150 research outputs found
Lie groups of conformal motions acting on null orbits
Space-times admitting a 3-dimensional Lie group of conformal motions
acting on null orbits are studied. Coordinate expressions for the metric and
the conformal Killing vectors (CKV) are provided (irrespectively of the matter
content) and then all possible perfect fluid solutions are found, although none
of these verify the weak and dominant energy conditions over the whole
space-time manifold.Comment: 5 pages, Late
A Lorentzian Gromov-Hausdoff notion of distance
This paper is the first of three in which I study the moduli space of
isometry classes of (compact) globally hyperbolic spacetimes (with boundary). I
introduce a notion of Gromov-Hausdorff distance which makes this moduli space
into a metric space. Further properties of this metric space are studied in the
next papers. The importance of the work can be situated in fields such as
cosmology, quantum gravity and - for the mathematicians - global Lorentzian
geometry.Comment: 20 pages, 0 figures, submitted to Classical and quantum gravity,
seriously improved presentatio
General non-rotating perfect-fluid solution with an abelian spacelike C_3 including only one isometry
The general solution for non-rotating perfect-fluid spacetimes admitting one
Killing vector and two conformal (non-isometric) Killing vectors spanning an
abelian three-dimensional conformal algebra (C_3) acting on spacelike
hypersurfaces is presented. It is of Petrov type D; some properties of the
family such as matter contents are given. This family turns out to be an
extension of a solution recently given in \cite{SeS} using completely different
methods. The family contains Friedman-Lema\^{\i}tre-Robertson-Walker particular
cases and could be useful as a test for the different FLRW perturbation
schemes. There are two very interesting limiting cases, one with a non-abelian
G_2 and another with an abelian G_2 acting non-orthogonally transitively on
spacelike surfaces and with the fluid velocity non-orthogonal to the group
orbits. No examples are known to the authors in these classes.Comment: Submitted to GRG, Latex fil
The state space and physical interpretation of self-similar spherically symmetric perfect-fluid models
The purpose of this paper is to further investigate the solution space of
self-similar spherically symmetric perfect-fluid models and gain deeper
understanding of the physical aspects of these solutions. We achieve this by
combining the state space description of the homothetic approach with the use
of the physically interesting quantities arising in the comoving approach. We
focus on three types of models. First, we consider models that are natural
inhomogeneous generalizations of the Friedmann Universe; such models are
asymptotically Friedmann in their past and evolve fluctuations in the energy
density at later times. Second, we consider so-called quasi-static models. This
class includes models that undergo self-similar gravitational collapse and is
important for studying the formation of naked singularities. If naked
singularities do form, they have profound implications for the predictability
of general relativity as a theory. Third, we consider a new class of
asymptotically Minkowski self-similar spacetimes, emphasizing that some of them
are associated with the self-similar solutions associated with the critical
behaviour observed in recent gravitational collapse calculations.Comment: 24 pages, 12 figure
Kerr-Schild Symmetries
We study continuous groups of generalized Kerr-Schild transformations and the
vector fields that generate them in any n-dimensional manifold with a
Lorentzian metric. We prove that all these vector fields can be intrinsically
characterized and that they constitute a Lie algebra if the null deformation
direction is fixed. The properties of these Lie algebras are briefly analyzed
and we show that they are generically finite-dimensional but that they may have
infinite dimension in some relevant situations. The most general vector fields
of the above type are explicitly constructed for the following cases: any
two-dimensional metric, the general spherically symmetric metric and
deformation direction, and the flat metric with parallel or cylindrical
deformation directions.Comment: 15 pages, no figures, LaTe
Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle-Pock algorithm
The primal-dual optimization algorithm developed in Chambolle and Pock (CP),
2011 is applied to various convex optimization problems of interest in computed
tomography (CT) image reconstruction. This algorithm allows for rapid
prototyping of optimization problems for the purpose of designing iterative
image reconstruction algorithms for CT. The primal-dual algorithm is briefly
summarized in the article, and its potential for prototyping is demonstrated by
explicitly deriving CP algorithm instances for many optimization problems
relevant to CT. An example application modeling breast CT with low-intensity
X-ray illumination is presented.Comment: Resubmitted to Physics in Medicine and Biology. Text has been
modified according to referee comments, and typos in the equations have been
correcte
Bi-conformal vector fields and their applications
We introduce the concept of bi-conformal transformation, as a generalization
of conformal ones, by allowing two orthogonal parts of a manifold with metric
\G to be scaled by different conformal factors. In particular, we study their
infinitesimal version, called bi-conformal vector fields. We show the
differential conditions characterizing them in terms of a "square root" of the
metric, or equivalently of two complementary orthogonal projectors. Keeping
these fixed, the set of bi-conformal vector fields is a Lie algebra which can
be finite or infinite dimensional according to the dimensionality of the
projectors. We determine (i) when an infinite-dimensional case is feasible and
its properties, and (ii) a normal system for the generators in the
finite-dimensional case. Its integrability conditions are also analyzed, which
in particular provides the maximum number of linearly independent solutions. We
identify the corresponding maximal spaces, and show a necessary geometric
condition for a metric tensor to be a double-twisted product. More general
``breakable'' spaces are briefly considered. Many known symmetries are
included, such as conformal Killing vectors, Kerr-Schild vector fields,
kinematic self-similarity, causal symmetries, and rigid motions.Comment: Replaced version with some changes in the terminology and a new
theorem. To appear in Classical and Quantum Gravit
Solvegeometry gravitational waves
In this paper we construct negatively curved Einstein spaces describing
gravitational waves having a solvegeometry wave-front (i.e., the wave-fronts
are solvable Lie groups equipped with a left-invariant metric). Using the
Einstein solvmanifolds (i.e., solvable Lie groups considered as manifolds)
constructed in a previous paper as a starting point, we show that there also
exist solvegeometry gravitational waves. Some geometric aspects are discussed
and examples of spacetimes having additional symmetries are given, for example,
spacetimes generalising the Kaigorodov solution. The solvegeometry
gravitational waves are also examples of spacetimes which are indistinguishable
by considering the scalar curvature invariants alone.Comment: 10 pages; v2:more discussion and result
Kinematic Self-Similarity
Self-similarity in general relativity is briefly reviewed and the differences
between self-similarity of the first kind and generalized self-similarity are
discussed. The covariant notion of a kinematic self-similarity in the context
of relativistic fluid mechanics is defined. Various mathematical and physical
properties of spacetimes admitting a kinematic self-similarity are discussed.
The governing equations for perfect fluid cosmological models are introduced
and a set of integrability conditions for the existence of a proper kinematic
self-similarity in these models is derived. Exact solutions of the irrotational
perfect fluid Einstein field equations admitting a kinematic self-similarity
are then sought in a number of special cases, and it is found that; (1) in the
geodesic case the 3-spaces orthogonal to the fluid velocity vector are
necessarily Ricci-flat and (ii) in the further specialisation to dust the
differential equation governing the expansion can be completely integrated and
the asymptotic properties of these solutions can be determined, (iii) the
solutions in the case of zero-expansion consist of a class of shear-free and
static models and a class of stiff perfect fluid (and non-static) models, and
(iv) solutions in which the kinematic self-similar vector is parallel to the
fluid velocity vector are necessarily Friedmann-Robertson-Walker (FRW) models.Comment: 29 pages, AmsTe
A physical application of Kerr-Schild groups
The present work deals with the search of useful physical applications of
some generalized groups of metric transformations. We put forward different
proposals and focus our attention on the implementation of one of them.
Particularly, the results show how one can control very efficiently the kind of
spacetimes related by a Generalized Kerr-Schild (GKS) Ansatz through
Kerr-Schild groups. Finally a preliminar study regarding other generalized
groups of metric transformations is undertaken which is aimed at giving some
hints in new Ans\"atze to finding useful solutions to Einstein's equations.Comment: 18 page
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