Spherically symmetric systems of fields and black holes. I. Definition and properties of apparent horizon

We investigate three model field theories: a minimally coupled charged scalar field together with gravity and electromagnetism, a minimally coupled SO(3) Yang-Mills field and gravity, and the Callan-Coleman-Jackiw scalar field. We restrict ourselves to spherically symmetric configurations; the corresponding dimensional reduction leads to an action functional on a two-dimensional spacetime which contains a metric, a neutral scalar, a charged scalar, and an electromagnetic field. The action is written in the second-order, covariant and gauge-invariant form. We generalize the definition of the future and past apparent horizon so that it will not be visible from the future and past null infinity, respectively, and will form a nontimelike surface, both also in the case of the Callan-Coleman-Jackiw model. We prove an inequality relating the surface area and the charges of the apparent horizon. We study the boundary conditions for the fields at the horizon, at the regular center, and at infinity. Finally, we speculate on the existence of static spherically symmetric solutions, where a black hole is surrounded by a matter shell; in two-dimensional spacetime, this looks like a kink.