Null cone preserving maps, causal tensors and algebraic Rainich theory

A rank-n tensor on a Lorentzian manifold whose contraction with n arbitrary causal future directed vectors is non-negative is said to have the dominant property. These tensors, up to sign, are called causal tensors, and we determine their general mathematical properties in arbitrary dimension N . Then, we prove that rank-2 tensors which map the null cone on itself are causal tensors. Previously it has been shown that, to any tensor field A on a Lorentzian manifold there is a corresponding “superenergy” tensor field T{A} (defined as a quadratic sum over all Hodge duals of A) which always has the dominant property. Here we prove that, conversely, any symmetric rank-2 tensor with the dominant property can be written in a canonical way as a sum of N superenergy tensors of simple forms. We show that the square of any rank-2 superenergy tensor is proportional to the metric in dimension N ≤ 4, and that the square of the superenergy tensor of any simple form is proportional to the metric in arbitrary dimension. Conversely, we prove in arbitrary dimension that any symmetric rank-2 tensor T whose square is proportional to the metric must be a causal tensor and, up to sign, the superenergy of a simple p-form, and that the trace of T determines the rank p of the form. This generalises, both with respect to the dimension N and the rank