Anisotropic Homogeneous Cosmologies in the Post-Newtonian Approximation

In this paper we explore how far the post-Newtonian theory, [9] goes in overcoming the difficulties associated with anisotropic homogeneous cosmologies in the Newtonian approximation. It will be shown that, unlike in the Newtonian case, the cosmological equations of the post-Newtonian approximation are much more in the spirit of general relativity with regard to the nine Bianchi types and issues of singularities.The situations of vanishing rotation and vanishing shear are treated separately. The homogeneous Bianchi I model is considered as an example of a rotation-free cosmology with anisotropy. It is found in the Newtonian approximation that there are arbitrary functions that need to be given for all time if the initial value problem is to be well-posed, while in the post-Newtonian case there is no such need. For the general case of a perfect fluid only the post-Newtonian theory can satisfactorily describe the effects of pressure. This is in accordance with findings in [7] where the post-Newtonian approximation was applied to homogeneous cosmologies.For a shear-free anisotropic homogeneous cosmology the Newtonian theory of Heckmann and Schücking, [2] is explored. Comparisons with its relativistic and post-Newtonian counterparts are made. In the Newtonian theory solutions exist to which there are no analogues in general relativity. The post-Newtonian approximation may provide a way out.     

Shear free solutions in general relativity theory

The Goldberg-Sachs theorem is an exact result on shear-free null geodesics in a vacuum spacetime. It is compared and contrasted with an exact result for pressure-free matter: shear-free flows cannot both expand and rotate. In both cases, the shear-free condition restricts the way distant matter can influence the local gravitational field. This leads to intriguing discontinuities in the relation of the General Relativity solutions to Newtonian solutions in the timelike case, and of the full theory to the linearised theory in the null case.     

Isotropic Singularities in Shear-free Perfect Fluid Cosmologies

We investigate barotropic perfect fluid cosmologies which admit an isotropic singularity. From the General Vorticity Result of Scott, it is known that these cosmologies must be irrotational. In this paper we prove, using two different methods, that if we make the additional assumption that the perfect fluid is shear-free, then the fluid flow must be geodesic. This then implies that the only shear-free, barotropic, perfect fluid cosmologies which admit an isotropic singularity are the FRW models.     

Spherically symmetric perfect fluid solutions of Einstein's equations in noncomoving coordinates

Almost all known spherically symmetric perfect fluid solutions of Einstein's equations have been obtained in comoving coordinates and nearly all are shear-free. In this paper we study two solutions in noncomoving coordinates and show that they contain shear.     

The GHP II Package with Applications

We present an advanced version of the Maple package GHP called GHPII. In it we provide a number of additional sophisticated tools to assist with problems formulated in the Geroch-Held-Penrose (ghp) formalism. The first part of this article discusses these new tools while in the second part we shall apply the ghp formalism, using the GHPII routines, to vacuum Petrov type D spacetimes and shear-free perfect fluids. We prove that for all shear-free perfect fluids with a barotropic equation of state, where two of the principal null directions are coplanar with the fluid four-velocity and vorticity then either the expansion or vorticity of the fluid must be zero.     

Perfect fluid spheres admitting flat 3-dimensional slices

We give the general spherically symmetric perfect fluid or dust solution with flat three-spaces orthogonal to the four-velocity. The case of dust corresponds to a subset of the Tolman solution. The shear-free case corresponds to the Friedmann line element.     

Observationally homogeneous shear-free perfect fluids

It has been conjectured that, in general relativity, shear-free perfect fluids which obey any reasonable barotropic equation of state are necessarily either non-expanding or nonrotating. We prove that this is valid in the restricted case when the fluid's expansion and energy density are assumed to be functionally dependent. In a cosmological context, this condition of functional dependence is of interest, because it is closely related to a recently proposed criterion of observational (spatial) homogeneity, which has been enunciated in the Postulate of Uniform Thermal Histories (indeed, the two are equivalent when the fluid's expansion is nonzero). Our result on shear-free fluids may be readily specialized to the case of hypersurface-homogeneous spacetimes, and in particular to that of spatially homogeneous cosmological models. We briefly examine all subcases in which the fluid's expansion is nonzero and focus attention on the one-parameter family of solutions which are not hypersurface-homogeneous.     

Static observers in curved spaces and non-inertial frames in Minkowski spacetime

Static observers in curved spacetimes may interpret their proper acceleration as the opposite of a local gravitational field (in the Newtonian sense). Based on this interpretation and motivated by the equivalence principle, we are led to investigate congruences of timelike curves in Minkowski spacetime whose acceleration field coincides with the acceleration field of static observers of curved spaces. The congruences give rise to non-inertial frames that are examined. Specifically, we find, based on the locality principle, the embedding of simultaneity hypersurfaces adapted to the non-inertial frame in an explicit form for arbitrary acceleration fields. We also determine, from the Einstein equations, a covariant field equation that regulates the behavior of the proper acceleration of static observers in curved spacetimes. It corresponds to an exact relativistic version of the Newtonian gravitational field equation. In the specific case in which the level surfaces of the norm of the acceleration field of the static observers are maximally symmetric two-dimensional spaces, the energy?Cmomentum tensor of the source is analyzed.     

How unique are the Friedmann-Robertson-Walker models of the universe

By accepting the validity of certain conjectures in classical general relativity and kinetic theory, it is argued that, in a sense, the spatially homogeneous and isotropic Friedmann-Robertson-Walker (FRW) cosmological models are unique. This is accomplished in two steps. First, there is reason to believe that kinetic theory requires perfect fluids to be shear-free. Second, it seems that general relativity constrains expanding shear-free fluids to be irrotational. The uniqueness of the FRW models then follows, since it has already been established that they are the only space-times which represent an expanding shear-free irrotational perfect fluid that are physically reasonable on a global scale.This essay received an honorable mention (1986) from the Gravity Research Foundation—Ed.     

Faraday instability with a polymer solution

We present an experimental study of the Faraday instability in which we compare the behavior of a Newtonian fluid (water-glycerine mixture) with that of a semi-dilute non-Newtonian solution of high molecular weight polymer. We show that although the dispersion relation of surface waves, derived for a layer of inviscid fluid, remains valid in that particular non-Newtonian case, the behavior of the instability threshold with frequency strongly differs from the Newtonian case. We explain this effect as a result of a frequency-dependent viscosity. The linear stability analysis of the non-Newtonian case shows a perfect agreement with the experimental results both for the dispersion relation and for the reduction of the instability threshold. We discuss the use of the characteristics of the Faraday experiment as a measurement tool to determine frequency dependent properties of non-Newtonian fluids. Received 5 January 1999     

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.     

分析力学中加速度的去魅

利用经典力学的拉格朗日方法,分别讨论了静平衡的条件和连续介质动力学.利用哈密顿方法,介绍了相空间中独特的平衡点以及适用于统计力学的稳定系综分布.这些例子表明:在分析力学的框架内,加速度概念已经去魅,所谓的“平衡态”也具有不同于牛顿方法的实现方式.     

The Formation of Non-Keplerian Rings of Matter About Compact Stars

The formation of energetic rings of matter in a Kerr spacetime with an outward pointing acceleration field does not appear to have previously been noted as a relativistic effect. In this paper we show that such rings are a gravimagneto effect with no Newtonian analog, and that they do not occur in the static limit. The energy efficiency of these rings can (depending of the strength of the acceleration field) be much greater than that of Keplerian disks. Counter-intuitively these rings rotate in a direction opposite to that of compact star about which they form. The size and energy efficiency of the rings depend on the fundamental parameters of the spacetime as well as the strength the acceleration field.     

Vorticity- and shear-free space-time in general relativity

The paper studies the vorticity- and shear-free nonstatic space-times with perfect fluid source in genera] relativity and finds that such space-times are either spherically symmetric or pseudospherical or plane symmetric.     

关于矢量-张量引力相互作用模型

本文从粒子运动方程出发证明了矢量-张量引力模型中反引力的存在,这种反引力如果以长程力的形式出现,将使得粒子的运动偏离短程线。在牛顿近似和弱场近似下,相应于矢量场的反引力与相应于张量场的吸引力互相抵消,使得检验粒子在球对称静态引力场中的加速度为零,这显然与牛顿万有引力现象矛盾。 关键词：     

On perturbations of solutions of the Einstein-Maxwell equations

By using the expressions for the solutions of the Einstein-Maxwell equations in terms of potentials, valid in the case where the spacetime admits a shear-free geodesic null congruence and the electromagnetic field is aligned to it, we show that a pair of complex potentials generates simultaneous perturbations of the gravitational and the electromagnetic fields. We also show that if the background electromagnetic field is null, then the pair of complex potentials is determined by a pair of coupled, linear, second-order differential equations.     

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.     

Shear-free Perfect Fluids in General Relativity: Gravito-magnetic Spacetimes

We investigate shear-free, perfect fluid solutions of Einstein's field equations in which the perfect fluid satisfies a barotropic equation of state p = p(w) such that w + p 0. We find that if the electric part of the Weyl tensor (with respect to the fluid flow) vanishes and the spacetime is not conformally flat then the fluid volume expansion is zero but the vorticity is necessarily nonzero. In addition, we show that if p = –w/3 then necessarily either the fluid expansion is zero or the fluid vorticity is zero.     

General relativity in Newtonian form

It is shown how the use of coordinates where time is measured with clocks moving radially in a spherically symmetric gravitational field leads to general relativistic dynamical expressions that are exactly identical to corresponding expressions in Newtonian theory. The general formalism is developed for the case where the stress-energy tensor is that of a perfect fluid. Expressions like the Newtonian inverse square gravitational law, the Newtonian equation of continuity for fluid flow, Newtonian conservation of energy, etc., follow quite naturally from the fully-fledged exact general relativistic equations. Specific examples involving cosmology and gravitational collapse are given.     

An evolution of adiabatic matter: a case for the quasistatic regime

We establish the connection between the standard ADM 3+1 treatment of matter with its characteristic equivalent, in the context of spherical symmetry. The flux-conservative rendition of the fluid equations are obtained. Considering adiabatic distributions of perfect fluid, we evolve the system using the so-called post-quasi-static approximation in radiation coordinates. We obtain an adiabatic matter evolution in the quasi-static regime or slow motion, which is not shear-free nor geodesic.