Superenergy tensors and their applications

In Lorentzian manifolds of any dimension the concept of causal tensors is introduced. Causal tensors have positivity properties analogous to the so-called “dominant energy condition”. Further, it is shown how to build, from any given tensor A, a new tensor quadratic in A and “positive”, in the sense that it is causal. These tensors are called superenergy tensors due to historical reasons because they generalize the classical energy-momentum and Bel-Robinson constructions. Superenergy tensors are basically unique and with multiple and diverse physical and mathematical applications, such as: a) definition of new divergencefree currents, b) conservation laws in propagation of discontinuities of fields, c) the causal propagation of fields, d) null-cone preserving maps, e) generalized Rainich-like conditions, f) causal relations and transformations, and g) generalized symmetries. Among many others. 1 Causal tensors In this contribution1 V will denote a differentiable N -dimensional manifold V endowed with a metric g of Lorentzian signature N − 2. The solid Lorentzian cone at x will be denoted by Θx = Θ + x ∪Θ − x , where Θ ± x ⊂ Tx(V ) are the future (+) and past (–) half-cones. The null cone ∂Θx is the boundary of Θx and its elements are the null vectors at x. An arbitrary point x ∈ V is usually taken, but all definitions and results translate immediately to tensor fields if a time orientation has been chosen. The x-subscript is then dropped. Definition 1.1 [1] A rank-r tensor T has the dominant property at x ∈ V if T (~u1, . . . , ~ur) ≥ 0 ∀~u1, . . . , ~ur ∈ Θ + x . The set of rank-r tensor (fields) with the dominant property will be denoted by DPr . We also put DP − r ≡ {T : −T ∈ DP + r }, DPr ≡ DP + r ∪DP − r , DP ± ≡ ⋃ r DP ± r , DP ≡ DP + ∪ DP. It is worthwhile to check also my joint contribution with Garćıa-Parrado, as well as that of Bergqvist’s, in this volume, with related results. Notice that signature convention here is opposite to those contributions and to [1]. 2 Superenergy tensors By a natural extension R+ = DP0 ⊂ DP . Rank-1 tensors in DP are simply the past-pointing causal 1-forms (while those in DP−1 are the future-directed ones). For rank-2 tensors, the dominant property was introduced by Plebański [2] in General Relativity and is usually called the dominant energy condition [3] because it is a requirement for physically acceptable energy-momentum tensors. The elements of DP will be called “causal tensors”. As in the case of pastand future-pointing vectors, any statement concerning DP has its counterpart concerning DP, and they will be generally omitted. Trivially one has Property 1.1 If T (i) ∈ DPr and αi ∈ R + (i = 1, ..., n) then n