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A weighted de Rham operator acting on arbitrary tensor fields
, 2008
"... and their local potentials. ..."
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Dimensionally Dependent Tensor Identities by Double Antisymmetrisation
 J. Math. Phys
"... Some years ago, Lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure, and could all be considered consequences in ndimensional space of a pair of fundamental identities involving tracefree (p,p)forms where 2p ≥ n. We generalise Lovelock’s results ..."
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Some years ago, Lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure, and could all be considered consequences in ndimensional space of a pair of fundamental identities involving tracefree (p,p)forms where 2p ≥ n. We generalise Lovelock’s results, and by using the fact that associated with any tensor in ndimensional space there is associated a fundamental tensor identity obtained by antisymmetrising over n + 1 indices, we establish a very general ’master ’ identity for all tracefree (k, l)forms. We then show how various other special identities are direct and simple consequences of this master identity; in particular we give direct application to Maxwell, Lanczos, Ricci, Bel and BelRobinson tensors, and also demonstrate how relationships between scalar invariants of the Riemann tensor can be investigated in a systematic manner. I.
The Lanczos potential for Weylcandidate tensors exists only in four dimensions”, Gen
 Rel. Grav
, 2000
"... We prove that a Lanczos potential Labc for the Weyl candidate tensor Wabcd does not generally exist for dimensions higher than four. The technique is simply to assume the existence of such a potential in dimension n, and then check the integrability conditions for the assumed system of differential ..."
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We prove that a Lanczos potential Labc for the Weyl candidate tensor Wabcd does not generally exist for dimensions higher than four. The technique is simply to assume the existence of such a potential in dimension n, and then check the integrability conditions for the assumed system of differential equations; if the integrability conditions yield another nontrivial differential system for Labc and Wabcd, then this system’s integrability conditions should be checked; and so on. When we find a nontrivial condition involving only Wabcd and its derivatives, then clearly Weyl candidate tensors failing to satisfy that condition cannot be written in terms of a Lanczos potential Labc. 1
A local potential for the Weyl tensor in all dimensions
 Class. Quantum Grav
, 2004
"... 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 Cabcd, and more generally for all tensors Wabcd with the symmetry properties of the Weyl tensor. The classical fourdimensional Lancz ..."
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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 Cabcd, and more generally for all tensors Wabcd with the symmetry properties of the Weyl tensor. The classical fourdimensional 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 1form potential Aa for the 2form electromagnetic field Fab 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 1 A a;a = 0 [12]: ∇ 2 A a = F ab;b. In four dimensions, Lanczos [11] proposed the existence of a double (2, 1)form potential H ab c = H [ab] c for the double (2, 2)form Weyl tensor C ab cd (see e.g. [14] for the definition
LOCAL EXISTENCE OF SPINOR POTENTIALS
, 1999
"... We present a new, simple proof of existence for the Lanczos spinor potential in 3+1 dimensions that introduces a potential TABCD = T (ABC)D of the Lanczos potential together with several generalizations to other index configurations and metric signatures. The potential TABCD can also be used to expr ..."
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We present a new, simple proof of existence for the Lanczos spinor potential in 3+1 dimensions that introduces a potential TABCD = T (ABC)D of the Lanczos potential together with several generalizations to other index configurations and metric signatures. The potential TABCD can also be used to express, in a concise way, the gauge freedom left in the Lanczos potential after the differential gauge has been specified. We consider Einstein spacetimes and prove that in those spacetimes any symmetric (3,1)spinor possesses a symmetric potential HABA ′ B ′. Potentials of this type have earlier occurred in some special cases investigated e.g., by Torres del Castillo, Bergqvist and ourselves. 1
DOI 10.1007/s1077300899077 Lanczos Potential for the van Stockung SpaceTime
"... Abstract The Lanczos Potential is a theoretical useful tool to find the conformal Weyl curvature tensor Cabcd of a given relativistic metric. In this paper we find the Lanczos potential Labc for the van Stockung vacuum gravitational field. Also, we show how the wave equation can be combined with sp ..."
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Abstract The Lanczos Potential is a theoretical useful tool to find the conformal Weyl curvature tensor Cabcd of a given relativistic metric. In this paper we find the Lanczos potential Labc for the van Stockung vacuum gravitational field. Also, we show how the wave equation can be combined with spinor methods in order to find this important three covariant index tensor.
SPINORS, SPIN COEFFICIENTS AND LANCZOS POTENTIALS
, 1998
"... It has been demonstrated, in a number of special situations, that the spin coefficients of a canonical spinor dyad can be used to define a Lanczos potential of the Weyl curvature spinor. In this paper we explore some of these potentials and show that they can be defined directly from the spinor dyad ..."
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It has been demonstrated, in a number of special situations, that the spin coefficients of a canonical spinor dyad can be used to define a Lanczos potential of the Weyl curvature spinor. In this paper we explore some of these potentials and show that they can be defined directly from the spinor dyad in a very simple way, but that the results do not generalize significantly, in any direct manner. A link to metric, asymmetric, curvaturefree connections, which suggests a more natural relationship between the Lanczos potential and spin coefficients, is also considered. 1
LOCAL EXISTENCE OF SPINOR AND TENSOR POTENTIALS
, 2000
"... We give new simple direct proofs in all spacetimes for the existence of asymmetric (n, m + 1)spinor potentials for completely symmetric (n + 1, m)spinors and for the existence of symmetric (n, 1)spinor potentials for symmetric (n + 1, 0)spinors. These proofs introduce a ‘superpotential’, i.e., a ..."
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We give new simple direct proofs in all spacetimes for the existence of asymmetric (n, m + 1)spinor potentials for completely symmetric (n + 1, m)spinors and for the existence of symmetric (n, 1)spinor potentials for symmetric (n + 1, 0)spinors. These proofs introduce a ‘superpotential’, i.e., a potential of the potential, which also enables us to get explicit statements of the gauge freedom of the original potentials. The main application for these results is the Lanczos potential LABCA ′, of the Weyl spinor and the electromagnetic vector potential AAA ′. We also investigate the possibility of existence of a symmetric potential HABA ′ B ′ for the Lanczos potential, and prove that in all Einstein spacetimes any symmetric (3,1)spinor LABCA ′ possesses a symmetric potential HABA ′ B ′. Potentials of this type have been found earlier in investigations of some very special spinors in restricted classes of spacetimes. All of the new spinor results are translated into tensor notation, and where possible given also for four dimensional spaces of arbitrary signature. 1
The nonexistence of a Lanczos potential for the Weyl curvature tensor in dimensions n ≥ 7
, 2002
"... In this paper it is shown that a Lanczos potential for the Weyl curvature tensor does not exist for all spaces of dimension n≥7. ..."
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In this paper it is shown that a Lanczos potential for the Weyl curvature tensor does not exist for all spaces of dimension n≥7.
GR15 Workshop A3 Mathematical Studies of Field Equations
, 2008
"... In this report I have tried to describe the main streams of research presented at the workshop. Occasionally I have described some contributions which were presented in poster form, whenever they were directly related to the subjects ..."
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In this report I have tried to describe the main streams of research presented at the workshop. Occasionally I have described some contributions which were presented in poster form, whenever they were directly related to the subjects