Geometry and fixed data

Let

with

Fix

Let $E={\mathrm{Sym}}^2{T}^{\ast }M$. The tensor arguments are smooth perturbations in $C^{\infty }(M;E)$, not necessarily decaying at spatial infinity. The scalar is real and smooth unless an algebraic complexification of a test pairing is explicitly used.

Choose a fixed real smooth switch

Its coefficients are ambient smooth and flat at the plateau endpoints. No discontinuous extension at $t=0$ is introduced.

Use the fixed volume density ${\mathrm{d}}^{4}x$ and global Minkowski frames to identify tensor-valued distributional densities with their finitely many component distributions. All pairings are bilinear, without complex conjugation.

Trace reversal is

The selected free tensor operator is

The inverse coefficient remains $-1$, not $-4$. The action-equation convention $r_{\mathrm{D}}=-u_{\mathrm{D}}$ does not rename the response or kernel used here XVIA,XVID.

The frequency is $\lambda$. Repeated spacetime indices are denoted by $\alpha ,\mu ,\nu ,\rho ,\ell$ rather than by $\lambda$.