Third-order tensor potentials for the Riemann and Weyl tensors

Bampi, F.; Caviglia, Giacomo

doi:10.1007/bf00759166

Cited by 73 publications

(97 citation statements)

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“…[14] for the definition and properties of r-fold forms), and this result was confirmed in [2] (for any double 1 A semicolon indicates covariant derivative with respect to the canonical connection. As usual, we use round and square brackets to denote symmetrization and antisymmetrization of indices, respectively.…”

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confidence: 75%

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A local potential for the Weyl tensor in all dimensions

Edgar

Senovilla

2004

Class. Quantum Grav.

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In all dimensions n ≥ 4 and arbitrary signature, we demonstrate the existence of a new local potential -a double (2, 3)-form, P ab cde -for the Weyl curvature tensor C abcd , and more generally for all tensors W abcd with the symmetry properties of the Weyl tensor. The classical four-dimensional Lanczos potential for a Weyl tensor -a double (2, 1)-form, H ab c -is proven to be a particular case of the new potential: its double dual.PACS Numbers: 02.40. Ky, 04.20Cv In n dimensions, the existence of a 1-form potential A a for the 2-form electromagnetic field F ab enables the electromagnetic field equations to be written as a wave equation (in Lorentz signature) for the potential, which is particularly simple in the differential gauge

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confidence: 75%

“…It has already been noted that if n = 4 the Weyl tensor, and more generally all Weyl candidates, have a so-called Lanzcos potential [11], [2], [10], [1]. This is a double (2,1) …”

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confidence: 99%

A local potential for the Weyl tensor in all dimensions

Edgar

Senovilla

2004

Class. Quantum Grav.

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“…It was subsequently showed by Bampi and Caviglia [6] that the Weyl-Lanczos equations are always locally solvable for analytic metrics in four dimensions. The existence question for smooth, but not necessarily analytic, Lorentzian metrics has been eventually settled by Illge [7], who proved global existence of the Weyl-Lanczos potential on, say smooth, globally hyperbolic four-dimensional space-times.…”

Section: Introductionmentioning

confidence: 98%

Some potentials for the curvature tensor on three-dimensional manifolds

Chruściel

Gerber

2005

Gen Relativ Gravit

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We study equations of Riemann-Lanczos type on three dimensional manifolds. Obstructions to global existence for global Lanczos potentials are pointed out. We check that the imposition of the original Lanczos symmetries on the potential leads to equations which do not have a determined type, leading to problems when trying to prove global existence. We show that elliptic equations can be obtained by relaxing those symmetry requirements in at least two different ways, leading to global existence of potentials under natural conditions. A second order potential for the Ricci tensor is introduced.

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“…Lanczos [1][2][3][4][5][6] showed, for any R 4 , the existence of a potential K abc with the properties:…”

Section: Introductionmentioning

confidence: 99%

“…Given the Weyl tensor, it may be very difficult to obtain a Lanczos superpotential by integrating directly (2), but here we shall show three ways to deduce one solution of (2) for Gödel spacetime [21,22], with the interesting structure:…”

Section: Introductionmentioning

confidence: 99%