Instability of marginally outer trapped surfaces from initial data set symmetry

Abstract

Let $(\tilde{\Sigma },h_{ab},K_{ab})$ be an initial data set and let $x^a$ be a symmetry vector of $\tilde{\Sigma }$. Consider a MOTS $\mathcal{S}$ in $\tilde{\Sigma }$ and let the symmetry vector be decomposable along the unit normal to $\mathcal{S}$ in $\tilde{\Sigma }$, and along $\mathcal{S}$. In this note we present some basic results with regards to the stability of $\mathcal{S}$. The vector decomposition allows us to characterize the instability of $\mathcal{S}$ by the nature of the zero set of the normal component to $\mathcal{S}$ and the divergence of the component along $\mathcal{S}$. Further observations are made under the assumption of $\mathcal{S}$ having a constant mean curvature, and $\tilde{\Sigma }$ being an Einstein manifold.

7 2-column pages, previously unstated assumptions on the spacetime geometry included to ensure the validity of some of the results, typos fixed, accepted in CQG