Einstein's equations and the embedding of 3-dimensional CR manifolds

Hill, C. Denson; Lewandowski, Jerzy; Nurowski, Paweł

doi:10.1512/iumj.2008.57.3473

Cited by 20 publications

(61 citation statements)

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“…At this point it is worthwhile to mention that a similar problem was studied by one of us some years ago in [14]; see also the more modern treatment in [6]. Using the results of [6,14] and the symbolic calculation program Mathematica, we reduced the Einstein equation (11.5) to the following form:…”

Section: Reduction Of the Einstein Equationsmentioning

confidence: 99%

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Intrinsic geometry of oriented congruences in three dimensions

Hill¹,

Nurowski²

2009

Journal of Geometry and Physics

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a b s t r a c tStarting from the classical notion of an oriented congruence (i.e. a foliation by oriented curves) in R 3 , we abstract the notion of an oriented congruence structure. This is a 3-dimensional CR manifold (M, H, J) with a preferred splitting of the tangent space TM = V ⊕ H. We find all local invariants of such structures using Cartan's equivalence method refining Cartan's classification of 3-dimensional CR structures. We use these invariants and perform Fefferman like constructions, to obtain interesting Lorentzian metrics in four dimensions, which include explicit Ricci-flat and Einstein metrics, as well as not conformally Einstein Bach-flat metrics.

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Section: Reduction Of the Einstein Equationsmentioning

confidence: 99%

“…Using the results of [6,14] and the symbolic calculation program Mathematica, we reduced the Einstein equation (11.5) to the following form:…”

Section: Reduction Of the Einstein Equationsmentioning

confidence: 99%

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Intrinsic geometry of oriented congruences in three dimensions

Hill¹,

Nurowski²

2009

Journal of Geometry and Physics

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“…Good sources of background on the relationship between spacetimes and CR structures can be found, for instance, in the thesis of Nurowski [5]. Recently, Hill, Lewandowski and Nurowski [6] generalized earlier work of [7,5] to provide a new formulation of twisting algebraically special spacetimes with cosmological constant. It allows a classification of algebraically special spacetimes according to Cartan's classification of 3-dimensional CR structures.…”

Section: Introductionmentioning

confidence: 99%

“…Building on the work of Hill, Lewandowski, and Nurowski [6], we first present, in Section 2, the twisting type II vacuum metric formulated according to CR geometry, together with our calculated Weyl scalars. Then in Section 3, we establish the transformation from the CR formalism to the canonical frame that is widely used, e.g., in [12].…”

Section: Introductionmentioning

confidence: 99%

CR structures and twisting vacuum spacetimes with two Killing vectors and cosmological constant: type II and more special

Zhang¹,

Finley²

2013

Class. Quantum Grav.

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Based on the CR formalism of algebraically special spacetimes by Hill, Lewandowski and Nurowski, we derive a nonlinear system of two real ODEs, of which the general solution determines a twisting type II (or more special) vacuum spacetime with two Killing vectors (commuting or not) and at most seven real parameters in addition to the cosmological constant Λ. To demonstrate a broad range of interesting spacetimes that these ODEs can capture, special solutions of various Petrov types are presented and described as they appear in this approach. They include Kerr-NUT, Kerr and Debney/Demiański's type II, Lun's type II and III (subclasses of Held-Robinson), MacCallum and Siklos' type III (Λ < 0), and the type N solutions (Λ = 0) we found in an earlier paper, along with a new class of type II solutions as a nontrivial limit of Kerr and Debney's type II solutions. Also, we discuss a situation in which the two ODEs can be reduced to one. However, constructing the general solution still remains an open problem.

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