Positivity and conservation of superenergy tensors

Abstract

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...

New construction of causal tensors

“…To continue with the metric, one can easily verify that for an arbitrary C p the multivector G = (−1) deg(I) e I ⊗ e I , where the sum is over all multiindices, satisfies G = G and G 2 = dim(C p )G. So In the final section we investigate how the concept of principal null directions of causal tensors [15] can be treated within r C p .…”

“…In [15] sufficient and necessary conditions for a nullvector l to be a PND of the superenergy tensor T {A}, A ∈ r C 1,p , were obtained. It is nice that we can also do this for the superenergies of multivectors in r C p , Thus we have a simple characterization of the PND's of T {A}.…”

“…Especially we will prove that this construction contains all earlier ones, obtained with tensors or with Clifford algebra. In C 3 this construction also bears a natural relation to the superenergy of a Dirac spinor [15] and it is natural to treat a general multivector in r C p as a spinor. It could also be interesting to look at the divergence and conservation of these new causal tensors in our construction, as well as to look further at the subject of principal null directions.…”

“…A proof of the DP in general dimension was given in [17]. Another generalization is presented by Pozo and Parra in [14] and [15] (see also [13]) using Clifford algebras of a Lorentzian vector space. It turns out that the Clifford algebraic formulation is very simple and natural.…”

Alignment and the Classification of Lorentz-Signature Tensors

“…A discussion of some more theoretical aspects of alignment can be found in [7]. We note that a similar notion of an aligned null directions, valid for tensor products of multivectors but defined in terms of tensorial contractions, was previously introduced and discussed in [11].…”