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.     

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.     

Perfect fluid solution of Einstein equations with two spacelike killing fields

We present a perfect fluid solution of Einstein's equations, admitting a Killing tensor with Segre characteristics [(11)(11)] and two commuting spacelike Killing fields. The Equation of state has no physical meaning but is the same as that of the Wahlquist solution,e+3p=constant, which admits the same Killing tensor, as our solution, although the two Killing fields are timelike and spacelike, respectively.     

Perfect fluid space-times with a two-dimensional orthogonally transitive group of homothetic motions

We use theghp formalism to obtain perfect fluid space-times with a two-dimensional and orthogonally transitive group of proper homothetic motionsH ₂, with the additional condition that the four-velocity of the fluid either lies on the group orbits or is orthogonal to them. In the first case the orbits of theH ₂ are timelike and all possible solutions are explicitly given. They comprise (i) space-times of Petrov type I that admit a groupH ₃ containing two hypersurface orthogonal and commuting Killing vectors (when theH ₂ is abelian, the fluid has a stiff equation of state and the space-time is of type D), and (ii) a class of type D static space-times with a maximalH ₂ in which the two-spaces orthogonal to the group orbits have constant curvature. When the orbits of theH ₂ are spacelike, the fluid is necessarily stiff and different classes of solutions admitting maximalH ₂ andH ₃ are identified.     

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.     

Cauchy problems for the conformal vacuum field equations in general relativity

Cauchy problems for Einstein's conformal vacuum field equations are reduced to Cauchy problems for first order quasilinear symmetric hyperbolic systems. The “hyperboloidal initial value” problem, where Cauchy data are given on a spacelike hypersurface which intersects past null infinity at a spacelike two-surface, is discussed and translated into the conformally related picture. It is shown that for conformal hyperboloidal initial data of classH ^S,s≧4, there is a unique (up to questions of extensibility) development which is a solution of the conformal vacuum field equations of classH ^S. It provides a solution of Einstein's vacuum field equations which has a smooth structure at past null infinity.     

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.     

A method of reduction of Einstein's equations of evolution and a natural symplectic structure on the space of gravitational degrees of freedom

A program is outlined which addresses the problem of thereduction of Einstein's equations, namely, that of writing Einstein's vacuum equations in (3+1)-dimensions as anunconstrained dynamical system where the variables are thetrue degrees of freedom of the gravitational field. Our analysis is applicable for globally hyperbolic Ricci-flat spacetimes that admit constant mean curvature compact orientable spacelike Cauchy hypersurfaces M with degM=0 andM not diffeomorphic toF ₆, the underlying manifold of a certain compact orientable flat affine 3-manifold. We find that for these spacetimes, modulo the extended Poincaré conjecture and the use of local cross-sections rather than a global cross-section, (3+1)-reduction can be completed much as in the (2+1)-dimensional case. In both cases, one gets as the reduced phase space the cotangent bundleT ^* T _M of theTeichmüller space T _M of conformal structures onM, whereM is a given initial constant mean curvature compact orientable spacelike Cauchy hypersurface in a spacetime (V, g _V ), and one gets reduction of the full classical non-reduced Hamiltonian system with constraints to a reduced Hamiltonian system without constraints onT ^* T _M . For these reduced systems, the time parameter is the parameter of a family of monotonically increasing constant mean curvature compact orientable spacelike Cauchy hypersurfaces in a neighborhood of a given initial one. In the (2+1)-dimensional case, the Hamiltonian is the area functional of these hypersurfaces, and in the (3+1)-dimensional case, the Hamiltonian is the volume functional of these hypersurfaces.     

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.     

Double Structures and Multiple Symmetry Groups for the Reduced 4-Dimensional String Background Equations

By using the extended double complex method proposed previously, the 1-loop string background equations with axion and dilaton fields in 4 dimensions with two commuting Killing symmetries and c = 0 are reduced essentially to two double Ernst -models. Then the string Geroch group acting on the solution space is extended to a multi-fold version of a semidirect product of the string Geroch group and the Virasoro group. The usual string Geroch group is just one component of a subgroup in the multi-fold semidirect product group. It is also found that a dobule Z₂ symmetry exists for every case of the 4-dimensional reduced string equations and for most of the cases here, this Z₂ symmetry is new and cannot be obtained in the usual (non-double) scheme. These results show that the reduced string background equations considered possess more and richer symmetry structures than previously expected. Moreover, a double form of string soliton method is briefly described and, as an application, a multiple family of soliton solutions of the considered reduced string equations is given, which shows that the double method is more effective.     

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：     

Real Kasner and related complex “windmill” vacuum spacetime metrics

The Kasner family of vacuum solutions of Einstein's field equations admits a simply-transitiveH ₄, a four-parameter local homothetic group of motions which has an AbelianG ₃ subgroup. It is shown that a complex transformation of coordinates and constants exists which maps this family from the normal Kasner form into a form of vacuum metrics whose Weyl tensors are each Petrov type I and which were published in 1932 by Lewis. These metrics also admit a similarH ₄; however for one particular metric (for one parameter value) theH ₄ becomes aG ₄ and the resultant metric is one which was rediscovered by Petrov in 1962. These Lewis metrics are thus shown to be Kasner metrics over complex fields. Here they are calledwindmill metrics because of the rotating relationship between the coordinates and the Killing vector fields admitted. The principal null directions of thereal Kasner and the windmill metrics are discussed; the two families then provide illustrations of two degenerate classes of spacetime metrics whose Weyl tensors are of Petrov type I, as discussed elsewhere by Arianrhod and McIntosh. An extension of the windmill-type generation of metrics to some other families of metrics is also discussed.     

Modular structure of the local algebras associated with the free massless scalar field theory

The modular structure of the von Neumann algebra of local observables associated with a double cone in the vacuum representation of the free massless scalar field theory of any number of dimensions is described. The modular automorphism group is induced by the unitary implementation of a family of generalized fractional linear transformations on Minkowski space and is a subgroup of the conformal group. The modular conjugation operator is the anti-unitary implementation of a product of time reversal and relativistic ray inversion. The group generated by the modular conjugation operators for the local algebras associated with the family of double cone regions is the group of proper conformal transformations. A theorem is presented asserting the unitary equivalence of local algebras associated with lightcones, double cones, and wedge regions. For the double cone algebras, this provides an explicit realization of spacelike duality and establishes the known typeIII ₁ factor property. It is shown that the timelike duality property of the lightcone algebras does not hold for the double cone algebras. A different definition of the von Neumann algebras associated with a region is introduced which agrees with the standard one for a lightcone or a double cone region but which allows the timelike duality property for the double cone algebras. In the case of one spatial dimension, the standard local algebras associated with the double cone regions satisfy both spacelike and timelike duality.Supported by the National Science Foundation under Grant No. PHY-79-23251Supported in part by C. N. R.     

Vacuum spacetimes with a two-dimensional orthogonally transitive group of homothetic motions

Vacuum spacetimes with a two-dimensional orthogonally transitive groupH ₂ of proper homothetic motions acting on nonnull orbits are investigated with the aid of the Geroch-Held-Penrose formalism. It is found that these spacetimes admit in general anH ₃ of homothetic motions containing two commuting and hypersurface orthogonal Killing vector fields. The metric equations are integrated, and the line elements of the spacetimes in question are explicitly given in a diagonal form.     

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.     

Holography at an extremal De Sitter horizon

Rotating maximal black holes in four-dimensional de Sitter space, for which the outer event horizon coincides with the cosmological horizon, have an infinite near-horizon region described by the rotating Nariai metric. We show that the asymptotic symmetry group at the spacelike future boundary of the near-horizon region contains a Virasoro algebra with a real, positive central charge. This is evidence that quantum gravity in a rotating Nariai background is dual to a two-dimensional Euclidean conformal field theory. These results are related to the Kerr/CFT correspondence for extremal black holes, but have two key differences: one of the black hole event horizons has been traded for the cosmological horizon, and the near-horizon geometry is a fiber over dS₂ rather than AdS₂.     

Einstein tensor and generalizations of Birkhoff's theorem

The Einstein tensors of metrics having a 3-parameter group of (global) isometries with 2-dimensional non-null orbitsG ₃(2,s/t) are studied in order to obtainalgebraic conditions guaranteeing an additional normal Killing vector. It is shown that Einstein spaces withG ₃(2,s/t) allow aG ₄. A critical review of some of the literature on Birkhoff's theorem and its generalizations is given.This work was started at the Department of Physics, Temple University, Philadelphia, Pa., and supported there by the Aerospace Research Laboratories of the Office of Aerospace Research, U.S.A.F.     

Homogeneous solutions of the Einstein equations with an ideal charged fluid

The space-times V₄ with an ideal charged fluid as source allowing the motion group G_r, r 4, are investigated. It is assumed that the fluid velocity vector is directed along the timelike vector of the Killing group. In the case of groups G₄, acting on V₄, as well as groups of higher mobility, a complete investigation is performed of the space-times by using the system of Einstein-Maxwell equations. Exact solutions are found with fourth- and fifth-order groups.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 74–79, December, 1984.     

Spinor methods in conformal Killing transport

The equations of conformal Killing transport are discussed using tensor and spinor methods. It is shown that, in Minkowski space-time, the equations for a null conformal Killing vector ξ ^a are completely determined by the corresponding spinor ω ^A and its covariant derivative, which defines a spinor π _A′ . In conformally flat space-time, the covariant derivative of π _A′ is also involved. Some applications to twistor theory are briefly mentioned.