Invariant differential operators and the Karlhede classification of type N vacuum solutions

Ramos, M. P. Machado; Vickers, James

doi:10.1088/0264-9381/13/6/023

Cited by 18 publications

(17 citation statements)

References 9 publications

(18 reference statements)

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“…However, it is not known whether these type D and type O bounds are sharp. Detailed analysis of vacuum type N solutions yields q 5 [9]. A similar analysis of non-vacuum type N solutions produced a claim of q 5 [8].…”

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confidence: 81%

The type N Karlhede bound is sharp

Pelavas¹

2007

Class. Quantum Grav.

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We present a family of four-dimensional Lorentzian manifolds whose invariant classification requires the seventh covariant derivative of the curvature tensor. The spacetimes in questions are null radiation, type N solutions on an antide Sitter background. The large order of the bound is due to the fact that these spacetimes are properly CH 2 , i.e., curvature homogeneous of order 2 but non-homogeneous. This means that tetrad components of R, ∇R, ∇ (2) R are constant, and that essential coordinates first appear as components of ∇ (3) R. Covariant derivatives of orders 4, 5, 6 yield one additional invariant each, and ∇ (7) R is needed for invariant classification. Thus, our class proves that the bound of 7 on the order of the covariant derivative, first established by Karlhede, is sharp. Our finding corrects an outstanding assertion that invariant classification of four-dimensional Lorentzian manifolds requires at most ∇ (6) R.

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confidence: 81%

The type N Karlhede bound is sharp

Pelavas¹

2007

Class. Quantum Grav.

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“…We analyse case I and II mentioned in the previous section separately. The terms relating to the covariant derivative of the Weyl spinor up to fifth order can be found in [4].…”

Section: First Covariant Derivativementioning

confidence: 99%

“…However, one uses a different process to fix the frame. The idea is similar to the one used in the vacuum case [4]. We analyse each case separately.…”

Section: Lowering the Boundmentioning

confidence: 99%

“…In a recent paper by Machado Ramos and Vickers [4], the bound on differentiation in the vacuum type-N case was reduced from the Collins bound [5] of six to five. The approach used here consisted in using the generalized GHP formalism [6] to express the components of the Weyl spinor and its derivatives.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we use a similar approach to the one used in the vacuum case [4]. However, in non-vacuum we must consider the contribution of the Ricci spinor and the Ricci scalar as well as their successive covariant derivatives since they no longer vanish.…”

Section: Introductionmentioning

confidence: 99%

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