Journal ArticleDOI
The Cauchy problem in general relativity. III. On locally imbedding a family of null hypersurfaces
TLDR
In this paper, the problem of locally imbedding a null hypersurface in a Riemannian manifold was studied and the generalized Gauss-Codazzi equations were derived.Abstract:
This paper is concerned with the problem of locally imbedding a null hypersurface in a Riemannian manifold. More precisely, on a one‐parameter family of null hypersurfaces, rigged by an arbitrary null vector field, in a four‐dimensional space–time manifold, a particular symmetric affine connection is used to derive the corresponding generalized Gauss–Codazzi equations. In addition, expressions are obtained for the projections of the Ricci tensor, which are relevant to the characteristic initial‐value problem of general relativity.read more
Citations
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Null Surface Geometrodynamics
TL;DR: In this article, the dynamical structure of the theory of gravity using a time parameter whose level surfaces are null hypersurfaces in spacetime is investigated using a spacelike foliation of codimension two.
Journal ArticleDOI
Lorentzian Geometry of Globally Framed Manifolds
TL;DR: In this article, a new class of globally framed manifolds (carrying a Lorentz metric) is introduced to establish a relation between the spacetime geometry and framed structures, and it is shown that strongly causal (in particular, globally hyperbolic) spacetimes can carry a regular framed structure.
Journal ArticleDOI
Particles and strings in degenerate metric spaces
L. A. Cabral,Victor O. Rivelles +1 more
TL;DR: In this paper, the authors consider relativistic and non-relativistic particles and strings in spaces (or spacetimes) with a degenerate metric and show that the resulting dynamics is described by a rich structure of constraints.
Journal ArticleDOI
Observer-dependent Gauss–Codazzi formalism for null hypersurfaces in the space–time
TL;DR: In this article, a Gauss-Codazzi framework for null hypersurfaces in the space-time was introduced, with the use of space time splitting techniques, and working within the framework of general coordinates of the ambient space.
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Review: The Mathematical Theory of Gravitational Discontinuity Hypersurfaces
TL;DR: In this paper, a new observer-dependent Gauss-Codazzi analogue is introduced, and the definition of conservative solution is extended to the characteristic case and a result by Smoller and Temple is generalized.
References
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Journal ArticleDOI
Gravitational Waves in General Relativity. VII. Waves from Axi-Symmetric Isolated Systems
TL;DR: In this paper, it is shown that the flow of information to infinity is controlled by a single function of two variables called the news function, together with initial conditions specified on a light cone, which fully defines the behaviour of the system.
Journal ArticleDOI
Gravitational Waves in General Relativity. VIII. Waves in Asymptotically Flat Space-Time
TL;DR: In this paper, a convenient way of splitting the metric tensor and the Einstein field equations, applicable in any space-time, is first introduced, and suitable boundary conditions are set.
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Gravitational Fields in Finite and Conformal Bondi Frames
TL;DR: In this paper, the authors generalize the Bondi-Sachs treatment of the initial value problem using null coordinate systems, which is applicable in both finite and asymptotic regions of space whose sources are bounded by a finite world tube.
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On the Characteristic Initial Value Problem in Gravitational Theory
TL;DR: In this paper, the problem of determining a solution of the Einstein field equations for the gravitational field from data set on a pair of intersecting characteristic (that is, null) hypersurfaces and on their intersection is considered.