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  • 1.
    Andersson, Fredrik
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Edgar, S.B.
    Existence of Lanczos potentials and superpotentials for the Weyl spinor/tensor2001In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 18, no 12, p. 2297-2304Article in journal (Refereed)
    Abstract [en]

    A new and concise proof of existence - emphasizing the very natural and simple structure - is given for the Lanczos spinor potential LABCA' of an arbitrary symmetric spinor WABCD defined by WABCD = 2?(AA' LBCD)A', this proof is easily translated into tensors in such a way that it is valid in four-dimensional spaces of any signature. In particular, this means that the Weyl spinor ?ABCD has Lanczos potentials in all spacetimes, and furthermore that the Weyl tensor has Lanczos potentials on all four-dimensional spaces, irrespective of signature. In addition, two superpotentials for WABCD are identified: the first TABCD (= T(ABC)D) is given by LABCA' = ?A'DTABCD, while the second HABA'B' (= H(AB)(A'B')) (which is restricted to Einstein spacetimes) is given by LABCA' = ? (AB' HBC)A'B'. The superpotential TABCD is used to describe the gauge freedom in the Lanczos potential.

  • 2.
    Bergqvist, Göran
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Hoglund, A.
    Algebraic Rainich theory and antisymmetrization in higher dimensions2002In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 19, no 12, p. 3341-3355Article in journal (Refereed)
    Abstract [en]

    The classical Rainich(-Misner-Wheeler) theory gives necessary and sufficient conditions on an energy-momentum tensor T to be that of a Maxwell field (a 2-form) in four dimensions. Via Einstein's equations, these conditions can be expressed in terms of the Ricci tensor, thus providing conditions for a spacetime geometry to be an Einstein-Maxwell spacetime. One of the conditions is that T2 is proportional to the metric, and it has previously been shown in arbitrary dimension that any tensor satisfying this condition is a superenergy tensor of a simple p-form. Here we examine algebraic Rainich conditions for general p-forms in higher dimensions and their relations to identities by antisymmetrization. Using antisymmetrization techniques we find new identities for superenergy tensors of these general (non-simple) forms, and we also prove in some cases the converse: that the identities are sufficient to determine the form. As an example we obtain the complete generalization of the classical Rainich theory to five dimensions.

  • 3.
    Bergqvist, Göran
    et al.
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Lankinen, Paul
    Unique characterization of the Bel-Robinson tensor2004In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 21, no 14, p. 3499-3503Article in journal (Refereed)
    Abstract [en]

    We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel-Robinson-type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors.

  • 4.
    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.

  • 5.
    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.

  • 6.
    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.

  • 7.
    Garcia-Parrado, Alfonso
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    Dynamical laws of superenergy in General Relativity2008In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 25, p. 015006-015032Article in journal (Refereed)
    Abstract [en]

    The Bel and Bel-Robinson tensors were introduced nearly 50 years ago in an attempt to generalize to gravitation the energy-momentum tensor of electromagnetism. This generalization was successful from the mathematical point of view because these tensors share mathematical properties which are remarkably similar to those of the energy-momentum tensor of electromagnetism. However, the physical role of these tensors in general relativity has remained obscure and no interpretation has achieved wide acceptance. In principle, they cannot represent energy and the term superenergy has been coined for the hypothetical physical magnitude lying behind them. In this work, we try to shed light on the true physical meaning of superenergy by following the same procedure which enables us to give an interpretation of the electromagnetic energy. This procedure consists in performing an orthogonal splitting of the Bel and Bel-Robinson tensors and analyzing the different parts resulting from the splitting. In the electromagnetic case such splitting gives rise to the electromagnetic energy density, the Poynting vector and the electromagnetic stress tensor, each of them having a precise physical interpretation which is deduced from the dynamical laws of electromagnetism (Poynting theorem). The full orthogonal splitting of the Bel and Bel-Robinson tensors is more complex but, as expected, similarities with electromagnetism are present. Also the covariant divergence of the Bel tensor is analogous to the covariant divergence of the electromagnetic energy-momentum tensor and the orthogonal splitting of the former is found. The ensuing equations are to the superenergy what the Poynting theorem is to electromagnetism. Some consequences of these dynamical laws of superenergy are explored, among them the possibility of defining superenergy radiative states for the gravitational field.

  • 8.
    Gomez-Lobo, A.G.-P.
    et al.
    Linköping University, Department of Mathematics. Linköping University, The Institute of Technology.
    Kroon, J.A.V.
    School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, United Kingdom.
    Kerr initial data2008In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 25, no 20Article in journal (Refereed)
    Abstract [en]

    Exploiting a (3+1) analysis of the Mars-Simon tensor, conditions on a vacuum initial data set ensuring that its development is isometric to a subset of the Kerr spacetime are found. These conditions are expressed in terms of the vanishing of a positive scalar function defined on the initial data hypersurface. Applications of this result are discussed. © 2008 IOP Publishing Ltd.

  • 9.
    Herberthson, Magnus
    Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
    Static spacetimes with prescribed multipole moments: a proof of a conjecture by Geroch2009In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 26, no 21, p. 1-26Article in journal (Refereed)
    Abstract [en]

    In this paper we give sufficient conditions on a sequence of multipole moments for a static spacetime to exist with precisely these moments. The proof is constructive in the sense that a metric having prescribed multipole moments up to a given order can be calculated. Since these sufficient conditions agree with already known necessary conditions, this completes the proof of a long standing conjecture due to Geroch.

  • 10.
    Herberthson, Magnus
    Linköping University, The Institute of Technology. Linköping University, Department of Mathematics, Applied Mathematics.
    The gravitational dipole and explicit multipole moments of static axisymmetric spacetimes2004In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 21, no 22, p. 5121-5138Article in journal (Refereed)
    Abstract [en]

    The multipole moments of static axisymmetric asymptotically flat spacetimes are considered. The usual set of recursively defined tensors is replaced with one real-valued function m defined on R+ ∪ {0}, where the moments are given by the derivatives of m at 0. As examples of applications, we show that the Schwarzschild spacetime is the gravitational monopole and derive the metric for the gravitational dipole.

  • 11.
    Ludwig, G
    et al.
    Univ Alberta, Dept Math Sci, Edmonton, AB T6G 1S6, Canada Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden.
    Edgar, SB
    Univ Alberta, Dept Math Sci, Edmonton, AB T6G 1S6, Canada Linkoping Univ, Dept Math, S-58183 Linkoping, Sweden.
    A generalized Lie derivative and homothetic or Killing vectors in the Geroch-Held-Penrose formalism2000In: Classical and quantum gravity, ISSN 0264-9381, E-ISSN 1361-6382, Vol. 17, no 7, p. 1683-1705Article in journal (Refereed)
    Abstract [en]

    A generalized Lie derivative operator suitable for use within the GHP formalism and the notion of preferred GHP tetrads relative to a vector are introduced. The usual homothetic or Killing equations are then replaced by an equivalent but much more manageable set of equations involving the commutators of this new operator with the four GHP derivative operators. This allows for an efficient treatment of the homothetic or Killing condition when constructing new solutions of Einstein's field equations or when obtaining the homothetic and/or Killing vectors for a given metric. Two applications are given. The first sheds new light on the vacuum twisting type N problem with one or two homothetic/Killing vectors. In the second we find the subclass of ail type N, and of all conformally flat, pure radiation metrics (with tau not equal 0) which possess one or more homothetic or Killing vectors.

  • 12.
    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.

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