2,927 research outputs found
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 - and -properties, and show that if the curvature tensors of the space possess the
-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 -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/
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