The symmetric-tensor curl symbol is skew-adjoint with norm at most the wave number, giving positivity of the dust principal block

Let $V$ be the real vector space of symmetric $3\times 3$ matrices with the positive Frobenius product $⟨B,B^{\mathrm{\prime }}⟩=B^{ij}{B}_{ij}^{\mathrm{\prime }}$, and for $\xi \in {\mathbb{R}}^3$ define $C(\xi ):V\to V,(C(\xi )B{)}_{ij}={ϵ}_{kl(i}{\xi }^k{B}_{j)}{}^l,$ with indices raised and lowered by the Euclidean metric and $ϵ$ the Levi-Civita symbol on the spatial frame of Dust reduction variables: signature flip, frame, connection, kinematics, comoving-parallel gauge and unknowns. Then:

(a) $C(\xi )$ is skew-adjoint: $⟨C(\xi )B,B^{\mathrm{\prime }}⟩=-⟨B,C(\xi )B^{\mathrm{\prime }}⟩$;

(b) $\mathrm{\parallel }C(\xi )\mathrm{\parallel }\le \mathrm{\mid }\xi \mathrm{\mid }$ for the operator norm induced by the Frobenius product;

(c) the block operator $\left(\begin{array}{cc}0& C(\xi )\\ -C(\xi )& 0\end{array}\right)$ on $V\oplus V$ is symmetric (self-adjoint) with operator norm at most $\mathrm{\mid }\xi \mathrm{\mid }$;

(d) if $\zeta$ is a covector with $\zeta (U)=z$ and spatial part ${\zeta }_{\perp }$, the symbol matrix $\left(\begin{array}{cc}zI& C({\zeta }_{\perp })\\ -C({\zeta }_{\perp })& zI\end{array}\right)$ is positive definite whenever $z>\mathrm{\mid }{\zeta }_{\perp }\mathrm{\mid }$, in particular for every future-positive timelike conormal $\zeta$. Thus the coupled electric/magnetic Weyl principal block has a positive time matrix on every spacelike initial surface. This is a statement about that block; it makes no assertion here about the other evolution equations of the full reduced dust system.