Geometric properties of a certain class of compact dynamical horizons in locally rotationally symmetric class II spacetimes

Abstract

In this paper, we study the geometry of a certain class of compact dynamical horizons with a time-dependent induced metric in locally rotationally symmetric class II spacetimes. We first obtain a compactness condition for embedded [Formula: see text]-manifolds in these spacetimes, satisfying the weak energy condition, with non-negative isotropic pressure [Formula: see text]. General conditions for a [Formula: see text]-manifold to be a dynamical horizon are imposed, as well as certain genericity conditions, which in the case of locally rotationally symmetric class II spacetimes reduces to the statement that “the weak energy condition is strictly satisfied or otherwise violated”. The compactness condition is presented as a spatial first-order partial differential equation in the sheet expansion [Formula: see text], in the form [Formula: see text], where [Formula: see text] is the Gaussian curvature of [Formula: see text]-surfaces in the spacetime and [Formula: see text] is a real number parametrizing the differential equation, where [Formula: see text] can take on only two values, [Formula: see text] and [Formula: see text]. Using geometric arguments, it is shown that the case [Formula: see text] can be ruled out and the [Formula: see text] ([Formula: see text]-dimensional sphere) geometry of compact dynamical horizons for the case [Formula: see text] is established. Finally, an invariant characterization of this class of compact dynamical horizons is also presented.