On quadratic first integrals of the geodesic equations for type {22} spacetimes

It is shown that every type {22} vacuum solution of Einstein's equations admits a quadratic first integral of the null geodesic equations (conformal Killing tensor of valence 2), which is independent of the metric and of any Killing vectors arising from symmetries. In particular, the charged Kerr solution (with or without cosmological constant) is shown to admit a Killing tensor of valence 2. The Killing tensor, together with the metric and the two Killing vectors, provides a method of explicitly integrating the geodesics of the (charged) Kerr solution, thus shedding some light on a result due to Carter.     

Fields,statistics and non-Abelian gauge groups

We examine field theories with a compact groupG of exact internal gauge symmetries so that the superselection sectors are labelled by the inequivalent irreducible representations ofG. A particle in one of these sectors obeys a parastatistics of orderd if and only if the corresponding representation ofG isd-dimensional. The correspondence between representations of the observable algebra and representations ofG extends to a mapping of the intertwining operators for these representations preserving linearity, tensor products and conjugation. Although we assume no explicit commutation property between fields, the commutation relations of fields of the same irreducible tensor character underG at spacelike separations are largely determined by the statistics parameter of the corresponding sector. For fields of conjugate irreducible tensor character the observable part of the commutator (anticommutator) vanishes at spacelike separations if the corresponding sector has para-Bose (para-Fermi) statistics.     

LETTER TO THE EDITOR: General Non-Rotating Perfect-Fluid Solution with an Abelian Spacelike C3 Including Only One Isometry

The general solution for non-rotating perfect-fluid spacetimes admitting one Killing vector and two conformal (non-isometric) Killing vectors spanning an abelian three-dimensional conformal algebra (C₃) acting on spacelike hypersurfaces is presented. It is of Petrov type D; some properties of the family such as matter contents are given. This family turns out to be an extension of a solution recently given in [9] using completely different methods. The family contains Friedman-Lemaître-Robertson-Walker particular cases and could be useful as a test for the different FLRW perturbation schemes. There are two very interesting limiting cases, one with a non-abelian G₂ and another with an abelian G₂ acting non-orthogonally transitively on spacelike surfaces and with the fluid velocity non-orthogonal to the group orbits. No examples are known to the authors in these classes.     

Perfect-fluid space-times with type-D Weyl tensor

We consider solutions of the Einstein field equations for which the Weyl tensor is of Petrov typeD, and whose source is a perfect fluid with equation of statep=p(w), wherep andw are the energy density and pressure of the fluid, respectively. We also impose two additional restrictions which are satisfied by most of the known solutions, namely, that the fluid 4-velocityu lies in the 2-space spanned by the two repeated principal null directions of the Weyl tensor, and that the Weyl tensor has zero magnetic part relative tou. Our main result is that for this class of solutions, the equation of state satisfies eitherdp/dw=0 ordp/dw= 1, or else the solution admits three or more Killing vector fields.     

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.     

Horizons and singularities in static,spherically symmetric spacetimes

We make a thorough study of the regions near finite-order metric-singularity boundaries of static, spherically symmetric spacetimes. After distinguishing curvature singularities from other types of metric breakdown, we examine the eigenvalues of the energy tensor near the singularities for positivity and energy dominance, find the causal class of the t-translation (static) Killing field, and ascertain the presence or absence of timelike, null, and spacelike geodesic incompleteness for each spacetime. For a certain subclass of spacetimes, we also show the completeness of all timelike and spacelike curves despite the superficial failure of the metric.     

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.     

PLANE SYMMETRIC GENERAL SOLUTION TO EINSTEIN-MAXWELL EQUATIONS IN HIGHER DIMENSIONS

The plane symmetric general solution to the Einstein-Maxwell equations in D =n+2 dimensions is presented. In addition to the n(n+1)/2 spacelike Killing vector fields characterizing the higher dimensional plane symmetry, there is also an extra Killing vector field in the solution, suggesting that the generalized Birkhoff theorem proved for 4-dimensional spacetimes might also be valid in higher dimensions.     

A note on Killing tensors

The connection between symmetric and skew-symmetric Killing tensors is studied. Some theorems on skew-symmetric Killing tensors are generalized, and it is shown that in all type-D vacuum metrics admitting a symmetric Killing tensor, this Killing tensor can be given in terms of a skew-symmetric Killing tensor.     

AG 2 II algebraically general twistfree vacuum spacetime

Several authors, e.g., Kerr and Debney (1970), Lun (1978), have obtained severalG ₂ II algebraically special vacuum solutions. NoG ₂ II algebraically general vacuum solutions in explicit form have been found before. In this paper, we start from a system of first order partial differential equations, obtained by using a triad formalism, which determines twistfree vacuum metrics with a spacelike Killing vector. The method of group-invariant solutions is then used and aG ₂ II algebraically general twistfree vacuum solution is obtained. The solution also admits a homothetic Killing vector and is non-geodesic. It is believed to be new. The following explicit solutions are also obtained: (1) A Petrov type II with aG ₁-group of motions solution which belongs to Kundt's class. (2) A Petrov type III,G ₃ Robinson-Trautman solution. All these solutions are known.     

Vacuum spacetimes with two-parameter spacelike isometry groups and compact invariant hypersurfaces: Topologies and boundary conditions

Spacetimes with closed spacelike hypersurfaces and spacelike two-parameter isometry groups are investigated to determine their possible global structures. It is shown that the two spacelike Killing vectors always commute with each other. Connected group-invariant spacelike hypersurfaces must be homeomorphic to S¹ ? S¹ ? S¹ (three-torus), S¹ ? S² (three-handle), S³ (three-sphere), or to a manifold which is covered by one of these. The spacetime metric and Einstein equations are simplified in the absence of nongravitational sources and used to establish the impossibility of topology change as well as other limitations on global structure. Regularity conditions for spacetimes of this type are derived and shown to be compatible with Einstein's equations.     

超Killing方程与超对称变换

在超空间(x,θ)上定义了度规张量场G^AB后,计算了四阶曲率张量R^D_ABC并找出其推广的循环性(cyclicity)。推导了超空间上保度量变换所应满足的条件,即超Killing方程:ξ_A:B+η_abξ_B:A=0。在零曲率情形,求出了超Killing方程的通解,及其相应生成元间的对易关系。在常曲率情形,找出了超Killing方程的特解。 关键词：     

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.     

Classification of conformal steckel spaces in the vaidia problem

The classification of isotropic conformal Steckel spaces satisfying a system of Einstein equations in which the right-hand side is the energy—momentum tensor of an isotropic ideal liquid is considered. The complete solution of the problem is found for the case of a conformal Steckel space admitting of one isotropic Killing vector field and two Killing tensor fields, when these objects form a complete set. Tomsk State University. Tomsk State Pedagogical University. Institute of High-Power Electronics, Siberian Branch, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 48–53, August, 1996.     

On the riemannian metrics in which admit all hyperplanes as minimal hypersurfaces

Inspired by a result of Bekkar (1991), Robert Lutz raised the following problem: determine the riemannian metrics in domains of ⁿ which admit all hyperplanes as minimal hypersurfaces. We solve the problem giving a formula which expresses its solutions in terms of the non-degenerate quadratic first integrals of the geodesic motion in the euclidean space (second-order Killing tensor fields). Then, we prove that for n = 3 the non-flat polynomial solutions of the problem are the left invariant riemannian metrics on the Heisenberg group.     

An asymptotically flat vacuum solution

An asympototically flat algebraically general vacuum metric is obtained. The solution is characterized by two commuting spacelike Killing vectors with flat integral surfaces and depends on one arbitrary function.     

On conformal Killing 2-form of the electromagnetic field

Let D:C^∞Λ^pM→C^∞(T^*MΛ^pM) be the first order linear differential operator on an n-dimensional (1≤p≤n−1) pseudo-Riemannian manifold (M,g). We have by the representation theory of orthogonal group, that the tangent bundle of this operation decomposes into the orthogonal and irreducible sum of forms of degree p+1 (which gives the exterior differential d), the forms of degree p−1 (defining the codifferential d^*) and the trace-free part of the partial symmetrization (the corresponding first order operator is denoted by D). The general forms in the kernel of D are closely related to conformal Killing vector fields, called conformal Killing p-forms, while those in kernel of d are called closed conformal Killing p-forms or, according to another terminology, planar p-forms. In particular an arbitrary planar 1-form ω is dual (by g) to the special concircular vector field ξ. We consider some local properties for the closed conformal Killing p-forms. As an application we present examples of decomposition into irreducible components for the electromagnetic field 2-form ω and its covariant derivative in four-dimensional space–time. In particular, we prove that the energy–momentum tensor T of the electromagnetic field is a symmetric conformal Killing tensor if the electromagnetic field 2-form ω is a conformal Killing form.     

The general stationary gravitational vacuum field of cylindrical symmetry

The general stationary vacuum gravitational field of cylindrical symmetry as recently found by Davies and Caplan is even static. The possible Petrov types of the Riemann tensor areI,D orO. In spacelike infinity the spacetime becomes necessarily flat.     

On Spacelike Congruences in Riemann-Cartan Space-time

The theory of spacelike congruences is briefly reviewed. Expressions for the expansion, the rotation and the shear tensor of the spacelike curves and their corresponding natural transport laws in Riemann-Cartan space-time are derived. We consider the Maxwell’s equations with torsion and established the Helmholtz theorems on vortex tubes, magnetic flux tubes and electric flux tubes. The fluid in magnetohydrodynamics provides E ^a =0, hence counterparts of Helmholtz theorems and the strength of the magnetic flux tube can be measured.     

A comment on the symmetries of Kerr black holes

Conditions are given for the linear dependence of the two Killing vectors, found by Hughston and Sommers to exist in a class of Einstein-Maxwell fields of Petrov typeD. The Killing tensors associated with these fields are shown to be contracted products of Killing Yano tensors.