Involute, minimal, outer, and increasingly trapped spheres

Seven different refinements of trapped surfaces are proposed, each intended as potential stability conditions. This article concerns spherical symmetry, but each condition can be generalized. Involute trapped spheres satisfy a similar condition to minimal trapped spheres, which are strictly minimal with respect to the Kodama vector. There is also a weaker version of involute trapped spheres. Outer trapped spheres have positive surface gravity. Increasingly (future, respectively, past) trapped spheres generate spheres which are more trapped in a (future, respectively, past) causal direction, with three types: in any such causal direction, along the dual Kodama vector, and in some such causal direction. Assuming the null energy condition, the seven conditions form a strict hierarchy, in the above order. In static space-times, they reduce to three inequivalent definitions, namely, minimal, outer, and increasingly trapped spheres. For a widely considered class of so-called nice (or nondirty) black holes, minimal trapped and outer trapped become equivalent. Reissner-Nordström black holes provide examples of this, and that the increasingly trapped differs. Examples where all three refinements differ are provided by a simple family of dirty black holes parametrized by mass and singularity area.