ar X iv : g r - qc / 0 60 80 65 v 1 1 2 A ug 2 00 6 The universal ‘ energy ’ operator

The " positive square " of any tensor is presented in a universal and unified manner , valid in Lorentzian manifolds of arbitrary dimension, and independently of any (anti)-symmetry properties of the tensor. For rank-m tensors, the positive square has rank 2m. Positive here means future, that is to say, satisfying the dominant property. The standard energy-momentum and super-energy tensors are recovered as appropriate parts of the general square. A richer structure of principal null directions arises. The energy-momentum tensors of a 2-form (such as the electromagnetic field) or of a 1-form (the gradient of the scalar field), or the Bel [4] and Bel-Robinson [3] tensors ('super-energy tensor' of the gravitational field, see also [9]), or the super-energy tensors of the electromagnetic [10, 24] and scalar fields [27, 24, 28], etc., have either different expressions or different number of indices, or both. A unified, algebraic, perspective of all these tensors was presented in [24] for Lorentzian manifolds of any dimension, where a general method to construct the positive square 1 of any tensor, called generically its 'superenergy' tensor, was put forward. This method recovered the standard expressions for the energy-momentum and super-energy tensors of the physical fields, keeping therefore its diversity in rank and structure, and allows to understand that these differences arise because of, and depend on, (i) the number of anti-symmetric sets into which the indices of the fields fall, and (ii) the number of indices within each of these antisymmetric blocks. In this short Note, I would like to call attention to a very useful positive square of a tensor that does not seem to have been noticed before. It is the most general —up to index permutations– future tensor quadratic on the given tensor. 1 By " positive square " of a tensor I mean a future tensor quadratic in the original one. Future tensors are those satisfying the dominant property, see (5) below; see section 4 in [24], section 2 in [6], section 2.1 in [18], section 7 in [22], and references therein for further details.