A spacetime not characterised by its invariants is of aligned type II

TL;DR

This paper demonstrates that higher-dimensional Lorentzian metrics not characterized by their invariants are necessarily of aligned type II, linking invariant theory with algebraic classification and proving related conjectures.

Contribution

It establishes a fundamental connection between invariant characterization and algebraic type, extending the understanding of Lorentzian metrics in higher dimensions.

Findings

Metrics not characterized by invariants are of aligned type II

All positive boost-weight components vanish in such metrics

Supports the algebraic VSI conjecture

Abstract

By using invariant theory we show that a (higher-dimensional) Lorentzian metric that is not characterised by its invariants must be of aligned type II; i.e., there exists a frame such that all the curvature tensors are simultaneously of type II. This implies, using the boost-weight decomposition, that for such a metric there exists a frame such that all positive boost-weight components are zero. Indeed, we show a more general result, namely that any set of tensors which is not characterised by its invariants, must be of aligned type II. This result enables us to prove a number of related results, among them the algebraic VSI conjecture.