Null cone preserving maps, causal tensors and algebraic Rainich theory

Abstract

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 1 p, the classical algebraic Rainich conditions, which are necessary and sufficient conditions for a metric to originate algebraically in some physical field. Furthermore, it has the important geometric interpretation that the set of superenergy tensors of simple forms is precisely the set of tensors which leave the null cone invariant and preserve its time orientation. It also means that all involutory Lorentz transformations can be represented as superenergy tensors of simple forms, and that any rank-2 superenergy tensor is the sum of at most N conformally involutory Lorentz transformations. Nonsymmetric null cone preserving maps are shown to have a symmetric part with the dominant property and are classified according to the null eigenvectors of the skewsymmetric part. We therefore obtain a complete classification of all conformal Lorentz transformations and singular null cone preserving maps on any Lorentzian manifold of any dimension.2

Unique characterization of the Bel–Robinson tensor

“…In [3] it was shown in arbitrary dimension that a symmetric tensor T ab which satisfies the dominant energy condition can be written as the superenergy tensor of a simple p-form [8], T ab = 4 g ab T cd T cd . That a p-form is simple means that it is a wedge product of p 1-forms.…”

“…It was also shown that the dominant energy condition could be removed since T ac T b c = 1 4 g ab T cd T cd implies that either T ab or −T ab satisfies the dominant energy condition. Therefore the conclusion without the dominant energy condition is ±T ab = In [2] superenergy tensors of more general p-forms were considered and the results of [3] were generalized in the way that the condition T ac T b c = 1 4 g ab T cd T cd was replaced by a third-order equation for T ab .…”

“…It seems that the tensorial methods used in [2] and [3] are very complicated to generalize to the higher rank case.…”

General study and basic properties of causal symmetries

“…Furthermore we can even extend it to higher rank tensors by means of the following definition [49,10] Definition A.1 A tensor T ∈ T 0 r (x) is said to be future (respectively past) if T( u 1 , . .…”

The causal hierarchy of spacetimes

“…By the way, it is also known that any of the following conditions is equivalent to items 1, 2 (this yields bounds on the possible curvatures): (a) ∃a In fact, any of these items implies item 2, by using that any lightlike vector can be approximated by both, timelike and spacelike ones. For some algebraic extensions to higher order tensors, see [3].…”