Classical and quantum gravity in 1 + 1 dimensions. II: The universal coverings

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Abstract
A set of simple rules for constructing the maximal (e.g. analytic) extensions for any metric with a Killing field in an (effectively) two-dimensional spacetime is formulated. The application of these rules is extremely straightforward, as is demonstrated by various examples and illustrated with numerous figures. Despite the resulting simplicity we also comment on some subtleties concerning the concept of Penrose diagrams. Most noteworthy among these, perhaps, is that (smooth) spacetimes which have both degenerate and non-degenerate (Killing) horizons do not allow for globally smooth Penrose diagrams. Physically speaking this obstruction corresponds to an infinite relative red/blueshift between observers moving across the two horizons. The present work provides a further step in the classification of all global solutions of the general class of two-dimensional gravity - Yang - Mills systems introduced in part I, comprising, for example, all generalized (linear and nonlinear) dilaton theories. In part I we constructed the local solutions, which were found to always have a Killing field; in this paper we provide all universal covering solutions (the simply connected maximally extended spacetimes). A subsequent part III will treat the diffeomorphism inequivalent solutions for all other spacetime topologies. Part II is entirely self-contained; prior reading of part I is not necessary.