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

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.

14pages, CQG to appear

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• Wick rotations and real GIT
• On conformally recurrent manifolds of dimension greater than 4
• $\mathcal{I}$-Preserving Diffeomorphisms of Lorentzian Manifolds
• Teleparallel geometries not characterized by their scalar polynomial torsion invariants
• Electric and magnetic Weyl tensors in higher dimensions