Spacetime, fields, and zero-Cauchy response

Let

Indices are raised and lowered with $\eta$, and

The metric perturbation $h_{\mu \nu }$ is a smooth symmetric covariant tensor. Write

The symbol $h$ without indices denotes the trace only where no confusion can arise.

Let $E_{\mathrm{R}}$ and $E_{\mathrm{A}}$ denote the retarded and advanced Green operators for $\square$, respectively:

The zero-Cauchy solution operator ${\mathsf{E}}_0$ is defined by

For smooth $s$, it can be realized for $t>0$ as

The right-hand side is understood as the retarded solution of the past-compact distributional source $\theta s$. Its restriction to $M_{+}$ has the prescribed zero Cauchy data.

The first coordinate response is

This operator is not switched.

For a compact set $K⋐\{t>0\}$, set

In Minkowski spacetime this is compact. Finite propagation speed implies that $w[h]{|}_K$ depends only on $h$ in ${\mathcal{D}}_K$, including its trace on the initial surface.