Proposition: Plane and section

With

one has ${\sigma }_{\mathrm{plane}}(e,r)=1$. Every $k=qe+\pi r$ is a linearized vacuum solution, and no nonzero such $k$ is pure gauge.

For

the decomposition $h=k+K\xi$ is unique, and the retarded harmonic map

satisfies $w(h)=\xi$ and $h-Kw(h)=k$.

Proof. The Cauchy data are $(e,\dot{e})=(A,0)$ and $(r,\dot{r})=(0,A/M_A)$, giving the symplectic pairing. TT wave evolution gives $D(k)=0$, zero trace, and $\square k=0$; substitution into the linearized Einstein tensor gives vacuum.

Linearized curvature is unchanged by $K\xi$. For a temporal-gauge TT field, ${R^{(1)}}_{0i0j}(k)=-\frac{1}{2}{{\partial }_t}^2{k}_{ij}$. A pure gauge member of the plane would therefore satisfy $\Delta k=0$. A harmonic Schwartz function vanishes, so $k=0$.

For the section, (eq:gauge-identities) gives $D(h)=\square {\xi }^{♭}$. Since $\xi$ is compactly supported, the retarded zero-past solution of that equation is ${\xi }^{♭}$. A retarded solution can be constructed using zero Cauchy data before the compact source; Lemma lem:wave with $m=0$ gives existence and uniqueness. Uniqueness of $k$ follows from the curvature argument, and a compactly supported Minkowski Killing field is zero. Thus the decomposition and the section are unique.