Causal dynamics of null horizons under linear perturbations

We study the causal dynamics of an embedded null horizon foliated by marginally outer trapped surfaces (MOTS) for a locally rotationally symmetric background spacetime subjected to linear perturbations. We introduce a simple procedure which characterizes the transition of the causal character of the null horizon. We apply our characterization scheme to nondissipative perturbations of the Schwarzschild and spatially homogeneous backgrounds. For the latter, a linear equation of state was imposed. Assuming a harmonic decomposition of the linearized field equations, we clarify the variables of a formal solution to the linearized system that determine how the null horizon evolves. For both classes of backgrounds, the shear and vorticity 2-vectors are essential to the characterization, and their roles are made precise. Finally, we discuss aspects of the relationship between the characterizing conditions and the MOTS stability operator. Various properties related to the self-adjointness of the MOTS stability operator are extensively discussed.