Two essential properties of energy-momentum tensors T µν are their positivity and conservation. This is mathematically formalized by, respectively, an energy condition, as the dominant energy condition, and the vanishing of their divergence ∇ µ T µν = 0. The classical Bel and Bel-Robinson superenergy tensors, generated from the Riemann and Weyl tensors, respectively, are rank-4 tensors. But they share these two properties with energy-momentum tensors: the dominant property (DP) and the divergence-free property in the absence of sources (vacuum). Senovilla [2, 3] defined a universal algebraic construction which generates a basic superenergy tensor T {A} from any arbitrary tensor A. In this construction, the seed tensor A is structured as an r-fold multivector, which can always be done. The most important feature of the basic superenergy tensors is that they satisfy automatically the DP, independently of the generating tensor A. In [8], we presented a more compact definition of T {A} using the r-fold Clifford algebra r Cℓ p,q . This form for the superenergy tensors allowed us to obtain an easy proof of the DP valid for any dimension. In this paper we include this proof. We explain which new elements appear when we consider the tensor T {A} generated by a non-degree-defined r-fold multivector A and how orthogonal Lorentz transformations and bilinear observables of spinor fields are included as particular cases of superenergy tensors. We find some sufficient conditions for the seed tensor A, which guarantee that the generated tensor T {A} is divergence-free. These sufficient conditions are satisfied by some physical fields, which are presented as examples. PACS number: 0460T µν is its 'positivity'. For most fields, T µν satisfies the dominant energy condition, that is, the energy flux or momentum vector ‡ j µ = −T µν u ν , measured by any future-pointing causal observer u, is also a future-pointing causal vector. When this is not satisfied, there arise problems of interpretation or rules of selection, as for the Tetrode tensor of a Dirac field. Indeed, this positivity condition (or another as the weak or the strong energy condition), is usually required for the Einstein tensor of any physically acceptable spacetime.The name superenergy was first applied to the Bel-Robinson (BR) and Bel tensors [1], which are defined from the conformal Weyl tensor and the Riemann tensor, respectively. The motivation for this name is that they share some properties with energy-momentum tensors. The prefix super appears because they are rank-4 tensors instead of rank-2. In [2], Senovilla defined an algebraic construction which generates a basic superenergy tensor T {A}, from an arbitrary seed tensor A. A much more extensive treatment discussing properties and applications is found in [3]. This construction unifies, in a single procedure, the BR and Bel tensors and many energymomentum tensors from different physical fields. The most important feature of the basic superenergy tensors T {A} is that, independently of the seed tensor A, they...
The concepts of rigid body and reference frame are so deeply intertwined in newtonian physics that it is impossible to distinguish clearly between them. As a matter of fact, a reference frame is a rigid body taken as a basis for surveying. Consequently, the so called 'angular velocity' can be understood as a quantity relating two arbitrary reference frames. But it also appears as a typical concept in rigid-body kinematics. The purpose of the paper is to try to clarify both aspects, which the authors believe are obscure and mishandled in most textbooks on mechanics, by using a quaternion formulation for rotations as a pedagogical alternative to the usual matrix version.
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