Sharp version of the Goldberg–Sachs theorem

Gover, A. Rod; Hill, C. Denson; Nurowski, Paweł

doi:10.1007/s10231-010-0151-4

Cited by 28 publications

(46 citation statements)

References 19 publications

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“…The conformal invariance of the theorem itself is less straightforward, but can be checked from the tranformation laws of the Bianchi identity. This property is well-known in four dimensions [PR86,GHN10], and we defer its generalisation to higher dimensions for a future publication [TCa].…”

Section: A Five-dimensional Goldberg-sachs Theoremmentioning

confidence: 97%

Optical structures, algebraically special spacetimes, and the Goldberg–Sachs theorem in five dimensions

Taghavi-Chabert¹

2011

Class. Quantum Grav.

112

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Optical (or Robinson) structures are one generalisation of four-dimensional shearfree congruences of null geodesics to higher dimensions. They are Lorentzian analogues of complex and CR structures. In this context, we extend the Goldberg-Sachs theorem to five dimensions. To be precise, we find a new algebraic condition on the Weyl tensor, which generalises the Petrov type II condition, in the sense that it ensures the existence of such congruences on a five-dimensional spacetime, vacuum or under weaker assumptions on the Ricci tensor. This results in a significant simplification of the field equations. We discuss possible degenerate cases, including a five-dimensional generalisation of the Petrov type D condition. We also show that the vacuum black ring solution is endowed with optical structures, yet fails to be algebraically special with respect to them. We finally explain the generalisation of these ideas to higher dimensions, which has been checked in six and seven dimensions.

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Section: A Five-dimensional Goldberg-sachs Theoremmentioning

confidence: 97%

Optical structures, algebraically special spacetimes, and the Goldberg–Sachs theorem in five dimensions

Taghavi-Chabert¹

2011

Class. Quantum Grav.

112

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“…The above diagram depicts the classification over C. In the real case (such is ours) there are more sub-cases, as some of the intersection points might be complex. See for example [15] for the complete classification.…”

Section: 22mentioning

confidence: 99%

The dancing metric, ${G}_2$-symmetry and projective rolling

Bor

Lamoneda

Nurowski

2018

Trans. Amer. Math. Soc.

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The "dancing metric" is a pseudo-riemannian metric g of signature (2,2) on the space M 4 of non-incident point-line pairs in the real projective plane RP 2 . The null-curves of (M 4 , g) are given by the "dancing condition": the point is moving towards a point on the line, about which the line is turning. We establish a dictionary between classical projective geometry (incidence, cross ratio, projective duality, projective invariants of plane curves. . . ) and pseudo-riemannian 4-dimensional conformal geometry (null-curves and geodesics, parallel transport, self-dual null 2-planes, the Weyl curvature,. . . ). There is also an unexpected bonus: by applying a twistor construction to (M 4 , g), a G 2 -symmetry emerges, hidden deep in classical projective geometry. To uncover this symmetry, one needs to refine the "dancing condition" by a higher-order condition, expressed in terms of the osculating conic along a plane curve. The outcome is a correspondence between curves in the projective plane and its dual, a projective geometry analog of the more familiar "rolling without slipping and twisting" for a pair of riemannian surfaces.

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“…Algebraic classification of totally symmetric 4-index spinors (like C ABCD and CȦḂĊḊ) has been presented in [14]. [26] the authors used the following symbols for these types: G r , SG, G, II r , II, D r , D, III r , N r and 0, respectively. We do not present here the details of this classification.…”

Section: Formalismmentioning

confidence: 99%

On geometry of congruences of null strings in 4-dimensional complex and real pseudo-Riemannian spaces

Chudecki

2017

Journal of Mathematical Physics

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Abstract.4-dimensional spaces equipped with 2-dimensional (complex holomorphic or real smooth) completely integrable distributions are considered. The integral manifolds of such distributions are totally null and totally geodesics 2-dimensional surfaces which are called the null strings. Properties of congruences (foliations) of such 2-surfaces are studied. Some relations between properties of congruences of null strings, Petrov-Penrose type of SD Weyl spinor and algebraic types of the traceless Ricci tensor are analyzed.

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Sharp version of the Goldberg–Sachs theorem

Cited by 28 publications

References 19 publications

Optical structures, algebraically special spacetimes, and the Goldberg–Sachs theorem in five dimensions

Optical structures, algebraically special spacetimes, and the Goldberg–Sachs theorem in five dimensions

The dancing metric, ${G}_2$-symmetry and projective rolling

On geometry of congruences of null strings in 4-dimensional complex and real pseudo-Riemannian spaces

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