Third Rank Killing Tensors in General Relativity. The (1 + 1)-dimensional Case

Third rank Killing tensors in (1 +1)-dimensional geometries are investigated andclassified. It is found that a necessary and sufficientcondition for such a geometry to admit a third rankKilling tensor can always be formulated as a quadratic PDE, oforder three or lower, in a Kahler type potential for themetric. This is in contrast to the case of first andsecond rank Killing tensors for which the integrability condition is a linear PDE. The motivation for studying higher rank Killing tensors in (1 +1)-geometries, is the fact that exact solutions of theEinstein equations are often associated with a first orsecond rank Killing tensor symmetry in the geodesicflow formulation of the dynamics. This is in particulartrue for the many models of interest for which thisformulation is (1 + 1)-dimensional, where just one additional constant of motion suffices forcomplete integrability. We show that new exact solutionscan be found by classifying geometries admitting higherrank Killing tensors.     

Kundt solutions of Einstein's equations with one non-null Killing vector

The Kundt class of algebraically special solutions of Einstein's field equations, representing vacuum and electromagnetic null fields with one twisting, non-null Killing vector, is discussed. This generalizes the case with a hypersurface-orthogonal Killing vector field which is discussed by Kramer and Neugebauer [3]. The solutions are shown to be equivalent to the Hoenselaers (vacuum) and Hoenselaers-Skea (electromagnetic null) solutions, once some small corrections and the relevant coordinate transformations are made to the latter solutions.     

Complete Classification of Conformal Killing Symmetries of Plane-Symmetric Static Spacetimes in Teleparallel Theory

We have studied the conformal, homothetic and Killing vectors in the context of teleparallel theory of gravitation for plane-symmetric static spacetimes. We have solved completely the non-linear coupled teleparallel conformal Killing equations. This yields the general form of teleparallel conformal vectors along with the conformal factor for all possible cases of metric functions. We have found four solutions which are divided into one Killing symmetries and three conformal Killing symmetries. One of these teleparalel conformal vectors depends on x only and other is a function of all spacetime coordinates. The three conformal Killing symmetries contain three proper homothetic symmetries where the conformal factor is an arbitrary non-zero constant.     

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 developments are of Petrov type N or D.     

Solution Generating with Perfect Fluids

We apply a technique, due to Stephani, for generating solutions of the Einstein-perfect-fluid equations. This technique is similar to the vacuum solution generating techniques of Ehlers, Harrison, Geroch and others. We start with a seed solution of the Einstein-perfect-fluid equations with a Killing vector. The seed solution must either have (i) a spacelike Killing vector and equation of state P = or (ii) a timelike Killing vector and equation of state + 3P = 0. The new solution generated by this technique then has the same Killing vector and the same equation of state. We choose several simple seed solutions with these equations of state and where the Killing vector has no twist. The new solutions are twisting versions of the seed solutions.     

Cohomological analysis of bosonic D-strings and 2d sigma models coupled to abelian gauge fields

We analyze completely the BRST cohomology on local functionals for two-dimensional sigma models coupled to abelian world-sheet gauge fields, including effective bosonic D-string models described by Born-Infeld actions. In particular we prove that the rigid symmetries of such models are exhausted by the solutions to generalized Killing vector equations which we have presented recently, and provide all the consistent first order deformations and candidate gauge anomalies of the models under study. For appropriate target space geometries we find nontrivial deformations both of the abelian gauge transformations and of the world-sheet diffeomorphisms, and antifield-dependent candidate anomalies for both types of symmetries separately, as well as mixed ones.     

The verification of Killing tensor components for metrics in general relativity using the computer algebra system SHEEP

We report on a program, written in the computer algebra system SHEEP, for verifying the components of Killing tensors and conformal Killing tensors. We give some examples, including the components of the Killing tensor admitted by the Kerr metric. We also note that the explicit form of all conformal Killing tensors for a subclass of the Petrov typeD solutions is known.     

Stekkel space of electrovacuum with isotropic complete sets of the type (1.1)

A complete classification of the Stekkel spaces of the electrovacuum, admitting one isotropic Killing vector field and two Killing tensor fields, is presented. All solutions of the Einstein-Maxwell equations found refer to the Petrov type N.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 79–83, October, 1988.In conclusion we note that all solutions presented above are of the Petrov type N.     

Killing and Noether Symmetries of Plane Symmetric Spacetime

This paper is devoted to investigate the Killing and Noether symmetries of static plane symmetric spacetime. For this purpose, five different cases have been discussed. The Killing and Noether symmetries of Minkowski spacetime in cartesian coordinates are calculated as a special case and it is found that Lie algebra of the Lagrangian is 10 and 17 dimensional respectively. The symmetries of Taub’s universe, anti-deSitter universe, self similar solutions of infinite kind for parallel perfect fluid case and self similar solutions of infinite kind for parallel dust case are also explored. In all the cases, the Noether generators are calculated in the presence of gauge term. All these examples justify the conjecture that Killing symmetries form a subalgebra of Noether symmetries (Bokhari et al. in Int. J. Theor. Phys. 45:1063, 2006).     

Strings in Anisotropic Cosmological Spacetimes

We present a general method to reduce the full set of equations of motion and constraints in the conformal gauge for the bosonic string moving in a four-dimensional curved spacetime manifold with two spacelike Killing vector fields, to a set of six coupled first-order partial differential equations in six unknown functions. By an explicit transformation the constraints are solved identically and one ends up with only the equations of motion and integrability conditions. We apply the method to the family of inhomogeneous, non-singular cosmological models of Senovilla possessing two spacelike Killing vector fields, and show how one can extract classes of special exact solutions, even for this highly complicated metric. For the case of the same family of exact cosmological spacetimes, we give an explicit example, not previously encountered, where we have a direct and mutual transfer of energy between the string and the gravitational field.     

Invariant solutions of Liouville's equation in Robertson-Walker space-times

We find all solutions of Liouville's equation in Robertson-Walker space-times that are either spatially homogeneous or isotropic or both. Some of these solutions depend on constants of motion that are not generated by Killing vectors. We indicate how these solutions may be used to find Einstein-Liouville solutions.     

Coupling solutions of BGG-equations in conformal spin geometry

BGG-equations are geometric overdetermined systems of partial differential equations (PDEs) on parabolic geometries. Normal solutions of BGG-equations are particularly interesting, and we give a simple formula for the necessary and sufficient additional integrability conditions on a solution. We then discuss a procedure for coupling known solutions of BGG-equations to produce new ones. Employing a suitable calculus for conformal spin structures, this yields explicit coupling formulas and conditions between almost Einstein scales, conformal Killing forms, and twistor spinors. Finally, we discuss a class of generic twistor spinors that provides an invariant decomposition of conformal Killing fields.     

Spatially homogeneous and inhomogeneous cosmologies with equation of statep=μ

This paper gives an algorithm for generating solutions of the Einstein field equations which have an irrotational perfect fluid, with equation of statep=, as source, and which admit a two-parameter Abelian group of local isometries. The algorithm is used to derive a variety of new and known spatially homogeneous cosmological models, both tilted and nontilted. However, since the solutions in general only admit two Killing vectors, spatially inhomogeneous models are also obtained. Finally, it is pointed out that the solution generation technique used in this paper is closely related to solution generation techniques that have been used to generate solutions of the source-free Brans-Dicke field equations, and of the Einstein field equations with a massless scalar field as source.     

Perfect fluids with conformal motion

The axisymmetric rigidly rotating or static perfect fluid solutions admitting a proper conformai Killing vector orthogonal to the orbits of the two-dimensional group of motions are determined.     

Classification of Static Spherically Symmetric Spacetimes by Noether Symmetries

Noether symmetries of some of the well known spherically symmetric static solutions of the Einstein’s field equations are classified. The resulting Noether symmetries in each case are compared with conservation laws given by Killing vectors and collineations of the Ricci and Riemann tensors for corresponding solutions.     

Hidden supersymmetry of domain walls and cosmologies

We show that all domain-wall solutions of gravity coupled to scalar fields for which the world-volume geometry is Minkowski or anti-de Sitter admit Killing spinors, and satisfy corresponding first-order equations involving a superpotential determined by the solution. By analytic continuation, all flat or closed Friedmann-Lema?tre-Robertson-Walker cosmologies are shown to satisfy similar first-order equations arising from the existence of "pseudo Killing" spinors.     

Killing spinors for the bosonic string and Kaluza–Klein theory with scalar potentials

The paper consists mainly of two parts. In the first part, we obtain well-defined Killing spinor equations for the low-energy effective action of the bosonic string with the conformal anomaly term. We show that the conformal anomaly term is the only scalar potential that one can add into the action that is consistent with the Killing spinor equations. In the second part, we demonstrate that Kaluza–Klein theory can be gauged so that the Killing spinors are charged under the Kaluza–Klein vector. This gauging process generates a scalar potential with a maximum that gives rise to an AdS spacetime. We also construct solutions of these theories.     

Invariant solutions to the conformal Killing–Yano equation on Lie groups

We search for invariant solutions of the conformal Killing–Yano equation on Lie groups equipped with left invariant Riemannian metrics, focusing on 2-forms. We show that when the Lie group is compact equipped with a bi-invariant metric or 2-step nilpotent, the only invariant solutions occur on the 3-dimensional sphere or on a Heisenberg group. We classify the 3-dimensional Lie groups with left invariant metrics carrying invariant conformal Killing–Yano 2-forms.     

Newman-Penrose Formalism and Gravitational Impulsive and Shock Waves

Vacuum spacetimes endowed with two commuting spacelike Killing vector fields are considered. Subject to the hypothesis that there exists a shearfree null geodesic congruence orthogonal to the two-surface generated by the two commuting spacelike Killing vector fields,it is shown that, with a specific choice of null tetrad, the Newman-Penrose equations are reduced to an ordinary differential equation of Riccati type. fiom the consideration of this differential equation, exact solutions of the vacuum Einstein field equations with distribution valued Weyl curvature describing the propagation of gravitational impulsive and shock wave of variable polarization are then constructed.