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 $G_3$ 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.