Letter: Spatial Metrics Which are Static in Many Ways

We consider Riemannian 3-metrics which can form the spatial part of vacuum solutions of the Einstein equations, possibly with a cosmological constant, in more than one way (in a sense made precise). The locally rotationally symmetric (LRS) Kasner metric gives the simplest example, and we find that the resulting space-time metrics are always of Petrov type D.     

On shear free normal flows of a perfect fluid

Flows of a perfect fluid in which the flow-lines form a time-like shear-free normal congruence are investigated. The space-time is quite severely restricted by this condition on the flow: it must be of Petrov Type I and is either static or degenerate. All the degenerate fields are classified and the field equations solved completely, except in one class where one ordinary differential equation remains to be solved. This class contains the spherically symmetric non-uniform density fields and their analogues with planar or hyperbolic symmetry. The type D fields admit at least a one-parameter group of local isometries with space-like trajectories. All vacuum fields which admit a time-like shear-free normal congruence are shown to be static. Finally, shear-free perfect fluid flows which possess spherical or a related symmetry are considered, and all uniform density solutions and a few non-uniform density solutions are found. The exact solutions are tabulated in section 7.Supported by a Science Research Council Research Studentship and by a Turner and Newall Research Fellowship.     

Homothetic motions with null homothetic bivectors in general relativity

It is shown that if a nonflat vacuum space-time admits a homothetic vector field with a null homothetic bivector then that space-time is algebraically special. If that homothetic vector field is a nontrivial one (not a Killing one) then the space-time is Petrov type III orN.     

半平面对称非类光电磁场产生的半平面对称静态时空的代数性质

本文求出半平面对称非类光电磁场产生的半平面对称静态时空的自旋系数和Weyl张量的复标架分量,结果表明这种时空是Petrov Ⅰ型,即代数一般的。对零矢量和光学标量的性质也作了讨论。 关键词：     

Mappings of empty space-times leaving the curvature tensor invariant

It is shown that if in some local coordinate system the componentsR ⁱ _jkl of the curvature tensor of an empty space-time are known, then, provided the space-time is not of Petrov typeN with hypersurface orthogonal geodesic rays, the components of the metric tensor are uniquely determined up to a trivial constant scaling factor. The Petrov type-N empty space-times with hypersurface orthogonal geodesic rays are investigated. The most general mappings leaving the curvature tensorR ⁱ _jkl invariant are found for each class of these space-times.     

Determination of all Petrov type-N space-times on which the conformally invariant scalar wave equation satisfies Huygen's principle

It is shown that the conformally invariant wave equation on a Petrov type-N space-time satisfies Huygens' principle if and only if the space-time is conformally related to a plane wave space-time.     

Petrov Types of Slowly Rotating Fluid Balls

Circularly rotating axisymmetric perfect fluid space-times are investigated to second order in the small angular velocity. The conditions of various special Petrov types are solved in a comoving tetrad formalism. A number of theorems are stated on the possible Petrov types of various fluid models. It is shown that Petrov type II solutions must reduce to the de Sitter spacetime in the static limit. Two space-times with a physically satisfactory energy-momentum tensor are investigated in detail. For the rotating incompressible fluid, it is proven that the Petrov type cannot be D. The equation of the rotation function can be solved for the Tolman type IV fluid in terms of quadratures. It is also shown that the rotating version of the Tolman IV space-time cannot be Petrov type D.     

Nonstatic analogues of Kohler-Chao solution of imbedding class one

The nonstatic analogues of the Kohler-Chao perfect fluid solution have been derived starting with a spherically symmetric flatV ₅. It is also established that all the Petrov type D perfect fluid solutions of imbedding class one do not possess a pressure-free surface at a finite radius and therefore cannot be fitted to the external Schwarzschild's model.     

整体正规Schwarzschild时空的Newman-Penrose表述

分别在Schwarzschild和Kruskal坐标中给出整体正规Schwarzschild时空的Newman-Penrose表述,结果表明这种时空有整体代数性质,它是Petrov D型,对Weyl张量的标架分量、自旋系数和零矢量的行为也作了讨论。     

On an Alignment Condition of the Weyl Tensor

We generalize an alignment condition of the Weyl tensor given by Barnes and Rowlingson. The alignment condition is then applied to Petrov type D perfect fluid spacetimes. In particular, purely magnetic, Petrov type D, shear-free perfect fluids are shown to be locally rotationally symmetric.     

Classification of Petrov Type D Generalized Kerr-Schild Transformations

We study the Petrov type D Generalized Kerr-Schild vacuum space-times. If the background is a Minkowski space-time the resulting Kerr-Schild metrics are classified, while if we start from a Petrov type D background we first put restrictions on the possible perturbation null directions. It is then shown that the GKS metrics are different from the background if the null direction has nonvanishing expansion or twist, otherwise the transformation is merely a change of coordinates.     

Space-time metrics inducing the radiation connection

It is shown that Lorentzian manifolds giving rise to the radiation connection, i.e. such that the canonical connection induces a unique connection on an isotropic hypersurface, are of type D or conformally flat in Petrov classification, generalizing the Robinson-Bertotti electromagnetic universe with cosmological constant. Under an holonomy hypothesis it is deduced that the underlying space-time manifold is a direct product of two two-dimensional spaces, one of them being space-like, and the other one time-like.     

A Cylindrically Symmetric Universe Filled with Viscous Fluid in a Modified Brans-Dicke Cosmology

Adding the cosmological term, which is assumed to be variable in Brans-Dicke theory we have discussed about a cylindrically symmetric model filled with viscous fluid with free gravitational field of Petrov type-D. The effect of viscosity on various kinematical parameters has been discussed. Finally, this model has been transformed to the original form (1961) of Brans-Dicke theory (including a variable cosmological term).     

Ricci recurrent space times

The algebraic restrictions on the Ricci tensor in a Ricci-recurrent space-time are determined. The restrictions imposed on the Petrov type of the Weyl tensor are also given.     

Vacuum Rank 1 Space-Time Perturbations

We study Kerr-Schild type perturbations with a non-null perturbation vector in the vacuum case. The perturbation equations are derived and it is shown that they lead to constraints on the background space-time which can be interpreted in terms of the curvature of 3-spaces. The first order perturbation equations are used to construct new Petrov type D solutions tangent to the Schwarzschild metric.     

Principle of event symmetry

To accommodate topology change, the symmetry of space-time must be extended from the diffeomorphism group of a manifold to the symmetric group acting on the discrete set of space-time events. This is the principle ofevent-symmetric space-time. I investigate a number of physical toy models with this symmetry to gain some insight into the likely nature of event-symmetric space-time. In the more advanced models the symmetric group is embedded into larger structures such as matrix groups which provide scope to unify space-time symmetry with the internal gauge symmetries of particle physics. I also suggest that the symmetric group of space-time could be related to the symmetric group acting to exchange identical particles, implying a unification of space-time and matter. I end with a definition of a new type of loop symmetry which is important in event-symmetric superstring theory.     

Separation of variables in Einstein spaces. II. No ignorable coordinates

We prove that the only Einstein spaces which admit a coordinate system with no ignorable coordinates which separates the Hamilton-Jacobi equation are certain symmetric spaces of Petrov typeD due to Kasner and the constant-curvature de Sitter spaces. We also show that a space admitting a coordinate system with no ignorable coordinates which separates the Helmholtz (Schrödinger) equation must be of Petrov type     

静态球对称黑洞Dirac场的Stefan-Boltzmann定律

利用静态球对称黑洞Dirac场的统计熵，导出静态球对称黑洞的Stefan-Boltzmann定律，得 到黑洞的辐出度与视界温度的四次方成正比的结论.发现Stefan-Boltzmann常数不同于平直 时空的值，并且在不同时空度规中该常数有不同的值. 关键词： 黑洞 统计熵 薄层模型 辐出度     

Discrete symmetries of Petrov type-I space-times

The arbitrariness in the definition of the canonical frame in a Petrov type-I space-time is examined and shown to imply that the Newman-Penrose formalism is invariant under a 24-element discrete symmetry group. The transformations of the N-P scalars under the operations of the group are obtained, and conditions are formulated which characterize space-times that maintain this symmetry globally.     

A Petrov type-III perfect fluid solution of Einstein's equations

An exact solution of Einstein's perfect fluid field equations is presented. The Weyl tensor of this space-time is of Petrov type III.This work was carried out while the author was on an SRC grant at the Department of Mathematics, Queen Elizabeth College, University of London.