.

The actual response coefficients are

The surface term (eq:surface) pairs to zero with a spatial-TT tensor, but that observation alone does not prove conservation.

Define the electric curvature

It annihilates $K\xi$, and $\mathcal{E}(v_a)=-\Delta v_a/2$.

For compact smooth symmetric $C^{ij}$, define

Then