Probe solution and future processing test

Choose a spacelike-compact free probe solution $v_{\mathrm{bulk}}$ equal to the constant solution $1$ on a neighborhood of $K$. Such a solution can be constructed explicitly.

At $t_0=-T-1$, prescribe physical probe Cauchy position equal to one on the spatial projection of $J^{-}(K)$ at that time, extend it smoothly with compact radial support, and prescribe zero velocity. The constant function solves $P1=0$. Finite propagation and Cauchy uniqueness imply equality with one on $K$.

Write

Thus $v=\sqrt{4\pi }r(x)$ on $K$.

Choose ${\eta }_{+}$ equal to zero for $t\le T+1$ and one for $t\ge T+2$, and put

This is a real compact test in the future processing slab. Its advanced solution satisfies

and therefore equals one on $K$.