CAUSAL TENSORS AND SIMPLE FORMS

Abstract

A rank-r tensor on a Lorentzian manifold of dimension N is causal if the contraction with r arbitrary causal future-directed vectors is non-negative. General superenergy tensors1, such as the Bel and Bel-Robinson tensors, are examples of even ranked causal tensors1,2, and may therefore be useful when defining norms for geometric evolution equations3. We here show that any symmetric rank-2 causal tensor (energy-momentum tensors satisfying the dominant energy condition) can be written as a sum of at most N superenergy tensors of simple forms. If N=4 this can be expressed in an elegant way as the sum of four spinors squared. Since, for arbitrary N, the superenergy of any simple form is a self-map of the cone (its square is proportional to the metric) this leads to new representations and classifications of all conformal Lorentz transformations and to generalisations of the classical four dimensional Rainich-Misner-Wheeler (RMW) theory of determining the space-time physics from its geometry4. For non-simple forms more complicated equations are satisfied by the superenergy tensors, but also in this case we are able to generalise the RMW theory, and as an example the complete algebraic RMW theory in five dimensions is obtained5.