Classical and Quantum Gravity in 1+1 Dimensions, Part II: The Universal Coverings

TL;DR

This paper presents simple rules for constructing maximal extensions of two-dimensional spacetimes with Killing fields, illustrating their application with examples, and discusses subtleties in Penrose diagrams related to horizons and redshift effects.

Contribution

It introduces universal covering solutions for 2D gravity-Yang-Mills systems, extending local solutions to maximal globally extended spacetimes, without requiring prior knowledge of Part I.

Findings

Provided straightforward rules for maximal extensions of 2D spacetimes.

Demonstrated that smooth spacetimes with mixed horizons lack globally smooth Penrose diagrams.

Identified physical redshift effects preventing smooth Penrose diagrams in certain cases.

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 at 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, maybe, 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, e.g., 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 kept entirely self-contained; a prior reading of Part I is not necessary.