Higher dimensional bivectors and classification of the Weyl operator (original) (raw)
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Refinements of the Weyl tensor classification in five dimensions
Classical and Quantum Gravity, 2012
We refine the null alignment classification of the Weyl tensor of a five-dimensional spacetime. The paper focusses on the algebraically special alignment types N, III, II and D, while types I and G are briefly discussed. A first refinement is provided by the notion of spin type of the components of highest boost weight. Second, we analyze the Segre types of the Weyl operator acting on bivector space and examine the intersection with the spin type classification. We present a full treatment for types N and III, and illustrate the classification from different viewpoints (Segre type, rank, spin type) for types II and D, paying particular attention to possible nilpotence, which is a new feature of higher dimensions. We also point out other essential differences with the four-dimensional case. In passing, we exemplify the refined classification by mentioning the special subtypes associated to certain important spacetimes, such as Myers-Perry black holes, black strings, Robinson-Trautman spacetimes, and purely electric/magnetic type D spacetimes.
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Classical and Quantum Gravity, 2008
We discuss the algebraic classification of the Weyl tensor in higher-dimensional Lorentzian manifolds. This is done by characterizing algebraically special Weyl tensors by means of the existence of aligned null vectors of various orders of alignment. Further classification is ...
Algebraic Classification of the Weyl Tensor
Alignment classification of tensors on Lorentzian manifolds of arbitrary dimension is summarized. This classification scheme is then applied to the case of the Weyl tensor and it is shown that in four dimensions it is equivalent to the well known Petrov classification. The approaches using Bel-Debever criteria and principal null directions of the superenergy tensor are also discussed.
Electric and Magnetic Weyl Tensors in Higher Dimensions
Springer Proceedings in Physics, 2014
Recent results on purely electric (PE) or magnetic (PM) spacetimes in n dimensions are summarized. These include: Weyl types; diagonalizability; conditions under which direct (or warped) products are PE/PM.
The necessity of Lorentz transforming the Dirac matrices is an ongoing issue with contradicting opinions. The Lorentz transformation of Dirac spinors is clear but for the Dirac adjoint, the combination of a spinor and the 'time-like' zeroth gamma-matrix, the situation is fussy again. In the Feynman slash objects, the gamma matrix four vector connects to the dynamic four vectors without really becoming one itself. The Feynman slash objects exist in 4-D Minkowsky space-time on the one hand, the gamma matrices are often taken as inert objects like the Minkowski metric itself on the other hand. To be short, a slumbering confusion exists in RQM's roots. In this paper, first a Pauli-level biquaternion environment equivalent to Minkowski space-time is presented. Then the Weyl-Dirac environment is produced as a PT doubling of the biquaternion Pauli-environment. It is the production process from basic elements that produces some clarification regarding the mentioned RQM foundational fussiness.
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Almost all presentations of Dirac theory in first or second quantization in Physics (and Mathematics) textbooks make use of covariant Dirac spinor fields. An exception is the presentation of that theory (first quantization) offered originally by Hestenes and now used by many authors. There, a new concept of spinor field (as a sum of non homogeneous even multivectors fields) is used. However, a carefully analysis (detailed below) shows that the original Hestenes definition cannot be correct since it conflicts with the meaning of the Fierz identities. In this paper we start a program dedicated to the examination of the mathematical and physical basis for a comprehensive definition of the objects used by Hestenes. In order to do that we give a preliminary definition of algebraic spinor fields (ASF ) and Dirac-Hestenes spinor fields (DHSF ) on Minkowski spacetime as some equivalence classes of pairs (Ξu, ψ Ξu ), where Ξu is a spinorial frame field and ψ Ξu is an appropriate sum of multivectors fields (to be specified below). The necessity of our definitions are shown by a carefull analysis of possible formulations of Dirac theory and the meaning of the set of Fierz identities associated with the 'bilinear covariants' (on Minkowski spacetime) made with ASF or DHSF. We believe that the present paper clarifies some misunderstandings (past and recent) appearing on the literature of the subject. It will be followed by a sequel paper where definitive definitions of ASF and DHSF are given as appropriate sections of a vector bundle called the left spin-Clifford bundle. The bundle formulation is essential in order to be possible to produce a coherent theory for the covariant derivatives of these fields on arbitrary Riemann-Cartan spacetimes. The present paper contains also Appendices (A-E) which exhibits a truly useful collection of results concerning the theory of
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A review of some facts concerning classical spacetime geometry is presented together with a description of the most elementary aspects of the two-component spinor formalisms of Infeld and van der Waerden. Special attention is concentrated upon the gauge characterization of the basic geometric objects borne by the formalisms. It is pointed out that spin-affine configurations may be naively defined by carrying out parallel displacements of null world vectors within the framework of the γ-formalism. The standard result that assigns a covariant gauge behaviour to the symmetric parts of any admissible spin connexions is deduced out of building up a generalized version of spin transformation laws. A fairly complete algebraic description of curvature splittings is carried out on the basis of the construction of a set of spinor commutators for each formalism. The pertinent computations take up the utilization of some covariant differential prescriptions which facilitate specifying the action of the commutators on arbitrary spin tensors and densities. It turns out that the implementation of such commutators under certain circumstances gives rise to a system of wave equations for gravitons and Infeld-van der Waerden photons which possess in either formalism a gauge-invariance property associated with appropriate spinor-index configurations. The situation regarding the accomplishment of the couplings between Dirac fields and electromagnetic curvatures is entertained to a considerable extent.