| Popis: |
Spacetimes with stationary regions have always been among the most intensively studied ones, but by far the larger body of literature thereon has been inextricably connected with the investigation of stationary black holes arising as solutions of the Einstein field equations. Although the most important results, such as those pertaining to the issue of black hole uniqueness (Stationary black holes: uniqueness and beyond, Living Rev, Relativity, [8]), (Black hole uniqueness theorems, Cambridge University Press, Cambridge[19]) or rigidity theorems (Rigidity results in general relativity: a review [20]) do depend very sensitively on the dimension of the spacetime, assumptions of asymptotic flatness and/or the detailed analytical features of the field equations, many interesting results pertaining to stationary spacetimes and horizons therein can still be obtained by purely geometric methods. It is the purpose of this paper to give a review, without any attempt at comprehensiveness, of some global geometric consequences of the existence of a complete Killing vector field which becomes timelike at some open set, the connected components of which are referred to as stationary regions. If the Killing field changes causal character, horizons appear. I discuss their general structure and regularity under suitable assumptions on their causality and geodesic (in)completeness, but without assuming any field equations, asymptotic flatness/hyperbolicity or any dimensional restrictions. More specifically, the main focus is on presenting some old and new theorems giving descriptions of the global geometric structure of stationary regions, as well as regularity of the underlying horizons. These are meant to illustrate that a number of the extant results in the literature are not artifacts of solutions in General Relativity and/or asymptotic assumptions. Although many of the results presented are known in some form (maybe with slightly different assumptions), most are reworked in a (hopefully) didactic, unified fashion, and a number of them with new proofs. |
