136 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
What is General Relativity?
General relativity is a set of physical and geometric principles, which lead
to a set of (Einstein) field equations that determine the gravitational field,
and to the geodesic equations that describe light propagation and the motion of
particles on the background. But open questions remain, including: What is the
scale on which matter and geometry are dynamically coupled in the Einstein
equations? Are the field equations valid on small and large scales? What is the
largest scale on which matter can be coarse grained while following a geodesic
of a solution to Einstein's equations? We address these questions. If the field
equations are causal evolution equations, whose average on cosmological scales
is not an exact solution of the Einstein equations, then some simplifying
physical principle is required to explain the statistical homogeneity of the
late epoch Universe. Such a principle may have its origin in the dynamical
coupling between matter and geometry at the quantum level in the early
Universe. This possibility is hinted at by diverse approaches to quantum
gravity which find a dynamical reduction to two effective dimensions at high
energies on one hand, and by cosmological observations which are beginning to
strongly restrict the class of viable inflationary phenomenologies on the
other. We suggest that the foundational principles of general relativity will
play a central role in reformulating the theory of spacetime structure to meet
the challenges of cosmology in the 21st century.Comment: 18 pages. Invited article for Physica Scripta Focus issue on 21st
Century Frontiers. v2: Appendix amended, references added. v3: Small
corrections, references added, matches published versio
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
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
Lorentzian spacetimes with constant curvature invariants in three dimensions
In this paper we study Lorentzian spacetimes for which all polynomial scalar
invariants constructed from the Riemann tensor and its covariant derivatives
are constant (CSI spacetimes) in three dimensions. We determine all such CSI
metrics explicitly, and show that for every CSI with particular constant
invariants there is a locally homogeneous spacetime with precisely the same
constant invariants. We prove that a three-dimensional CSI spacetime is either
(i) locally homogeneous or (ii) it is locally a Kundt spacetime. Moreover, we
show that there exists a null frame in which the Riemann (Ricci) tensor and its
derivatives are of boost order zero with constant boost weight zero components
at each order. Lastly, these spacetimes can be explicitly constructed from
locally homogeneous spacetimes and vanishing scalar invariant spacetimes.Comment: 14 pages; Modified to match published versio
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