466 research outputs found
Spinor calculus on 5-dimensional spacetimes
Penrose's spinor calculus of 4-dimensional Lorentzian geometry is extended to
the case of 5-dimensional Lorentzian geometry. Such fruitful ideas in Penrose's
spinor calculus as the spin covariant derivative, the curvature spinors or the
definition of the spin coefficients on a spin frame can be carried over to the
spinor calculus in 5-dimensional Lorentzian geometry. The algebraic and
differential properties of the curvature spinors are studied in detail and as
an application we extend the well-known 4-dimensional Newman-Penrose formalism
to a 5-dimensional spacetime.Comment: Convention mismatch and minor typos fixed. To appear in Journal of
Mathematical Physic
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
On the invariant causal characterization of singularities in spherically symmetric spacetimes
The causal character of singularities is often studied in relation to the
existence of naked singularities and the subsequent possible violation of the
cosmic censorship conjecture. Generally one constructs a model in the framework
of General Relativity described in some specific coordinates and finds an ad
hoc procedure to analyze the character of the singularity. In this article we
show that the causal character of the zero-areal-radius (R=0) singularity in
spherically symmetric models is related with some specific invariants. In this
way, if some assumptions are satisfied, one can ascertain the causal character
of the singularity algorithmically through the computation of these invariants
and, therefore, independently of the coordinates used in the model.Comment: A misprint corrected in Theor. 4.1 /Cor. 4.
Causal Relationship: a new tool for the causal characterization of Lorentzian manifolds
We define and study a new kind of relation between two diffeomorphic
Lorentzian manifolds called {\em causal relation}, which is any diffeomorphism
characterized by mapping every causal vector of the first manifold onto a
causal vector of the second. We perform a thorough study of the mathematical
properties of causal relations and prove in particular that two given
Lorentzian manifolds (say and ) may be causally related only in one
direction (say from to , but not from to ). This leads us to the
concept of causally equivalent (or {\em isocausal} in short) Lorentzian
manifolds as those mutually causally related. This concept is more general and
of a more basic nature than the conformal relationship, because we prove the
remarkable result that a conformal relation \f is characterized by the fact
of being a causal relation of the {\em particular} kind in which both \f and
\f^{-1} are causal relations. For isocausal Lorentzian manifolds there are
one-to-one correspondences, which sometimes are non-trivial, between several
classes of their respective future (and past) objects. A more important feature
of isocausal Lorentzian manifolds is that they satisfy the same causality
constraints. This indicates that the causal equivalence provides a possible
characterization of the {\it basic causal structure}, in the sense of mutual
causal compatibility, for Lorentzian manifolds. Thus, we introduce a partial
order for the equivalence classes of isocausal Lorentzian manifolds providing a
classification of spacetimes in terms of their causal properties, and a
classification of all the causal structures that a given fixed manifold can
have. A full abstract inside the paper.Comment: 47 pages, 10 figures. Version to appear in Classical and Quantum
Gravit
Discrete Group Actions on Spacetimes: Causality Conditions and the Causal Boundary
Suppose a spacetime is a quotient of a spacetime by a discrete group
of isometries. It is shown how causality conditions in the two spacetimes are
related, and how can one learn about the future causal boundary on by
studying structures in . The relations between the two are particularly
simple (the boundary of the quotient is the quotient of the boundary) if both
and have spacelike future boundaries and if it is known that the
quotient of the future completion of is past-distinguishing. (That last
assumption is automatic in the case of being multi-warped.)Comment: 32 page
On the construction of a geometric invariant measuring the deviation from Kerr data
This article contains a detailed and rigorous proof of the construction of a
geometric invariant for initial data sets for the Einstein vacuum field
equations. This geometric invariant vanishes if and only if the initial data
set corresponds to data for the Kerr spacetime, and thus, it characterises this
type of data. The construction presented is valid for boosted and non-boosted
initial data sets which are, in a sense, asymptotically Schwarzschildean. As a
preliminary step to the construction of the geometric invariant, an analysis of
a characterisation of the Kerr spacetime in terms of Killing spinors is carried
out. A space spinor split of the (spacetime) Killing spinor equation is
performed, to obtain a set of three conditions ensuring the existence of a
Killing spinor of the development of the initial data set. In order to
construct the geometric invariant, we introduce the notion of approximate
Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the
initial hypersurface and satisfy a certain second order elliptic equation
---the approximate Killing spinor equation. This equation arises as the
Euler-Lagrange equation of a non-negative integral functional. This functional
constitutes part of our geometric invariant ---however, the whole functional
does not come from a variational principle. The asymptotic behaviour of
solutions to the approximate Killing spinor equation is studied and an
existence theorem is presented.Comment: 36 pages. Updated references. Technical details correcte
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
Sustainability perspectives: a new methodological approach for quantitative assessment
This paper proposes a new tool to assess sustainability and make the concept
of sustainable development operational. It considers its multi-dimensional
structure combining the information deriving from a selection of relevant
sustainability indicators belonging to economic, social and environmental
pillars. The main novelties of this approach are the modelling framework, a
recursive-dynamic computable general equilibrium used to calculate the trend
of all indicators over time throughout the world, and the aggregation
methodology to reconcile them in one aggregate index to measure overall
sustainability. The former allows capturing the sector and regional
interactions and higher-order effects driven by background assumptions on
relevant variables to depict future scenarios. The latter makes it possible to
compare sustainability performances, under alternative scenarios, across
countries and over time. Main results show that the current sustainability at
world level differs from what the traditional measure of well-being, the GDP,
depicts, highlighting the trade-offs among different components of
sustainability. Moreover, in the next decade a slight decrease in world
sustainability may occur, in spite of an expected increase in world domestic
product. Finally, dedicated policies increase overall sustainability, showing
that social and environmental benefits may be greater than the correlated
economic costs
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
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