The fixed-time scalar map

For a smooth symmetric tensor $h$, let

and define

For $(K\xi {)}_{\mu \nu }={\partial }_{\mu }{\xi }_{\nu }+{\partial }_{\nu }{\xi }_{\mu }$ and $X_{\xi }={\xi }^{\mu }{\partial }_{\mu }$,

These identities follow by differentiating the displayed formulas; the second also follows by differentiating scalar covariance under a diffeomorphism.

For a free scalar $P\varphi =0$, set

It solves