Linearisation instability of gravity waves?

Abstract

Gravity waves in irrotational dust spacetimes are characterised by nonzero magnetic Weyl tensor $H_{ab}$. In the linearised theory, the divergence of $H_{ab}$ is set to zero. Recently Lesame et al. [Phys. Rev. D {\bf 53}, 738 (1996)] presented an argument to show that, in the exact nonlinear theory, $div H=0$ forces $H_{ab}=0$, thus implying a linearisation instability for gravity waves interacting with matter. However a sign error in the equations invalidates their conclusion. Bianchi type V spacetimes are shown to include examples with $div H=0\neq H_{ab}$. An improved covariant formalism is used to show that in a generic irrotational dust spacetime, the covariant constraint equations are preserved under evolution. It is shown elsewhere that $\mbox{div} H=0$ does not generate further conditions.