258 research outputs found
Vacuum spacetimes with a spacelike, hypersurface-orthogonal Killing vector: reduced equations in a canonical frame
The Newman-Penrose equations for spacetimes having one spacelike Killing
vector are reduced -- in a geometrically defined "canonical frame'' -- to a
minimal set, and its differential structure is studied. Expressions for the
frame vectors in an arbitrary coordinate basis are given, and
coordinate-independent choices of the metric functions are suggested which make
the components of the Ricci tensor in the direction of the Killing vector
vanish.Comment: 13 pages, no figures, LaTeX, to be published in Class. Quantum
Gravity; v2: added/rephrased content, corrected typos, changed 1 referenc
On the Papapetrou field in vacuum
In this paper we study the electromagnetic fields generated by a Killing
vector field in vacuum space-times (Papapetrou fields). The motivation of this
work is to provide new tools for the resolution of Maxwell's equations as well
as for the search, characterization, and study of exact solutions of Einstein's
equations. The first part of this paper is devoted to an algebraic study in
which we give an explicit and covariant procedure to construct the principal
null directions of a Papapetrou field. In the second part, we focus on the main
differential properties of the principal directions, studying when they are
geodesic, and in that case we compute their associated optical scalars. With
this information we get the conditions that a principal direction of the
Papapetrou field must satisfy in order to be aligned with a multiple principal
direction of the Weyl tensor in the case of algebraically special vacuum
space-times. Finally, we illustrate this study using the Kerr, Kasner and pp
waves space-times.Comment: 24 pages, LaTeX2e, IOP style. To appear in Classical and Quantum
Gravit
Test particle motion in a gravitational plane wave collision background
Test particle geodesic motion is analysed in detail for the background
spacetimes of the degenerate Ferrari-Ibanez colliding gravitational wave
solutions. Killing vectors have been used to reduce the equations of motion to
a first order system of differential equations which have been integrated
numerically. The associated constants of the motion have also been used to
match the geodesics as they cross over the boundary between the single plane
wave and interaction zones.Comment: 11 pages, 6 Postscript figure
General approach to the study of vacuum space-times with an isometry
In vacuum space-times the exterior derivative of a Killing vector field is a
2-form (named here as the Papapetrou field) that satisfies Maxwell's equations
without electromagnetic sources. In this paper, using the algebraic structure
of the Papapetrou field, we will set up a new formalism for the study of vacuum
space-times with an isometry, which is suitable to investigate the connections
between the isometry and the Petrov type of the space-time. This approach has
some advantages, among them, it leads to a new classification of these
space-times and the integrability conditions provide expressions that determine
completely the Weyl curvature. These facts make the formalism useful for
application to any problem or situation with an isometry and requiring the
knowledge of the curvature.Comment: 24 pages, LaTeX2e, IOP style. To appear in Classical and Quantum
Gravit
Consequences of a Killing symmetry in spacetime's local structure
In this paper we discuss the consequences of a Killing symmetry on the local
geometrical structure of four-dimensional spacetimes. We have adopted the point
of view introduced in recent works where the exterior derivative of the Killing
plays a fundamental role. Then, we study some issues related with this approach
and clarify why in many circumstances its use has advantages with respect to
other approaches. We also extend the formalism developed in the case of vacuum
spacetimes to the general case of an arbitrary energy-momentum content.
Finally, we illustrate our framework with the case of spacetimes with a
gravitating electromagnetic field.Comment: 20 pages, LaTeX2e, IOP style. Revised version accepted for
publication in Classical and Quantum Gravit
Geometrical locus of massive test particle orbits in the space of physical parameters in Kerr space-time
Gravitational radiation of binary systems can be studied by using the
adiabatic approximation in General Relativity. In this approach a small
astrophysical object follows a trajectory consisting of a chained series of
bounded geodesics (orbits) in the outer region of a Kerr Black Hole,
representing the space time created by a bigger object. In our paper we study
the entire class of orbits, both of constant radius (spherical orbits), as well
as non-null eccentricity orbits, showing a number of properties on the physical
parameters and trajectories. The main result is the determination of the
geometrical locus of all the orbits in the space of physical parameters in Kerr
space-time. This becomes a powerful tool to know if different orbits can be
connected by a continuous change of their physical parameters. A discussion on
the influence of different values of the angular momentum of the hole is given.
Main results have been obtained by analytical methods.Comment: 26 pages, 12 figure
A new approach to spherically symmetric junction surfaces and the matching of FLRW regions
We investigate timelike junctions (with surface layer) between spherically
symmetric solutions of the Einstein-field equation. In contrast to previous
investigations this is done in a coordinate system in which the junction
surface motion is absorbed in the metric, while all coordinates are continuous
at the junction surface.
The evolution equations for all relevant quantities are derived. We discuss
the no-surface layer case (boundary surface) and study the behaviour for small
surface energies. It is shown that one should expect cases in which the speed
of light is reached within a finite proper time.
We carefully discuss necessary and sufficient conditions for a possible
matching of spherically symmetric sections.
For timelike junctions between spherically symmetric space-time sections we
show explicitly that the time component of the Lanczos equation always reduces
to an identity (independently of the surface equation of state).
The results are applied to the matching of FLRW models. We discuss `vacuum
bubbles' and closed-open junctions in detail. As illustrations several
numerical integration results are presented, some of them indicate that the
junction surface can reach the speed of light within a finite time.Comment: new version - corrected boundary surface discussion, improved
presentation, and corrected reference 22 pages, many figure
Vacuum type I spacetimes and aligned Papapetrou fields: symmetries
We analyze type I vacuum solutions admitting an isometry whose Killing
2--form is aligned with a principal bivector of the Weyl tensor, and we show
that these solutions belong to a family of type I metrics which admit a group
of isometries. We give a classification of this family and we study the
Bianchi type for each class. The classes compatible with an aligned Killing
2--form are also determined. The Szekeres-Brans theorem is extended to non
vacuum spacetimes with vanishing Cotton tensor.Comment: 19 pages; a reference adde
A spacetime characterization of the Kerr metric
We obtain a characterization of the Kerr metric among stationary,
asymptotically flat, vacuum spacetimes, which extends the characterization in
terms of the Simon tensor (defined only in the manifold of trajectories) to the
whole spacetime. More precisely, we define a three index tensor on any
spacetime with a Killing field, which vanishes identically for Kerr and which
coincides in the strictly stationary region with the Simon tensor when
projected down into the manifold of trajectories. We prove that a stationary
asymptotically flat vacuum spacetime with vanishing spacetime Simon tensor is
locally isometric to Kerr. A geometrical interpretation of this
characterization in terms of the Weyl tensor is also given. Namely, a
stationary, asymptotically flat vacuum spacetime such that each principal null
direction of the Killing form is a repeated principal null direction of the
Weyl tensor is locally isometric to Kerr.Comment: 23 pages, No figures, LaTeX, to appear in Classical and Quantum
Gravit
Spherically symmetric static solution for colliding null dust
The Einstein equations are integrated in the presence of two (incoming and
outgoing) streams of null dust, under the assumptions of spherical symmetry and
staticity. The solution is also written in double null and radiation
coordinates and it is reinterpreted as an anisotropic fluid. Interior matching
with a static fluid and exterior matching with the Vaidya solution along null
hypersurfaces is discussed. The connection with two-dimensional dilaton gravity
is established.Comment: 12 pages, 7 figures, to appear in Phys. Rev.
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