20 research outputs found
A new special class of Petrov type D vacuum space-times in dimension five
Using extensions of the Newman-Penrose and Geroch-Held-Penrose formalisms to
five dimensions, we invariantly classify all Petrov type vacuum solutions
for which the Riemann tensor is isotropic in a plane orthogonal to a pair of
Weyl alligned null directionsComment: 4 pages, 1 table, no figures. Contribution to the proceedings of the
Spanish Relativity Meeting 2010 held in Granada (Spain
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
Initial data sets for the Schwarzschild spacetime
A characterisation of initial data sets for the Schwarzschild spacetime is
provided. This characterisation is obtained by performing a 3+1 decomposition
of a certain invariant characterisation of the Schwarzschild spacetime given in
terms of concomitants of the Weyl tensor. This procedure renders a set of
necessary conditions --which can be written in terms of the electric and
magnetic parts of the Weyl tensor and their concomitants-- for an initial data
set to be a Schwarzschild initial data set. Our approach also provides a
formula for a static Killing initial data set candidate --a KID candidate.
Sufficient conditions for an initial data set to be a Schwarzschild initial
data set are obtained by supplementing the necessary conditions with the
requirement that the initial data set possesses a stationary Killing initial
data set of the form given by our KID candidate. Thus, we obtain an algorithmic
procedure of checking whether a given initial data set is Schwarzschildean or
not.Comment: 16 page
Killing spinor initial data sets
A 3+1 decomposition of the twistor and valence-2 Killing spinor equation is
made using the space spinor formalism. Conditions on initial data sets for the
Einstein vacuum equations are given so that their developments contain
solutions to the twistor and/or Killing equations. These lead to the notions of
twistor and Killing spinor initial data. These notions are used to obtain a
characterisation of initial data sets whose development are of Petrov type N or
D.Comment: 31 pages, submitted to J. Geom. Phy
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
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
Dynamical laws of superenergy in General Relativity
The Bel and Bel-Robinson tensors were introduced nearly fifty years ago in an
attempt to generalize to gravitation the energy-momentum tensor of
electromagnetism. This generalization was successful from the mathematical
point of view because these tensors share mathematical properties which are
remarkably similar to those of the energy-momentum tensor of electromagnetism.
However, the physical role of these tensors in General Relativity has remained
obscure and no interpretation has achieved wide acceptance. In principle, they
cannot represent {\em energy} and the term {\em superenergy} has been coined
for the hypothetical physical magnitude lying behind them. In this work we try
to shed light on the true physical meaning of {\em superenergy} by following
the same procedure which enables us to give an interpretation of the
electromagnetic energy. This procedure consists in performing an orthogonal
splitting of the Bel and Bel-Robinson tensors and analysing the different parts
resulting from the splitting. In the electromagnetic case such splitting gives
rise to the electromagnetic {\em energy density}, the Poynting vector and the
electromagnetic stress tensor, each of them having a precise physical
interpretation which is deduced from the {\em dynamical laws} of
electromagnetism (Poynting theorem). The full orthogonal splitting of the Bel
and Bel-Robinson tensors is more complex but, as expected, similarities with
electromagnetism are present. Also the covariant divergence of the Bel tensor
is analogous to the covariant divergence of the electromagnetic energy-momentum
tensor and the orthogonal splitting of the former is found. The ensuing {\em
equations} are to the superenergy what the Poynting theorem is to
electromagnetism. See paper for full abstract.Comment: 27 pages, no figures. Typos corrected, section 9 suppressed and more
acknowledgments added. To appear in Classical and Quantum Gravit
Piecewise Silence in Discrete Cosmological Models
20 pages, 1 figure20 pages, 1 figureWe consider a family of cosmological models in which all mass is confined to a regular lattice of identical black holes. By exploiting the reflection symmetry about planes that bisect these lattices into identical halves, we are able to consider the evolution of a number of geometrically distinguished surfaces that exist within each of them. We find that the evolution equations for the reflection symmetric surfaces can be written as a simple set of Friedmann-like equations, with source terms that behave like a set of interacting effective fluids. We then show that gravitational waves are effectively trapped within small chambers for all time, and are not free to propagate throughout the space-time. Each chamber therefore evolves as if it were in isolation from the rest of the universe. We call this phenomenon "piecewise silence"
On the characterization of non-degenerate foliations of pseudo-Riemannian manifolds with conformally flat leaves
Published online 4 June 2013A necessary and sufficient condition for the leaves of a non-degenerate foliation of
a pseudo-Riemannian manifold to be conformally flat is developed. The condition
mimics the classical condition of the vanishing of the Weyl or Cotton tensor establish-
ing the conformal flatness of a pseudo-Riemannian manifold in the sense that it is also
formulated in terms of the vanishing of certain tensors. These tensors play the role
of the Weyl or the Cotton tensors and they are defined in terms of the curvature of a
linear torsion-free connection (the bi-conformal connection).Fundação para a Ciência e a Tecnologia (FCT