Curvature operators and scalar curvature invariants

We continue the study of the question of when a pseudo-Riemannain manifold can be locally characterised by its scalar polynomial curvature invariants (constructed from the Riemann tensor and its covariant derivatives). We make further use of alignment theory and the bivector form of the Weyl operator in higher dimensions, and introduce the important notions of diagonalisability and (complex) analytic metric extension. We show that if there exists an analytic metric extension of an arbitrary dimensional space of any signature to a Riemannian space (of Euclidean signature), then that space is characterised by its scalar curvature invariants. In particular, we discuss the Lorentzian case and the neutral signature case in four dimensions in more detail.Comment: 26 pages, 2 figure

Pseudo-Riemannian VSI spaces

In this paper we consider pseudo-Riemannian spaces of arbitrary signature for which all of their polynomial curvature invariants vanish (VSI spaces). We discuss an algebraic classification of pseudo-Riemannian spaces in terms of the boost weight decomposition and define the ${\bf S}_i$- and ${\bf N}$-properties, and show that if the curvature tensors of the space possess the ${\bf N}$-property then it is a VSI space. We then use this result to construct a set of metrics that are VSI. All of the VSI spaces constructed possess a geodesic, expansion-free, shear-free, and twist-free null-congruence. We also discuss the related Walker metrics.Comment: 14 page

Lorentzian manifolds and scalar curvature invariants

We discuss (arbitrary-dimensional) Lorentzian manifolds and the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. Recently, we have shown that in four dimensions a Lorentzian spacetime metric is either $\mathcal{I}$-non-degenerate, and hence locally characterized by its scalar polynomial curvature invariants, or is a degenerate Kundt spacetime. We present a number of results that generalize these results to higher dimensions and discuss their consequences and potential physical applications.Comment: submitted to CQ

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

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.Comment: 14pages, CQG to appea

Brane Waves

In brane-world cosmology gravitational waves can propagate in the higher dimensions (i.e., in the `bulk'). In some appropriate regimes the bulk gravitational waves may be approximated by plane waves. We systematically study five-dimensional gravitational waves that are algebraically special and of type N. In the most physically relevant case the projected non-local stress tensor on the brane is formally equivalent to the energy-momentum tensor of a null fluid. Some exact solutions are studied to illustrate the features of these branes; in particular, we show explicity that any plane wave brane can be embedded into a 5-dimensional Siklos spacetime. More importantly, it is possible that in some appropriate regime the bulk can be approximated by gravitational plane waves and thus may act as initial conditions for the gravitational field in the bulk (thereby enabling the field equations to be integrated on the brane).Comment: 9 pages v3:revised version, to appear in CQ

Electric and magnetic Weyl tensors in higher dimensions

Recent results on purely electric (PE) or magnetic (PM) spacetimes in n dimensions are summarized. These include: Weyl types; diagonalizability; conditions under which direct (or warped) products are PE/PM.Comment: 4 pages; short summary of (parts of) arXiv:1203.3563. Proceedings of "Relativity and Gravitation - 100 Years after Einstein in Prague", Prague, June 25-29, 2012 (http://ae100prg.mff.cuni.cz/