Null cone preserving maps, causal tensors and algebraic Rainich theory

TL;DR

This paper characterizes causal tensors and superenergy tensors on Lorentzian manifolds, generalizes Rainich theory, and classifies null cone preserving maps, revealing deep geometric and algebraic structures related to Lorentz transformations.

Contribution

It extends Rainich theory to arbitrary dimensions and ranks, providing a canonical form for symmetric tensors with the dominant property and classifying null cone preserving maps.

Findings

Symmetric rank-2 tensors with the dominant property can be expressed as sums of superenergy tensors of simple forms.

The square of any such tensor is proportional to the metric in dimensions less than 5.

Complete classification of conformal Lorentz transformations and null cone preserving maps.

Abstract

A rank-n tensor on a Lorentzian manifold V 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 properties in dimension N. We prove that rank-2 tensors which map the null cone on itself are causal. It is known that, to any tensor A on V there is a corresponding ``superenergy'' (s-e) tensor T{A} which always has the dominant property. 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 s-e tensors of simple forms. We show that the square of any rank-2 s-e tensor is proportional to the metric if N<5, and that this holds for the s-e tensor of any simple form for arbitrary N. Conversely, we prove that any symmetric rank-2 tensor T whose square is proportional to the metric must be, up to sign, the s-e of a simple p-form, and that the trace of T determines the rank p of the form. This generalises, both with respect to N and the rank p, the classical algebraic Rainich conditions, which are necessary and sufficient conditions for a metric to originate in some physical field, and has a geometric interpretation: the set of s-e tensors of simple forms is precisely the set of tensors which preserve the null cone and its time orientation. It also means that all involutory Lorentz transformations (LT) can be represented as s-e tensors of simple forms, and that any rank-2 s-e tensor is the sum of at most N conformally involutory LT. Non-symmetric null cone preserving maps are shown to have a causal symmetric part and are classified according to the null eigenvectors of the skew-symmetric part. We thus obtain a complete classification of all conformal LT and singular null cone preserving maps on V.