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
A New Condition Implying the Existence of a Constant Mean Curvature Foliation
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
It is shown that if a non-flat spacetime (M, g) whose future c-boundary is a single point satisfies RabVaVb ≥ 0 for all timelike vectors Va, equality holding only if Rab = 0, then sufficiently close to the future c-boundary the spacetime can be uniquely foliated by constant mean curvature compact hypersurfaces. The uniqueness proof uses a variational method developed by Brill and Flaherty to establish the uniqueness of maximal hypersurfaces.
In 1976 Dieter Brill and Frank Flaherty (1976) published an extremely important paper, “Isolated Maximal Hypersurfaces in Spacetime”, establishing that maximal hypersurfaces are unique in closed universes with attractive gravity everywhere. That is, there is only one such hypersurface, if it exists at all. In an earlier paper, Brill had established that in three-torus universes, only suitably identified flat space possessed a maximal hypersurface, so the existence of a maximal hypersurface is not guaranteed. These results by Brill are important because maximal hypersurfaces are very convenient spacelike hypersurfaces upon which to impose initial data; on such hypersurfaces the constraint equations are enormously simplified. Furthermore, in asymptotically flat space, foliations of spacetime by maximal hypersurfaces often exist, and the simplifications of the constraint equations on such a foliation make it easy to numerically solve the full four-dimensional vacuum Einstein equations for physically interesting situations.
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- Directions in General RelativityProceedings of the 1993 International Symposium, Maryland: Papers in Honor of Dieter Brill, pp. 306 - 315Publisher: Cambridge University PressPrint publication year: 1956
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