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  • 1
    Electronic Resource
    Electronic Resource
    Erratum: "Lorentzian metrics from characteristic surfaces" [J. Math. Phys. 36, 4975–4983 (1995)] (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 1114-1114 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532710
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  • 2
    Electronic Resource
    Electronic Resource
    GR via characteristic surfaces (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 4984-5004 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We reformulate the Einstein equations as equations for families of surfaces on a four-manifold. These surfaces eventually become characteristic surfaces for an Einstein metric (with or without sources). In particular they are formulated in terms of two functions on R4×S2, i.e., the sphere bundle over space–time, one of the functions playing the role of a conformal factor for a family of associated conformal metrics, the other function describing an S2's worth of surfaces at each space–time point. It is from these families of surfaces themselves that the conformal metric, conformal to an Einstein metric, is constructed; the conformal factor turns them into Einstein metrics. The surfaces are null surfaces with respect to this metric. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531210
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  • 3
    Electronic Resource
    Electronic Resource
    Ill-posedness in the Einstein equations (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 5535-5549 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: It is shown that the formulation of the Einstein equations widely in use in numerical relativity, namely, the standard ADM form, as well as some of its variations (including the most recent conformally-decomposed version), suffers from a certain but standard type of ill-posedness. Specifically, the norm of the solution is not bounded by the norm of the initial data irrespective of the data. A long-running numerical experiment is performed as well, showing that the type of ill-posedness observed may not be serious in specific practical applications, as is known from many numerical simulations. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.533423
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  • 4
    Electronic Resource
    Electronic Resource
    Erratum: "GR via characteristic surfaces" [J. Math. Phys. 36, 4984–5004 (1995)] (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 1115-1115 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532700
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  • 5
    Electronic Resource
    Electronic Resource
    Lorentzian metrics from characteristic surfaces (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 4975-4983 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: The following issue is raised and discussed; when do families of foliations by hypersurfaces on a given four-dimensional manifold without further structure become the null surfaces of some unknown, but to be determined, metric gab(x). Explicit conditions for these surfaces are found, so that they do define a unique conformal metric with the surfaces themselves being characteristics of that metric. By giving an additional function (to be the conformal factor), full knowledge of the metric is determined. It is clear from these results that one can use these surfaces (and the conformal factor) as fundamental variables for describing any Lorentzian geometry and in particular for its use in general relativity. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531209
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  • 6
    Electronic Resource
    Electronic Resource
    Linearized Einstein theory via null surfaces (1995)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 36 (1995), S. 5005-5022 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: Recently there has been developed a reformulation of general relativity (GR)—referred to as the null surface version of GR—where instead of the metric field as the basic variable of the theory, families of three-surfaces in a four-manifold become basic. From these surfaces themselves, a conformal metric, conformal to an Einstein metric, can be constructed. A choice of conformal factor turns it into an Einstein metric. The surfaces are then automatically characteristic surfaces of this metric. In the present paper we explore the linearization of this null surface theory and compare it with the standard linear GR. This allows a better understanding of many of the subtle mathematical issues and sheds light on some of the obscure points of the null surface theory. It furthermore permits a very simple solution generating scheme for the linear theory and the beginning of a perturbation scheme for the full theory. © 1995 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.531211
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  • 7
    Electronic Resource
    Electronic Resource
    A spinor reformulation of the null surface treatment of GR (2000)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 41 (2000), S. 6300-6317 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: We begin with a four-dimensional manifold, M, that possesses a two parameter family of (local) foliations by three-surfaces, with the two parameters being the coordinates on the sphere of directions at each point of the manifold expressed via homogeneous coordinates πA and πA′. By then requiring that each foliation (of the two parameter set of foliations) be a one parameter family of null surfaces for some (as yet unkown) conformal Lorentzian metric—we derive an explicit expression, in terms of the foliation description, for this conformal metric. We then show (1) how a conformal factor can be chosen to convert the conformal metric into a metric and (2) how to impose on the foliation and conformal factor conditions so that the metric satisfies the vacuum Einstein equations. The material described here is very much connected to the null surface formulation (NSF) of GR developed earlier. The advantages of the present formulation are that one can much more easily see the logical structure of the NSF, one can calculate with much greater ease and finally it allows [because of the use of (spinor) index calculus] generalizations of the NSF so that the study of the evolutionary development of the null surface singularities (caustics, etc.) can be developed. © 2000 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.1288250
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  • 8
    Electronic Resource
    Electronic Resource
    Erratum: "On the dynamics of characteristic surfaces" [J. Math. Phys. 36, 6397–6416 (1995)] (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 1113-1113 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532702
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  • 9
    Electronic Resource
    Electronic Resource
    Erratum: "Linearized Einstein theory via null surfaces" [J. Math. Phys. 36, 5005–5022 (1995)] (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 1116-1116 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532711
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  • 10
    Electronic Resource
    Electronic Resource
    The Eikonal equation in asymptotically flat space–times (1999)
    College Park, Md. : American Institute of Physics (AIP)
    Journal of Mathematical Physics 40 (1999), S. 1041-1056 
    Details
    ISSN: 1089-7658
    Source: AIP Digital Archive
    Topics: Mathematics , Physics
    Notes: In an arbitrary Lorentzian manifold we provide a description for the construction of null surfaces and their associated singularities, via solutions of the Eikonal equation. In particular, we study the singularities of the past light-cones from points on null infinity, the future light-cones from arbitrary interior points and the intersection of these with null infinity and unifying relationships between the different singularities. The starting point for this work is the assumption of a known family of solutions to the Eikonal equation. The work is based on the standard theory of singularities of smooth maps by Arnold and his colleagues. Though the work is intended to stand on its own, it can be thought of as being closely related to the recently developed null surface reformulation of GR. © 1999 American Institute of Physics.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1063/1.532705
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