The type N Karlhede bound is sharp

Pelavas, Nicos

doi:10.1088/0264-9381/25/1/012001

Cited by 16 publications

(26 citation statements)

References 14 publications

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“…The relationship to curvature homogeneous spacetimes is addressed in [13]. Support for these conjectures in the 4D case is discussed in the next section.…”

Section: Discussionmentioning

confidence: 92%

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Spacetimes With Constant Scalar Invariants

Hervik

2008

The Eleventh Marcel Grossmann Meeting

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Abstract. We study Lorentzian spacetimes for which all scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant (CSI spacetimes). We obtain a number of general results in arbitrary dimensions. We study and construct warped product CSI spacetimes and higher-dimensional Kundt CSI spacetimes. We show how these spacetimes can be constructed from locally homogeneous spaces and V SI spacetimes. The results suggest a number of conjectures. In particular, it is plausible that for CSI spacetimes that are not locally homogeneous the Weyl type is II, III, N or O, with any boost weight zero components being constant. We then consider the four-dimensional CSI spacetimes in more detail. We show that there are severe constraints on these spacetimes, and we argue that it is plausible that they are either locally homogeneous or that the spacetime necessarily belongs to the Kundt class of CSI spacetimes, all of which are constructed. The four-dimensional results lend support to the conjectures in higher dimensions.

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“…The relationship to curvature homogeneous spacetimes is addressed in [13]. Support for these conjectures in the 4D case is discussed in the next section.…”

Section: Discussionmentioning

confidence: 92%

“…The relationship between CSI and curvature homogeneity in 4D is studied in [13]. In locally homogeneous spacetimes all of the sectional curvatures (Gaussian curvatures) are constant.…”

Section: D Csimentioning

confidence: 99%

Spacetimes With Constant Scalar Invariants

Hervik

2008

The Eleventh Marcel Grossmann Meeting

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“…In both cases the classes of solutions have been determined and shown to be parameterized by one function of one variable [12]. More recently, a study of four dimensional curvature homogeneous spacetimes has shown that CH 3 implies local homogeneity [15]; in addition, there exists a class of proper CH 2 spacetimes of Petrov type N, Plebanski-Petrov type N with a negative cosmological constant. The class of proper CH 2 spacetimes has been explicitly determined and shown to depend on one function of one variable [15].…”

Section: Discussionmentioning

confidence: 99%

Lorentzian spacetimes with constant curvature invariants in three dimensions

Coley¹,

Hervik²,

Pelavas³

2007

Class. Quantum Grav.

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Abstract. In this paper we study Lorentzian spacetimes for which all polynomial scalar invariants constructed from the Riemann tensor and its covariant derivatives are constant (CSI spacetimes) in three dimensions. We determine all such CSI metrics explicitly, and show that for every CSI with particular constant invariants there is a locally homogeneous spacetime with precisely the same constant invariants. We prove that a three-dimensional CSI spacetime is either (i) locally homogeneous or (ii) it is locally a Kundt spacetime. Moreover, we show that there exists a null frame in which the Riemann (Ricci) tensor and its derivatives are of boost order zero with constant boost weight zero components at each order. Lastly, these spacetimes can be explicitly constructed from locally homogeneous spacetimes and vanishing scalar invariant spacetimes.

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“…In this paper we are interested in curvature homogeneity for a different, but related reason. As was shown in [18], curvature homogeneity is also a key concept in the search for maximal order metrics. The relevant observation is that for a CH k geometry the rank r k is small because t k = 0, and this is exactly what is needed for maximal order.…”

Section: Curvature Homogeneity and Pseudo-stabilizationmentioning

confidence: 92%

Three-dimensional spacetimes of maximal order

Wylleman¹

2013

Class. Quantum Grav.

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We show that the equivalence problem for three-dimensional Lorentzian manifolds requires at most the fifth covariant derivative of the curvature tensor. We prove that this bound is sharp by exhibiting a class of 3D Lorentzian manifolds which realize this bound. The analysis is based on a three-dimensional analogue of the Newman-Penrose formalism, and spinorial classification of the three-dimensional Ricci tensor.

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The type N Karlhede bound is sharp

Cited by 16 publications

References 14 publications

Spacetimes With Constant Scalar Invariants

Spacetimes With Constant Scalar Invariants

Lorentzian spacetimes with constant curvature invariants in three dimensions

Three-dimensional spacetimes of maximal order

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