345 research outputs found
On the causal properties of warped product spacetimes
It is shown that the warped product spacetime P=M *_f H, where H is a
complete Riemannian manifold, and the original spacetime M share necessarily
the same causality properties, the only exceptions being the properties of
causal continuity and causal simplicity which present some subtleties. For
instance, it is shown that if diamH=+\infty, the direct product spacetime P=M*H
is causally simple if and only if (M,g) is causally simple, the Lorentzian
distance on M is continuous and any two causally related events at finite
distance are connected by a maximizing geodesic. Similar conditions are found
for the causal continuity property. Some new results concerning the behavior of
the Lorentzian distance on distinguishing, causally continuous, and causally
simple spacetimes are obtained. Finally, a formula which gives the Lorentzian
distance on the direct product in terms of the distances on the two factors
(M,g) and (H,h) is obtained.Comment: 22 pages, 2 figures, uses the package psfra
An anisotropic gravity theory
We study an action integral for Finsler gravity obtained by pulling back an
Einstein-Cartan-like Lagrangian from the tangent bundle to the base manifold.
The vacuum equations are obtained imposing stationarity with respect to any
section (observer) and are well posed as they are independent of the section.
They imply that in vacuum the metric is actually independent of the velocity
variable so the dynamics becomes coincident with that of general relativity.Comment: Latex, 15 pages. v2: we fixed some typos and added two more
appendices with proofs of formulas used in the work. To appear in the topical
volume `Singularity theorems, causality, and all that (SCRI21)'
https://link.springer.com/collections/hjjgajaag
A Note on Non-compact Cauchy surface
It is shown that if a space-time has non-compact Cauchy surface, then its
topological, differentiable, and causal structure are completely determined by
a class of compact subsets of its Cauchy surface. Since causal structure
determines its topological, differentiable, and conformal structure of
space-time, this gives a natural way to encode the corresponding structures
into its Cauchy surface
The causal boundary of wave-type spacetimes
A complete and systematic approach to compute the causal boundary of
wave-type spacetimes is carried out. The case of a 1-dimensional boundary is
specially analyzed and its critical appearance in pp-wave type spacetimes is
emphasized. In particular, the corresponding results obtained in the framework
of the AdS/CFT correspondence for holography on the boundary, are reinterpreted
and very widely generalized. Technically, a recent new definition of causal
boundary is used and stressed. Moreover, a set of mathematical tools is
introduced (analytical functional approach, Sturm-Liouville theory, Fermat-type
arrival time, Busemann-type functions).Comment: 41 pages, 1 table. Included 4 new figures, and some small
modifications. To appear in JHE
Further properties of causal relationship: causal structure stability, new criteria for isocausality and counterexamples
Recently ({\em Class. Quant. Grav.} {\bf 20} 625-664) the concept of {\em
causal mapping} between spacetimes --essentially equivalent in this context to
the {\em chronological map} one in abstract chronological spaces--, and the
related notion of {\em causal structure}, have been introduced as new tools to
study causality in Lorentzian geometry. In the present paper, these tools are
further developed in several directions such as: (i) causal mappings --and,
thus, abstract chronological ones-- do not preserve two levels of the standard
hierarchy of causality conditions (however, they preserve the remaining levels
as shown in the above reference), (ii) even though global hyperbolicity is a
stable property (in the set of all time-oriented Lorentzian metrics on a fixed
manifold), the causal structure of a globally hyperbolic spacetime can be
unstable against perturbations; in fact, we show that the causal structures of
Minkowski and Einstein static spacetimes remain stable, whereas that of de
Sitter becomes unstable, (iii) general criteria allow us to discriminate
different causal structures in some general spacetimes (e.g. globally
hyperbolic, stationary standard); in particular, there are infinitely many
different globally hyperbolic causal structures (and thus, different conformal
ones) on , (iv) plane waves with the same number of positive eigenvalues
in the frequency matrix share the same causal structure and, thus, they have
equal causal extensions and causal boundaries.Comment: 33 pages, 9 figures, final version (the paper title has been
changed). To appear in Classical and Quantum Gravit
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
The causal ladder and the strength of K-causality. I
A unifying framework for the study of causal relations is presented. The
causal relations are regarded as subsets of M x M and the role of the
corresponding antisymmetry conditions in the construction of the causal ladder
is stressed. The causal hierarchy of spacetime is built from chronology up to
K-causality and new characterizations of the distinction and strong causality
properties are obtained. The closure of the causal future is not transitive, as
a consequence its repeated composition leads to an infinite causal subladder
between strong causality and K-causality - the A-causality subladder. A
spacetime example is given which proves that K-causality differs from infinite
A-causality.Comment: 16 pages, one figure. Old title: ``On the relationship between
K-causality and infinite A-causality''. Some typos fixed; small change in the
proof of lemma 4.
Petrov D vacuum spaces revisited: Identities and Invariant Classification
For Petrov D vacuum spaces, two simple identities are rederived and some new
identities are obtained, in a manageable form, by a systematic and transparent
analysis using the GHP formalism. This gives a complete involutive set of
tables for the four GHP derivatives on each of the four GHP spin coefficients
and the one Weyl tensor component. It follows directly from these results that
the theoretical upper bound on the order of covariant differentiation of the
Riemann tensor required for a Karlhede classification of these spaces is
reduced to two.Comment: Proof about the Karlhede upper bound improved and discussion of case
IIIA re-written. Acknowledgments section expanded. To appear in Classical and
Quantum Gravit
Conformal geodesics in spherically symmetric vacuum spacetimes with cosmological constant
An analysis of conformal geodesics in the Schwarzschild-de Sitter and
Schwarzschild-anti de Sitter families of spacetimes is given. For both families
of spacetimes we show that initial data on a spacelike hypersurface can be
given such that the congruence of conformal geodesics arising from this data
cover the whole maximal extension of canonical conformal representations of the
spacetimes without forming caustic points. For the Schwarzschild-de Sitter
family, the resulting congruence can be used to obtain global conformal
Gaussian systems of coordinates of the conformal representation. In the case of
the Schwarzschild-anti de Sitter family, the natural parameter of the curves
only covers a restricted time span so that these global conformal Gaussian
systems do not exist.Comment: 51 pages, 12 figures. Minor changes. File updated. To appear in CQ
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