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  • 1.
    Bergman Ärlebäck, Jonas
    et al.
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Edgar, Brian
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Mathematics and Applied Mathematics. Linköping University, The Institute of Technology.
    The Bach tensor and other divergence-free tensors2005In: International Journal of Geometric Methods in Modern Physics (IJGMMP), ISSN 0219-8878, Vol. 2, no 1, p. 13-21Article in journal (Refereed)
    Abstract [en]

    In four dimensions, we prove that the Bach tensor is the only symmetric divergence-free 2-tensor which is also quadratic in Riemann and has good conformal behavior. In n > 4 dimensions, we prove that there are no symmetric divergence-free 2-tensors which are also quadratic in Riemann and have good conformal behavior, nor are there any symmetric divergence-free 2-tensors which are concomitants of the metric tensor gab together with its first two derivatives, and have good conformal behavior.

  • 2.
    Bradley, M.
    et al.
    Department of Physics, Umeå University.
    Edgar, Brian
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    M.P. MacHado, Ramos
    Departamento de Matemática e Aplicações, Universidade Do Minho, Guimarães, Portugal.
    Classification of a class of conformally flat pure radiation metrics with cosmological constant2011In: AIP Conference Proceedings: vol 1458, AIP American Institute of Physics , 2011, Vol. 1458, p. 331-334Conference paper (Refereed)
    Abstract [en]

    A comparison between the Cartan-Karlhede classification of the conformally flat pure radiation metrics with a negative cosmological constant, satisfying Λ+ττ=0, and the construction of them in terms of the Generalised Invariant Formalism (GIF) is made. © 2012 American Institute of Physics.

  • 3.
    Bradley, Michael
    et al.
    Umea Univ, Dept Phys, S-90187 Umea, Sweden.
    Edgar, Brian
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Machado Ramos, M P
    Univ Minho, Dept Matemat Ciencia and Tecnol, P-4800058 Azurem, Guimaraes, Portugal.
    Invariant classification and the generalised invariant formalism: conformally flat pure radiation metrics2010In: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 42, no 1, p. 155-182Article in journal (Refereed)
    Abstract [en]

    Metrics obtained by integrating within the generalised invariant formalism are structured around their intrinsic coordinates, and this considerably simplifies their invariant classification and symmetry analysis. We illustrate this by presenting a simple and transparent complete invariant classification of the conformally flat pure radiation metrics (except plane waves) in such intrinsic coordinates; in particular we confirm that the three apparently non-redundant functions of one variable are genuinely non-redundant, and easily identify the subclasses which admit a Killing and/or a homothetic Killing vector. Most of our results agree with the earlier classification carried out by Skea in the different Koutras-McIntosh coordinates, which required much more involved calculations; but there are some subtle differences. Therefore, we also rework the classification in the Koutras-McIntosh coordinates, and by paying attention to some of the subtleties involving arbitrary functions, we obtain complete agreement with the results obtained in intrinsic coordinates. We have corrected and completed statements and results by Edgar and Vickers, and by Skea, about the orders of Cartan invariants at which particular information becomes available.

  • 4.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    A tensor derivation of the new conservation law for electromagnetism2003In: Marcel Grossman Conference 10,2003, 2003Conference paper (Other academic)
  • 5.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Constructing necessary and sufficient conditions for n-dimensional conformally Einstein spaces2004In: GR17,2004, 2004Conference paper (Other academic)
  • 6.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Divergence-free tensors, Maxwell's equations and dimension2003In: BritgravIII,2003, 2003Conference paper (Other academic)
  • 7.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Irreducible Killing tensors from conformal Killing vectors.2004In: Symmetry in Nonlinear Mathematical Physics,2003, Proceedings of Institute of Mathematics of NAS of Ukraine, Vol. 50: Institute of Mathematics of NAS of Ukraine , 2004, p. 708-Conference paper (Refereed)
  • 8.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Necessary and sufficient conditions for n-dimensional conformal Einstein spaces via dimensionally dependent identities2005In: Journal of Mathematical Physics, ISSN 0022-2488, E-ISSN 1089-7658, Vol. 46, no 1Article in journal (Refereed)
    Abstract [en]

    Listing has recently extended results of Kozameh, Newman, and Tod for fourdimensional space-times and presented a set of necessary and sufficient conditions for a metric to be locally conformally equivalent to an Einstein metric in all semi-Riemannian spaces of dimension n≥4-subject to a nondegeneracy restriction on the Weyl tensor. By exploiting dimensionally dependent identities we demonstrate how to construct two alternative versions of these necessary and sufficient conditions which we believe will be useful in applications. The four-dimensional case is discussed in detail and examples are also given in five and six dimensions.

  • 9.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    On effective constraints for the Riemann-Lanczos system of eauations2003Article in journal (Refereed)
    Abstract [en]

    There have been conflicting points of view concerning the Riemann-Lanczos problem in three and four dimensions. Using direct differentiation on the defining partial differential equations, Massa and Pagani (in four dimensions) and Edgar (in dimensions n=3) have argued that there are effective constraints so that not all Riemann tensors can have Lanczos potentials, using Cartan's criteria of integrability of ideals of differential forms Bampi and Caviglia have argued that there are no such constraints in dimensions n=4, and that, in these dimensions, all Riemann tensors can have Lanczos potentials. In this article we give a simple direct derivation of a constraint equation, confirm explicitly that known exact solutions of the Riemann-Lanczos problem satisfy it, and argue that the Bampi and Caviglia conclusion must therefore be flawed. In support of this, we refer to the recent work of Dolan and Gerber on the three-dimensional problem, by a method closely related to that of Bampi and Caviglia, they have found an "internal identity" which we demonstrate is precisely the three-dimensional version of the effective constraint originally found by Massa and Pagani, and Edgar.

  • 10.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    On the structure of the new electromagnetic conservation laws2004In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 21, no 4Article in journal (Refereed)
    Abstract [en]

    New electromagnetic conservation laws have recently been proposed: in the absence of electromagnetic currents, the trace of the Chevreton superenergy tensor, Hab is divergence free in four-dimensional (a) Einstein spacetimes for test fields, and (b) Einstein-Maxwell spacetimes. Subsequently it has been pointed out, in analogy with flat spaces, that for Ricci-flat spacetimes the trace of the Chevreton superenergy tensor Hab can be rearranged in the form of a generalized wave operator □L acting on the energy-momentum tensor Tab of the test fields, i.e., H ab = □Ltab/2. In this letter we show, for Einstein-Maxwell spacetimes in the full nonlinear theory, that, although, the trace of the Chevreton superenergy tensor Hab can again be rearranged in the form of a generalized wave operator □G acting on the electromagnetic energy-momentum tensor, in this case the result is also crucially dependent on Einstein's equations, hence we argue that the divergence-free property of the tensor Hab = □GT ab/2 has significant independent content beyond that of the divergence-free property of Tab.

  • 11.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Proofs of existence of local potentials for trace-free symmetric 2-forms using dimensionally dependent identities2005In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 54, no 3, p. 251-261Article in journal (Refereed)
    Abstract [en]

    We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second-order differential equation, e.g., a wave equation in Lorentz signature. Furthermore, we exploit higher-dimensional tensor identities to obtain the analogous results for (m, m)-forms in 2m dimensions. © 2004 Elsevier B.V. All rights reserved.

  • 12.
    Edgar, Brian
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Bradley, Michael
    Umeå University, Sweden.
    Piedade Machado Ramos, M.
    University of Minho, Portugal.
    Symmetry analysis of radiative spacetimes with a null isotropy using GHP formalism2014In: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 46, no 10, p. 1797-Article in journal (Refereed)
    Abstract [en]

    A complete and simple invariant classification of the conformally flat pure radiation metrics with a negative cosmological constant that were obtained by integration using the generalised invariant formalism is presented. We show equivalence between these metrics and the corresponding type O subclass of the more general spacetime studied by Siklos. The classification procedure indicates that the metrics possess a one degree of null isotropy freedom which has very interesting repercussions in the symmetry analysis. The Killing and homothetic vector analysis in GHP formalism is then generalised to this case were there is only one null direction defined geometrically. We determine the existing Killing vectors for the different subclasses that arise in the classification and compare these results to those obtained in the symmetry analysis performed by Siklos for a larger class of metrics with Ricci tensor representing a pure radiation field and a negative cosmological constant. It is also shown that there are no homothetic Killing vectors present.

  • 13.
    Edgar, Brian
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Garcia-Parrado Gomez-Lobo, Alfonso
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Maria Martin-Garcia, José
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Petrov D vacuum spaces revisited: identities and invariant classification2009In: CLASSICAL AND QUANTUM GRAVITY, ISSN 0264-9381, Vol. 26, no 10, p. 105022-Article in journal (Refereed)
    Abstract [en]

    For Petrov D vacuum spaces, two simple identities are rederived and some new identities are obtained, in a manageable form, by a systematic and transparent analysis using the GHP formalism. This gives a complete involutive set of tables for the four GHP derivatives on each of the four GHP spin coefficients and the one Weyl tensor component. It follows directly from these results that the theoretical upper bound on the order of covariant differentiation of the Riemann tensor required for a Karlhede classification of these spaces is reduced to two.

  • 14.
    Edgar, Brian
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Ludwig, G
    Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden Univ Alberta, Dept Math Sci, Edmonton, AB T6G 2G1, Canada.
    Finding conformal killing vectors from the invariant classification scheme2002In: International Journal of Modern Physics A, ISSN 0217-751X, E-ISSN 1793-656X, Vol. 17, no 20, p. 2757-2757Article in journal (Refereed)
  • 15.
    Edgar, Brian
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Machado, Ramos M. P.
    Obtaining the pure radiation metrics using invariant operators.2004In: GR17,2004, 2004Conference paper (Other academic)
  • 16.
    Edgar, Brian
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Machado Ramos, Maria
    Departamento de Mathematica  Universidade do Minho, Guimaraes, Portugal.
    Obtaining a class of type N pure radiation metrics using invariant operators2005In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 22, no 5, p. 791-802Article in journal (Refereed)
    Abstract [en]

    We develop further the integration procedure in the generalized invariant formalism, and demonstrate its efficiency by obtaining a class of Petrov type N pure radiation metrics without any explicit integration, and with comparatively little detailed calculations. The method is similar to the one exploited by Edgar and Vickers when deriving the general conformally flat pure radiation metric. A major addition to the technique is the introduction of non-intrinsic elements in the generalized invariant formalism, which can be exploited to keep calculations manageable.

  • 17.
    Edgar, Brian
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Machado Ramos, Maria
    Departmento de Matematica para a Ciencia e Tecnologia Universidade do Minho.
    Type O pure radiation metrics with a cosmological constant2007In: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 39, no 11, p. 1749-1772Article in journal (Refereed)
    Abstract [en]

    In this paper we complete the integration of the conformally flat pure radiation spacetimes with a non-zero cosmological constant Λ, and τ ≠ 0, by considering the case Λ +ττ ≠ 0. This is a further demonstration of the power and suitability of the generalised invariant formalism (GIF) for spacetimes where only one null direction is picked out by the Riemann tensor. For these spacetimes, the GIF picks out a second null direction (from the second derivative of the Riemann tensor) and once this spinor has been identified the calculations are transferred to the simpler GHP formalism, where the tetrad and metric are determined. The whole class of conformally flat pure radiation spacetimes with a non-zero cosmological constant (those found in this paper, together with those found earlier for the case Λ +ττ = 0) have a rich variety of subclasses with zero, one, two, three, four or five Killing vectors. © 2007 Springer Science+Business Media, LLC.

  • 18.
    Edgar, Brian
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Ramos, Maria Piedade
    Mathematics Department Universidade do Minho, Guimaraes, Portugal.
    Obtaining a class of conformally flat pure radiation metrics with cosmological constant using invariant operators2006In: XXVIII Spanish Relativity Meeting E.R.E. A Century of Relativity Physics,2005, Melville, NY, USA: American Institute of Physics (AIP), 2006, p. 507-Conference paper (Other academic)
    Abstract [en]

    In a recent paper by B. Edgar and M. P. Ramos the method using invariant operators to obtain the type O pure radiation class of metrics was generalized to type N pure radiation solutions. In this work we further generalize the method in determining type O pure radiation metrics with cosmological constant.

  • 19.
    Edgar, Brian
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Ramos, Maria Piedade
    Departamento de Mathematica  Universidade do Minho, Guimaraes, Portugal.
    Obtaining a class of Type O pure radiation metrics with a negative cosmological constant, using invariant operators2007In: General Relativity and Gravitation, ISSN 0001-7701, E-ISSN 1572-9532, Vol. 39, p. 539-566Article in journal (Refereed)
    Abstract [en]

    Using the generalised invariant formalism we derive a special subclass of conformally flat spacetimes whose Ricci tensor has a pure radiation and a Ricci scalar component. The method used is a development of the methods used earlier for pure radiation spacetimes of Petrov types O and N, respectively. In this paper we demonstrate how to handle, in the generalised invariant formalism, spacetimes with isotropy freedom and rich Killing vector structure. Once the spacetimes have been constructed, it is straightforward to deduce their Karlhede classification: the Karlhede algorithm terminates at the fourth derivative order, and the spacetimes all have one degree of null isotropy and three, four or five Killing vectors.

  • 20.
    Edgar, Brian
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Ramos, Maria Piedade
    Departamento de Mathematica  Universidade do Minho, Guimaraes, Portugal.
    Using the generalised invariant formalism: a class of conformally flat pure radiation metrics with a negative cosmological constant2007In: The Spanish Relativity Meeting ERE 2006: Einsteins Legacy,2006, Institute of Physics Publishing (IOPP), 2007, p. 012013-Conference paper (Refereed)
    Abstract [en]

    We demonstrate an integration procedure for the generalised invariant formalism by obtaining a subclass of conformally flat pure radiation spacetimes with a negative cosmological constant. The method used is a development of the methods used earlier for pure radiation spacetimes of Petrov types O and N respectively. This subclass of spacetimes turns out to have one degree of isotropy freedom, so in this paper we have extended the integration procedure for the generalised invariant formalism to spacetimes with isotropy freedom. 

  • 21.
    Edgar, Brian
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Senovilla, J M M
    Physics Department University of the Basque Country, Bilbao.
    The new potential for the Weyl tensor in n dimensions: wave equation and gauge2008In: Spanish Relativity Meeting 2007,2007, Spain: EDR , 2008, p. 245-248Conference paper (Refereed)
    Abstract [en]

    Although the Lanczos potential (a (2,1) form, L) for the Weyl tensor does not exist in dimensions greater than four, a new potential (a (2,3) form, P, which coincides with the double dual of L in four dimensions) has recently been shown to exist in all dimensions. In this talk we discuss the structure of its wave equation which is obtained from the Bianchi identities, and the related question of gauge. 

  • 22.
    Edgar, Brian
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Senovilla, Jose
    Physics University of the Basque Country.
    A local potential for the Weyl tensor in all dimensions2004In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 21, no 22Article in journal (Refereed)
    Abstract [en]

    In all dimensions n ≥ 4 and arbitrary signature, we demonstrate the existence of a new local potential - a double (2, 3)-form, Pabcde - for the Weyl curvature tensor Cabcd, and more generally for all tensors Wabcd with the symmetry properties of the Weyl tensor. The classical four-dimensional Lanczos potential for a Weyl tensor - a double (2, 1)-form, Habc - is proven to be a particular case of the new potential: its double dual.

  • 23.
    Edgar, Brian
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Senovilla, Jose
    Physics Department University of teh Basque Country.
    The new potential for the Weyl tensor in all dimensions: gauge and wave equation2007In: Britgrav7,2007, 2007Conference paper (Refereed)
    Abstract [en]

     Although the Lanczos potential L for the Weyl tensor does not exist in dimensions greater than four, a new potential P, (which coincides with the double dual of L in four dimensions) has recently been shown to exist in all dimensions. In this talk we discuss the structure of its wave equation which is obtained from the Bianchi identities, and the related question of gauge.  

  • 24.
    Edgar, Brian
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Senovilla, Jose M M
    Physics University of the Basque Country.
    A local potential for the Weyl tensor in all dimensions2004In: BritGravIV,2004, 2004Conference paper (Refereed)
  • 25.
    Edgar, Brian
    et al.
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Senovilla, Jose M M
    Physics University of the Basque Country.
    A weighted de Rham operator leading to local potentials for Riemann and Weyl tensors2006In: XXVIII Spanish Relativity Meeting E.R.E. A Century of Relativity Physics,2005 / [ed] Mornas, L; Alonso, JD, Melville, NY, USA: American Institute of Physics (AIP), 2006, p. 291-Conference paper (Other academic)
    Abstract [en]

    By defining a weighted de Rham operator for r-fold forms we obtain an associated superpotential for all tensors (considered as r-fold forms), in all dimensions. From this superpotential we deduce, in a straightforward and natural manner, the existence of 2r potentials for all r-fold forms. By specialising this result, we are able to obtain a pair of potentials for the Riemann tensor, and a single (2,3)-form potential for the Weyl tensor. This latter potential is the n-dimensional version of the double dual of the classical four dimensional (2,1)-form Lanczos potential. We also demonstrate that the new weighted de Rham operator is the natural operator to use in the Laplace-like equation for the Riemann tensor.

  • 26.
    Edgar, Brian
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Wingbrant, Ola
    Old and new results for superenergy tensors from dimensionally dependent tensor identities2003Article in journal (Refereed)
    Abstract [en]

    It is known that some results for spinors, and in particular for superenergy spinors, are much less transparent and require a lot more effort to establish, when considered from the tensor viewpoint. In this paper we demonstrate how the use of dimensionally dependent tensor identities enables us to derive a number of 4-dimensional identities by straightforward tensor methods in a signature independent manner. In particular, we consider the quadratic identity for the Bel-Robinson tensor TabcxTabcy = dxy TabcdTabcd/4 and also the new conservation law for the Chevreton tensor, both of which have been obtained by spinor means, both of these results are rederived by tensor means for 4-dimensional spaces of any signature, using dimensionally dependent identities, and, moreover, we are able to conclude that there are no direct higher dimensional analogs. In addition we demonstrate a simple way to show the nonexistense of such identities via counter examples, in particular we show that there is no nontrivial Bel tensor analog of this simple Bel-Robinson tensor quadratic identity. On the other hand, as a sample of the power of generalizing dimensionally dependent tensor identities from four to higher dimensions, we show that the symmetry structure, trace-free and divergence-free nature of the 4-dimensional Bel-Robinson tensor does have an analog for a class of tensors in higher dimensions.

  • 27. Rani, Rafaelle
    et al.
    Barnes, Alan
    Computer Science Aston University.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Killing Tensors from Conformal Killing Vectors2004In: Proceedings of the XXV Spanish Relativity Meeting Encuentros Relativistas Espanoles EREs2002,2002, Spain: University of Barcelona Press , 2004, p. 248-Conference paper (Refereed)
  • 28.
    Rani, Raffaele
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Physics, Chemistry and Biology.
    Edgar, S. Brian
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Barnes, A.
    Sch. of Eng. and Applied Science, Aston University, Birmingham B4 7ET, United Kingdom.
    Killing tensors and conformai Killing tensors from conformal Killing vectors2003In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 20, no 11, p. 1929-1942Article in journal (Refereed)
    Abstract [en]

    Koutras has proposed some methods to construct reducible proper conformal Killing tensors and Killing tensors (which are, in general, irreducible) when a pair of orthogonal conformal Killing vectors exist in a given space. We give the completely general result demonstrating that this severe restriction of orthogonality is unnecessary. In addition, we correct and extend some results concerning Killing tensors constructed from a single conformal Killing vector. A number of examples demonstrate that it is possible to construct a much larger class of reducible proper conformal Killing tensors and Killing tensors than permitted by the Koutras algorithms. In particular, by showing that all conformal Killing tensors are reducible in conformally flat spaces, we have a method of constructing all conformal Killing tensors, and hence all the Killing tensors (which will in general be irreducible) of conformally flat spaces using their conformal Killing vectors.

  • 29.
    Senovilla, Jose M M
    et al.
    Physics University of the Basque Country.
    Edgar, Brian
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    A weighted de Rham operator acting on arbitrary tensor fields and their local potentials2006In: Journal of Geometry and Physics, ISSN 0393-0440, E-ISSN 1879-1662, Vol. 56, no 10, p. 2135-2162Article in journal (Refereed)
    Abstract [en]

    We introduce a weighted de Rham operator which acts on arbitrary tensor fields by considering their structure as r-fold forms. We can thereby define associated superpotentials for all tensor fields in all dimensions and, from any of these superpotentials, we deduce in a straightforward and natural manner the existence of 2 r potentials for any tensor field, where r is its form-structure number. By specialising this result to symmetric double forms, we are able to obtain a pair of potentials for the Riemann tensor, and a single (2, 3)-form potential for the Weyl tensor due to its tracelessness. This latter potential is the n-dimensional version of the double dual of the classical four-dimensional (2, 1)-form Lanczos potential. We also introduce a new concept of harmonic tensor fields, and demonstrate that the new weighted de Rham operator has many other desirable properties and, in particular, is the natural operator to use in the Laplace-like equation for the Riemann tensor. © 2005 Elsevier Ltd. All rights reserved.

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