2,618 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
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
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
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
General Kundt spacetimes in higher dimensions
We investigate a general metric of the Kundt class of spacetimes in higher
dimensions. Geometrically, it admits a non-twisting, non-shearing and
non-expanding geodesic null congruence. We calculate all components of the
curvature and Ricci tensors, without assuming any specific matter content, and
discuss algebraic types and main geometric constraints imposed by general
Einstein's field equations. We explicitly derive Einstein-Maxwell equations,
including an arbitrary cosmological constant, in the case of vacuum or possibly
an aligned electromagnetic field. Finally, we introduce canonical subclasses of
the Kundt family and we identify the most important special cases, namely
generalised pp-waves, VSI or CSI spacetimes, and gyratons.Comment: 15 page
SO(n + 1) Symmetric Solutions of the Einstein Equations in Higher Dimensions
A method of solving the Einstein equations with a scalar field is presented.
It is applied to find higher dimensional vacuum metrics invariant under the
group SO(n + 1) acting on n-dimensional spheres.Comment: 11 page
Cosmic No Hair for Collapsing Universes
It is shown that all contracting, spatially homogeneous, orthogonal Bianchi
cosmologies that are sourced by an ultra-stiff fluid with an arbitrary and, in
general, varying equation of state asymptote to the spatially flat and
isotropic universe in the neighbourhood of the big crunch singularity. This
result is employed to investigate the asymptotic dynamics of a collapsing
Bianchi type IX universe sourced by a scalar field rolling down a steep,
negative exponential potential. A toroidally compactified version of M*-theory
that leads to such a potential is discussed and it is shown that the isotropic
attractor solution for a collapsing Bianchi type IX universe is supersymmetric
when interpreted in an eleven-dimensional context.Comment: Extended discussion to include Kantowski-Sachs universe. In press,
Classical and Quantum Gravit
Ricci identities in higher dimensions
We explore connections between geometrical properties of null congruences and
the algebraic structure of the Weyl tensor in n>4 spacetime dimensions. First,
we present the full set of Ricci identities on a suitable "null" frame, thus
completing the extension of the Newman-Penrose formalism to higher dimensions.
Then we specialize to geodetic null congruences and study specific consequences
of the Sachs equations. These imply, for example, that Kundt spacetimes are of
type II or more special (like for n=4) and that for odd n a twisting geodetic
WAND must also be shearing (in contrast to the case n=4).Comment: 8 pages. v2: typo corrected between Propositions 2 and 3. v3: typo in
the last term in the first line of (11f) corrected, missing term on the
r.h.s. of (11p) added, first sentence between Propositions 2 and 3 slightly
change
Generalization of the Geroch-Held-Penrose formalism to higher dimensions
Geroch, Held and Penrose invented a formalism for studying spacetimes
admitting one or two preferred null directions. This approach is very useful
for studying algebraically special spacetimes and their perturbations. In the
present paper, the formalism is generalized to higher-dimensional spacetimes.
This new formalism leads to equations that are considerably simpler than those
of the higher-dimensional Newman-Penrose formalism employed previously. The
dynamics of p-form test fields is analyzed using the new formalism and some
results concerning algebraically special p-form fields are proved.Comment: 24 page
Space-times admitting a three-dimensional conformal group
Perfect fluid space-times admitting a three-dimensional Lie group of
conformal motions containing a two-dimensional Abelian Lie subgroup of
isometries are studied. Demanding that the conformal Killing vector be proper
(i.e., not homothetic nor Killing), all such space-times are classified
according to the structure of their corresponding three-dimensional conformal
Lie group and the nature of their corresponding orbits (that are assumed to be
non-null). Each metric is then explicitly displayed in coordinates adapted to
the symmetry vectors. Attention is then restricted to the diagonal case, and
exact perfect fluid solutions are obtained in both the cases in which the fluid
four-velocity is tangential or orthogonal to the conformal orbits, as well as
in the more general "tilting" case.Comment: Latex 34 page
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