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Classical and Quantum Gravity in 1+1 Dimensions, Part II: The Universal Coverings
General Relativity and Quantum Cosmology · Submitted 30 Nov 1995
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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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.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.92 | 0.94 | 0.50 | asserted 1.00 |
| s1 | The application of these rules is extremely straightforward, as is demonstrated at various examples and illustrated with numerous figures. | contribution 0.38 | 0.81 | 0.06 | asserted 1.00 |
| s2 | Despite the resulting simplicity we also comment on some subtleties concerning the concept of Penrose diagrams. | limitation 0.50 | 0.70 | 0.05 | asserted 0.82 |
| s3 | 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. | result 0.94 | 0.76 | 0.27 | asserted 0.69 |
| s4 | 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. | interpretation 0.57 | 0.76 | 0.24 | asserted 0.93 |
| s5 | 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). | contribution 0.61 | 0.91 | 0.29 | asserted 1.00 |
| s6 | 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. | future work 0.96 | 0.40 | 0.06 | asserted 1.00 |
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Original 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.
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6 arXiv papers cite this in 8 sentences
Papers citing it · 7 sentences without a confident label
Cites it as background 1 paper
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Lower dimensional – both (2+1)- and (1+1)-dimensional – theories for gravity have been formulated with a twofold purpose of clarifying the mathematical properties of the Einstein equations (since they are a system of coupled nonlinear partial derivative equations, their solution has been derived so far only in simplified cases requiring strong symmetry assumptions) in an analogue but formally simpler case, and for developing possible frameworks for a quantum formulation of the gravitational interaction [ref] .
Curvature Invariants and Lower Dimensional Black Hole Horizons · Introduction: Locating the Event Horizon of a Black Hole · 0.99
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