Directions in General Relativity Book contents
- Frontmatter
- Contents
- Contributors
- Symposium Program
- Papers from both Volumes Classified by Subjects
- Preface
- Dieter Brill: A Spacetime Perspective
- Thawing the Frozen Formalism: The Difference Between Observables and What We Observe
- Jacobi's Action and the Density of States
- Decoherence of Correlation Histories
- The Initial Value Problem in Light of Ashtekar's Variables
- Status Report on an Axiomatic Basis for Functional Integration
- Solution of the Coupled Einstein Constraints On Asymptotically Euclidean Manifolds
- Compact Cauchy Horizons and Cauchy Surfaces
- The Classical Electron
- Gauge (In)variance, Mass and Parity in D=3 Revisited
- Triality, Exceptional Lie Groups and Dirac Operators
- The Reduction of the State Vector and Limitations on Measurement in the Quantum Mechanics of Closed Systems
- Quantum Linearization Instabilities of de Sitter Spacetime
- What is the True Description of Charged Black Holes?
- Limits on the Adiabatic Index in Static Stellar Models
- On the Relativity of Rotation
- Recent Progress and Open Problems in Linearization Stability
- Brill Waves
- You Can't Get There from Here: Constraints on Topology Change
- Time, Measurement and Information Loss in Quantum Cosmology
- Impossible Measurements on Quantum Fields
- A New Condition Implying the Existence of a Constant Mean Curvature Foliation
- Maximal Slices in Stationary Spacetimes with Ergoregions
- (1 + 1)-Dimensional Methods for General Relativity
- Coalescence of Primal Gravity Waves to Make Cosmological Mass Without Matter
- Curriculum Vitae of Dieter Brill
- Ph. D. Theses supervised by Dieter Brill
- List of Publications by Dieter Brill
(1 + 1)-Dimensional Methods for General Relativity
Published online by Cambridge University Press: 06 January 2010
- Frontmatter
- Contents
- Contributors
- Symposium Program
- Papers from both Volumes Classified by Subjects
- Preface
- Dieter Brill: A Spacetime Perspective
- Thawing the Frozen Formalism: The Difference Between Observables and What We Observe
- Jacobi's Action and the Density of States
- Decoherence of Correlation Histories
- The Initial Value Problem in Light of Ashtekar's Variables
- Status Report on an Axiomatic Basis for Functional Integration
- Solution of the Coupled Einstein Constraints On Asymptotically Euclidean Manifolds
- Compact Cauchy Horizons and Cauchy Surfaces
- The Classical Electron
- Gauge (In)variance, Mass and Parity in D=3 Revisited
- Triality, Exceptional Lie Groups and Dirac Operators
- The Reduction of the State Vector and Limitations on Measurement in the Quantum Mechanics of Closed Systems
- Quantum Linearization Instabilities of de Sitter Spacetime
- What is the True Description of Charged Black Holes?
- Limits on the Adiabatic Index in Static Stellar Models
- On the Relativity of Rotation
- Recent Progress and Open Problems in Linearization Stability
- Brill Waves
- You Can't Get There from Here: Constraints on Topology Change
- Time, Measurement and Information Loss in Quantum Cosmology
- Impossible Measurements on Quantum Fields
- A New Condition Implying the Existence of a Constant Mean Curvature Foliation
- Maximal Slices in Stationary Spacetimes with Ergoregions
- (1 + 1)-Dimensional Methods for General Relativity
- Coalescence of Primal Gravity Waves to Make Cosmological Mass Without Matter
- Curriculum Vitae of Dieter Brill
- Ph. D. Theses supervised by Dieter Brill
- List of Publications by Dieter Brill
Summary
Abstract
We present the (1+1)-dimensional method for studying general relativity of 4-dimensions. We first discuss the general formalism, and subsequently draw attention to the algebraically special class of space-times, following the Petrov classification. It is shown that this class of space-times can be described by the (1+1)-dimensional Yang-Mills action interacting with matter fields, with the spacial diffeomorphisms of the 2-surface as the gauge symmetry. The (Hamiltonian) constraint appears polynomial in part, whereas the non-polynomial part is a non-linear sigma model type in (1+1)-dimensions. It is also shown that the representations of w∞-gravity appear naturally as special cases of this description, and we discuss briefly the w∞-geometry in term of the fibre bundle.
Introduction
For past years many 2-dimensional field theories have been intensively studied as laboratories for many theoretical issues, due to great mathematical simplicities that often exist in 2-dimensional systems. Recently these 2-dimensional field theories have received considerable attention, for different reasons, in connection with general relativistic systems of 4-dimensions, such as self-dual spaces [1] and the black-hole space-times [2, 3]. These 2-dimensional formulations of self-dual spaces and blackhole space-times of allow, in principle, many 2-dimensional field theoretic methods developed in the past relevant for the description of the physics of 4-dimensions.
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- Directions in General RelativityProceedings of the 1993 International Symposium, Maryland: Papers in Honor of Dieter Brill, pp. 323 - 338Publisher: Cambridge University PressPrint publication year: 1956