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

Ramos, M. P. Machado

doi:10.1088/0264-9381/15/2/016

Cited by 10 publications

(10 citation statements)

References 13 publications

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“…However, contrary to the above findings, we exhibit an example of a type N, nonvacuum solution that has IC order q = 7, and thereby show that the type N Karlhede bound is sharp. In our best estimation, the apparent discrepancy between our result and previous claims is due by a subtle error in the analysis of one subcase in [8].…”

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confidence: 99%

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The type N Karlhede bound is sharp

Milson,

Pelavas

2007

Preprint

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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 nonhomogeneous. 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: 99%

“…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]. Subsequently, examples of type N solutions with q = 5 were discovered [16].…”

mentioning

confidence: 99%

The type N Karlhede bound is sharp

Milson,

Pelavas

2007

Preprint

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“…It is straightforward to confirm that such a choice is consistent with the commutators (17) and creates no inconsistency with the other tables.…”

Section: 2mentioning

confidence: 76%

“…However, we know that we have not yet extracted all the information from the commutators (17), since they have only been applied to three zero-weighted coordinate candidates. So we closely examine the structure of the commutators (17) to determine whether they suggest the existence of a fourth zero-weighted scalar, functionally independent of the first three coordinate candidates, whose table is consistent with the commutators. In fact, we get a strong hint from the previous section, and consider the possibility of the existence of a real zero-weighted scalar T , which satisfies the table…”

Section: 2mentioning

confidence: 99%

Integration using invariant operators: conformally flat radiation metrics

Edgar

Vickers

1999

Class. Quantum Grav.

110

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Abstract.A new method is presented for obtaining the general conformally flat radiation metric by using the differential operators of Machado Ramos and Vickers (a generalisation of those of Geroch, Held and Penrose) which are invariant under null rotations and rescalings. The solution is found by constructing involutive tables of these derivatives applied to the quantities which arise in the Karlhede classification of this class of metrics.

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“…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].…”

mentioning

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.

show abstract

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

Cited by 10 publications

References 13 publications

The type N Karlhede bound is sharp

The type N Karlhede bound is sharp

Integration using invariant operators: conformally flat radiation metrics

The type N Karlhede bound is sharp

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