Two explicit response prescriptions

For a symmetric tensor $h$, write

In four dimensions ${\mathsf{r}}^2=I$, and

Let $\mathcal{Q}$ consist of smooth conserved symmetric sources $Q\in C^{\infty }(\mathbb{R};\mathcal{S}({\mathbb{R}}^3))$, with complexification allowed. Schwartz seminorms and all time derivatives are locally uniformly bounded. Conservation means ${\partial }^{\mu }{Q}_{\mu \nu }=0$.

Fix

For $b\in \mathcal{S}$, this is smooth, and

To verify these statements, split the integral into $|y|<|x|/2$ and its complement. In the first region expand or differentiate the kernel; in the second, use rapid decay and move derivatives onto $b$ near the integrable singularity. The resulting bounds use finitely many Schwartz seminorms. In particular $\nabla {\Delta }^{-1}b\in L^2$, although ${\Delta }^{-1}b$ need not be in $L^2$.

Put

Then $\Delta a=-2\rho$ and ${\partial }_j{B}_{ji}=2J_i$.

Define two trace-reversed data prescriptions:

For either row, with data $(a_{\mu \nu },b_{\mu \nu })$, evolve

Model 3 uses ${{\mathcal{R}}^{\mathrm{NW}}}_{t_0}$.