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.     

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.     

Killing vector fields and the Einstein-Maxwell field equations in general relativity

A number of theorems concerning non-null electrovac spacetimes, that is space-times whose metric satisfies the source-free Einstein-Maxwell equations for some non-null bivector F_ij, are presented. Firstly, we suppose that the metric is invariant under a one-parameter group of isornetries with Killing vector field ξ. It is proved that the electromagnetic field tensor F_ij is invariant under the group, in the sense that its Lie derivative with respect to ξ vanishes, if and only if the gradient α_ij of the complexion scalar is orthogonal to ξ. It is is also proved that if in addition ξ is hypersurface orthogonal, it is necessarily parallel to α,_i. These results are used to generalize theorems of Perjes and Majumdar concerning static electrovac space-times. Secondly, we suppose that the metric is invariant under a two-parameter othogonally transitive Abelian group of isometries. It is proved that in this case F_ij is necessarily invariant under the group. The above results can be used to simplify many derivations of exact solutions of the Einstein-Maxwell equations.     

Symmetry mappings in Einstein-Maxwell space-times

General properties of Einstein-Maxwell spaces, with both null and nonnull source-free Maxwell fields, are examined when these space-times admit various kinds of symmetry mappings. These include Killing, homothetic and conformal vector fields, curvature and Ricci collineations, and mappings belonging to the family of contracted Ricci collineations. In particular, the behavior of the electromagnetic field tensor is examined under these symmetry mappings. Examples are given of such space-times which admit proper curvature and proper Ricci collineations. Examples are also given of such space-times in which the metric tensor admits homothetic and other motions, but in which the corresponding Lie derivatives of the electromagnetic Maxwell tensor are not just proportional to the Maxwell tensor.On leave from Mathematics Department, Monash University, Clayton, Victoria, 3168, Australia.     

Proper Conformal Killing Vectors in Kantowski-Sachs Metric

This paper deals with the existence of proper conformal Killing vectors(CKVs) in Kantowski-Sachs metric.Subject to some integrability conditions, the general form of vector filed generating CKVs and the conformal factor is presented. The integrability conditions are solved generally as well as in some particular cases to show that the nonconformally flat Kantowski-Sachs metric admits two proper CKVs, while it admits a 15-dimensional Lie algebra of CKVs in the case when it becomes conformally flat. The inheriting conformal Killing vectors(ICKVs), which map fluid lines conformally, are also investigated.     

Stationary Riemannian space-times with self-dual curvature

Riemannian space-times with self-dual curvature and which admit at least one Killing vector field (stationary) are examined. Such space-times can be classified according to whether a certain scalar field (which is the difference between the Newtonian and NUT potentials) reduces to a constant or not. In the former category (called here KSD) are the multi-TaubNUT and multi-instanton space-times. Nontrivial examples of the latter category have yet to be discovered. It is proved here that the static self-dual metrics are flat. It is also proved that each stationary metric for which the Newtonian and nut potentials are functionally related admits a Killing vector field relative to which the metric is KSD. It has also been proved that the regularity of the field everywhere implies that the metric is KSD. Finally it is proved that for non-KSD space-times every regular compact level surface of the field encloses the total NUT charge, which must be proportional to the Euler number of the surface.The research reported here was done while the author was an NSERC Postdoctoral Fellow at Simon Fraser University.The author is also a member of the Theoretical Science Institute at Simon Fraser University, and preparation for publication was partially assisted NSERC Research Grant No. 3993.     

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.     

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.     

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.     

The trace anomaly in coloured black holes

The influence of the trace anomaly of the energy-momentum tensor of gauge theories is discussed in a spherically symmetric space-time. It is found that for pure electric and magnetic fields, the point β(g) = ?g represents a phase transition from any asymptotically flat metric to an asymptotically non-flat metric which turns out to be confining in the electric case.     

Third-order Killing tensors in the Schwarzschild space-time

The general solution for the third-order Killing tensor equation in the Schwarzschild space-time is written down. It follows that the Schwarzschild metric admits only redundant Killing tensors of order 3.This work was carried out under the auspices of the National Group for Mathematical Physics of C. N. R.     

Finding Collineations of Kimura Metrics

Kimura investigated static spherically symmetric metrics and found several to have quadratic first integrals. We use REDUCE and the package Dimsym to seek collineations for these metrics. For one metric we find that three proper projective collineations exist, two of which are associated with the two irreducible quadratic first integrals found by Kimura. The third projective collineation is found to have a reducible quadratic first integral. We also find that this metric admits two conformal motions and that the resulting reducible conformal Killing tensors also lead to Kimura's quadratic integrals. We demonstrate that when a Killing tensor is known for a metric we can seek an associated collineation by solving first order equations that give the Killing tensor in terms of the collineation rather than the second order determining equations for collineations. We report less interesting results for other Kimura metrics.     

Vaidya spacetime in the diagonal coordinates

We have analyzed the transformation from initial coordinates (v, r) of the Vaidya metric with light coordinate v to the most physical diagonal coordinates (t, r). An exact solution has been obtained for the corresponding metric tensor in the case of a linear dependence of the mass function of the Vaidya metric on light coordinate v. In the diagonal coordinates, a narrow region (with a width proportional to the mass growth rate of a black hole) has been detected near the visibility horizon of the Vaidya accreting black hole, in which the metric differs qualitatively from the Schwarzschild metric and cannot be represented as a small perturbation. It has been shown that, in this case, a single set of diagonal coordinates (t, r) is insufficient to cover the entire range of initial coordinates (v, r) outside the visibility horizon; at least three sets of diagonal coordinates are required, the domains of which are separated by singular surfaces on which the metric components have singularities (either g ₀₀ = 0 or g ₀₀ = ∞). The energy–momentum tensor diverges on these surfaces; however, the tidal forces turn out to be finite, which follows from an analysis of the deviation equations for geodesics. Therefore, these singular surfaces are exclusively coordinate singularities that can be referred to as false fire-walls because there are no physical singularities on them. We have also considered the transformation from the initial coordinates to other diagonal coordinates (η, y), in which the solution is obtained in explicit form, and there is no energy–momentum tensor divergence.     

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：     

Rigidity in vacuum under conformal symmetry

Motivated in part by Eardley et al. (Commun Math Phys 106(1):137–158, 1986), in this note we obtain a rigidity result for globally hyperbolic vacuum spacetimes in arbitrary dimension that admit a timelike conformal Killing vector field. Specifically, we show that if M is a Ricci flat, timelike geodesically complete spacetime with compact Cauchy surfaces that admits a timelike conformal Killing field X, then M must split as a metric product, and X must be Killing. This gives a partial proof of the Bartnik splitting conjecture in the vacuum setting.     

On the role of the Killing tensor in the Einstein-Maxwell theory

It is shown that when a four dimensional source-free Einstein-Maxwell space-time admits a group of motions leaving the electromagnetic field unchanged a linear relation exists between two Maxwell fields and the covariant derivative of a Killing vector. For the case in which the two electromagnetic fields are related by a duality rotation it is seen that a purely geometric form of Einstein's equations may be derived. The behaviour of these under a class of quasi conformal transformations of the metric is shown to lead to Harrison's theorem.     

Twistor Spinors with Zero on Lorentzian 5-Space

We present in this paper a C ¹-metric of Lorentzian signature (1,4) on an open neighbourhood of the origin in \(\mathbb{R}^{5}\), which admits a solution to the twistor equation for spinors with a unique isolated zero at the origin. The metric is not conformally flat in any neighbourhood of the origin and the associated conformal Killing vector to the twistor generates a one-parameter group of essential conformal transformations. The construction is based on the Eguchi-Hanson metric in dimension 4.     

Inheriting geodesic flows

We investigate the propagation equations for the expansion, vorticity and shear for perfect fluid space-times which are geodesic. It is assumed that space-time admits a conformal Killing vector which is inheriting so that fluid flow lines are mapped conformally. Simple constraints on the electric and magnetic parts of the Weyl tensor are found for conformal symmetry. For homothetic vectors the vorticity and shear are free; they vanish for nonhomothetic vectors. We prove a conjecture for conformal symmetries in the special case of inheriting geodesic flows: there exist no proper conformal Killing vectors (ψ _;ab ≠ 0) for perfect fluids except for Robertson-Walker space-times. For a nonhomothetic vector field the propagation of the quantity ln (R _ab u ^a u ^b ) along the integral curves of the symmetry vector is homogeneous.     

Isotropic coordinates and Schwarzschild metric

Written in terms of isotropic coordinatesr, t, the Schwarzschild metric as usually given is static, i.e., admits a timelike Killing vector for all values ofr andt. Therefore the region within the event horizon cannot be accounted for. This deficiency is remedied here, by finding the general spherically symmetric vacuum metric in isotropic coordinates.     

On ellipsoidal solutions of Liouville's equation in general relativity

A relativistic, collisionless gas of gravitating particles all having the same proper mass (possibly equal to zero) is studied under the assumption that the oneparticle distribution function is locally ellipsoidal in momentum space with respect to some timelike vector field (observer). Liouville's equation implies that the distribution function depends only on a quadratic form in the 4- momenta, whose coefficients are a Killing tensor in the case of non- vanishing proper mass, and a conformal Killing tensor in the case of vanishing rest mass of the particles. It is suggested that cosmological models of Bianchi-type I can be described in terms of ellipsoidal momentum distribution functions whose ellipsoidal tensor is built out of the Killing vectors associated with the spatial homogeneity.