Fixed geometry, supports, and backgrounds

Let

The quantum scalar is defined on $M$. The slab $M_T=[0,T]\times {\mathbb{R}}^3$ and $\Sigma =\{t=0\}$ belong to the coordinate prescription. Write ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$ for the Klein–Gordon Green operators and ${{\mathsf{E}}_0}^{\mathrm{R}},{{\mathsf{E}}_0}^{\mathrm{A}}$ for the massless ones.

The backgrounds $h,k$ are fixed smooth real symmetric tensors on $M$, without field equations or decay assumptions. Only their restrictions to finite causal regions will be used. Set

With $H={\mathrm{tr}}_{\eta }h$, abbreviate

Indices are moved with $\eta$.

Fix

and the same relatively compact plateau $U$ as upstream:

Choose $\delta >0$ such that

Take $0<\epsilon <\delta$. Thus $g=0$ on $0\le t\le \delta$, and the fixed algebra lies above every ramp in this tail of the family.

The frozen ramp is

It is smooth, takes values in $[0,1]$, and is flat at the endpoints.