Isotropic singularities and isotropization in a class of Bianchi type-VI h cosmologies

The evolution of a class of exact spatially homogeneous cosmological models of Bianchi type VI _h is discussed. It is known that solutions of type VI _h cannot approach isotropy asymptotically at large times. Indeed the present class of solutions become asymptotic to an anisotropic vacuum plane wave solution. Nevertheless, for these solutions the initial anisotropy can decay, leading to a stage of finite duration in which the model is close to isotropy. Depending on the choice of parameters in the solution, this quasi-isotropic stage can commence at the initial singularity, in which case the singularity is of the type known as isotropic or Friedmann-like. The existence of this quasi-isotropic stage implies that these models can be compatible in principle with the observed universe.     

Inhomogeneous mixmaster universes: Some exact solutions

Algorithms for generating new exact solutions of the Einstein-Klein-Gordon field equations, which describe inhomogeneous universes with S³ topology of spatial sections, are developed. The known exact vacuum and stiff-fluid solutions with S³ topology are used as an input. The methods developed are further applied to derive inhomogenous generalizations of Bianchi type IX solutions and inhomogeneous S³ Gowdy models with gravitational and scalar waves. It is shown that the new solutions, which are generalizations of the Bianchi type IX models, permit identification of the scalar field with the velocity potential of the stiff irrotational fluid. The latter result is further used to study the growth rate of density perturbations of the isotropic and anisotropic Bianchi type IX universes in a fully nonlinear relativistic regime. The role of anisotropy of the rate of growth of density perturbations is studied in detail.     

On Bianchi type-I vacuum solutions inR+R 2 theories of gravitation. II. The axially symmetric anisotropic case

The field equations following from a LagrangianL(1/L)(–g)^1/2[(1/2)R+l ²(R _lk R ^lk +R ²)] will be considered for Bianchi type-I homogeneous models. Thereby the special case,+3=0, is considered qualitatively for axially symmetric anisotropic metrics. Generically, the solutions have both past and future singularities, but it will be proven by topological arguments that the two-dimensional space of solutions possesses a one-parameter subspace of solutions with a behavior similar to the Kasner solution.     

Bianchi VI cosmological models representing perfect fluid and radiation with electric-type free gravitational fields

A homogeneous Bianchi type VI_h cosmological model filled with perfect fluid, null electromagnetic field and streaming neutrinos is obtained for which the free gravitational field is of the electric type. The barotropic equation of statep = (–1) is imposed in the particular case of Bianchi VI₀ string models. Various physical and kinematical properties of the models are discussed.     

Letter: On Tilted Perfect Fluid Bianchi Type VI0 Self-Similar Models

We show that the tilted perfect fluid Bianchi VI₀ family of self-similar models found by Rosquist and Jantzen [K. Rosquist and R. T. Jantzen, Exact power law solutions of the Einstein equations, 1985 Phys. Lett. 107A 29–32] is the most general class of tilted self-similar models but the state parameter lies in the interval (6/5, 3/2). The model has a four dimensional stable manifold indicating the possibility that it may be future attractor, at least for the subclass of tilted Bianchi VI₀ models satisfying n =0 in which it belongs. In addition the angle of tilt is asymptotically significant at late times suggesting that for the above subclasses of models the tilt is asymptotically extreme.     

Curvature tensors unified field equations on SEXn

We study the curvature tensors and field equations in then-dimensional SE manifold SEX_n. We obtain several basic properties of the vectorsS andU and then of the SE curvature tensor and its contractions, such as a generalized Ricci identity, a generalized Bianchi identity, and two variations of the Bianchi identity satisfied by the SE Einstein tensor. Finally, a system of field equations is discussed in SEX_n and one of its particular solutions is constructed and displayed.     

Ψ=We ±Φ Quantum cosmological solutions for class A Bianchi models

We find exact solutions to the Wheeler-DeWitt equation, for a certain factor ordering. They have the form =We ^± for class A Bianchi models, where is a solution to the classical Hamilton-Jacobi equation, generalizing the only known solution of Moncrief and Ryan for the Bianchi type IX model in standard quantum cosmology. The same kind of solution has also been found in supersymmetric quantum cosmology.     

Bianchi Type I Viscous Universe with Variable G and Λ

Exact solutions for an anisotropic Bianchi type I model with bulk viscosity and variable G and are obtained. We have found some solutions that correspond to our earlier work for the isotropic one. Unlike Kalligas et al., an inflationary solution with a variable energy density has been found where the anisotropy energy decreases exponentially with time. There is a period of hyper-inflation during which the energy density remains constant.     

Bianchi identities,electromagnetic waves,and charge conservation in theP(4) theory of gravitation and electromagnetism

The Bianchi identities for theP(4)=O(1, 3) ^4* theory of gravitation and electromagnetism are decomposed into the standardO(1, 3) Riemannian Bianchi identity plus an additional ^4* component. When combined with the Einstein-Maxwell affine field equations the ^4* components of theP(4) Bianchi identities imply conservation of magnetic charge and the wave equation for the Maxwell field strength tensor. These results are analyzed in light of the special geometrical postulates of theP(4) theory. We show that our development is the analog of the manner in which the Riemannian Bianchi identities, when combined with Einstein's field equations, imply conservation of stress-energy-momentum and the wave equation for the LanczosH-tensor.     

Ricci collineations of the Bianchi type II,VIII, and IX space-times

Ricci and contracted Ricci collineations of the Bianchi type II, VIII, and IX space-times, associated with the vector fields of the form (i) one component of ^a (x ^b ) is different from zero and (ii) two components of ^a (x ^b ) are different from zero, fora, b=1, 2, 3, 4, are presented. In subcase (i.b), which is ^a = (0, ¹(x ^a ),0,0), some known solutions are found, and in subcase (i.d), which is ^a =(0, 0, 0, ⁴(x ^a )), choosingS(t) = const.×R(t), the Bianchi type II, VIII, and IX spacetime is reduced to the Robertson-Walker metric.     

The affine geometry of the Lanczos H-tensor formalism

We identify the fiber-bundle-with-connection structure that underlies the Lanczos H-tensor formulation of Riemannian geometrical structure. We consider linear connections to be type (1,2) affine tensor fields, and we sketch the structure of the appropriate fiber bundle that is needed to describe the differential geometry of such affine tensors, namely the affine frame bundleA ₁ ² M with structure groupA ₁ ² (4) =GL(4) T ₁ ^{2 4} over spacetimeM. Generalized affine connections on this bundle are in 1-1 correspondence with pairs(, K) onM, where thegl(4)-component denotes a linear connection and the T ₁ ^{2 4}-componentK is a type (1,3) tensor field onM. We show that the Lanczos H-tensor arises from a gauge fixing condition on this geometrical structure. The resulting translation gauge, theLanczos gauge, is invariant under the transformations found earlier by Lanczos. The other Lanczos variablesQ _mandq are constructed in terms of the translational component of the generalized affine connection in the Lanczos gauge. To complete the geometric reformulation we reconstruct the Lanczos Lagrangian completely in terms of affine invariant quantities. The essential field equations derived from ourA ₁ ² (4)-invariant Lagrangian are the Bianchi and Bach-Lanczos identities for four-dimensional Riemannian geometry.     

Isotropization in Bianchi type IX vacuum cosmology

We find the most general Bianchi type IX solution in Brans-Dicke theory (BDT) for the vacuum case, with the local rotational symmetry. For BDT coupling parameterw > 500 the universe becomes isotropic for any amount of initial anisotropy. In the extended inflation scenario, the Brans-Dicke scalar field can avoid the inflation in one direction.     

Unification of electromagnetism and gravitation: The equations of electrodynamics derived from the Riemann tensor in general relativity

The equations of free-space electrodynamics are derived directly from the Riemann curvature tensor and the Bianchi identity of general relativity by contracting on two indices to give a novel antisymmetric Ricci tensor. Within a factore/h, this is the field-strength tensor G of free-space electrodynamics. The Bianchi identity for G describes free-space electrodynamics in a manner analogous to, but more general than, Maxwell's equations for electrodynamics, the critical difference being the existence in general and special relativity of the Evans-Vigier fieldB ⁽³⁾.     

Bianchi-I Cosmologies with Electromagnetic and Spinor Fields. Isotropization Problem

We consider Bianchi type I cosmologies with unidirectional magnetic and electric fields, assuming as well the existence of a global spinor field ψ(t) as one more possible source of gravity able to suppress the inevitable anisotropy accompanying a nonzero vector field. The field ψ(t) is assumed to contain a nonlinearity in the form s ⁿ , where and n=const (the special case n=1 corresponds to a Dirac massive field). The structure of the stress-energy tensor of the spinor field is shown to be the same as that of a perfect fluid with the equation of state p=w ρ where w=n−1. The Dirac massive spinor field and nonlinear fields with n<4/3 are shown to be able to provide isotropization. A numerical estimate shows that this isotropization could occur early enough to be compatible with observations.     

Isotropic Evolution of a JBD Anisotropic Bianchi Universe

I study the dynamical effects due to the Brans-Dicke scalar -field at the early stages of a supposedly anisotropic Universe expansion in the scalar-tensor cosmology of Jordan-Brans-Dicke. This is done considering the behaviour of the general solutions for the homogeneous model of Bianchi type VII in the vacuum case. I conclude that the Bianchi-VII₀ model shows an isotropic expansion and that its only physical solution is equivalent to a Friedman-Robertson-Walker spacetime whose evolution can, depending on the value of the JBD coupling constant, begin in a singularity and, after expanding (inflating, if > 0), shrink to another, or starting in a non-singular state, collapse to a singularity. I also conclude that the general Bianchi-VII _h (with h 0) models show strong curvature singularities producing a complete collapse of the homogeneity surfaces to 2D-manifolds, to 1D-manifolds or to single points. Our analysis depends crucially on the introduction of the so-called intrinsic time, , as the product of the JBD scalar field times a mean scale factor a ³ = a ₁ a ₂ a ₃, in which the finite-cosmological-time evolution of this universe unfolds into an infinite -range. These universes isotropize from an anisotropic initial state, thence I conclude that they are stable against anisotropic perturbations.     

Weyl equation in some anisotropic stiff fluid universes

The Weyl equation (massless Dirac equation) is studied in a family of exact solutions of the Einstein equations whose material content is a perfect fluid with stiff equation of state (p=) and which are of Bianchi type I. The field equation is solved exactly for some members of the family.     

Homogeneous anisotropic cosmological models with viscous fluid and magnetic field

The paper presents some exact solutions of Bianchi types I and III and Kantowski-Sachs cosmological models consisting of a dissipative fluid along with an axial magnetic field. A barytropic equation of state (p=), together with a pair of linear relations between the matter density (), the shear scalar (), and the expansion scalar () have been assumed for simplicity. The solutions are basically of two different types, one for the Bianchi I and the other for III and Kantowski-Sachs type. The presence of the magnetic field, however, does not change the fundamental nature of the initial singularity.     

Bianchi type I cosmological models with variableG and A

Einstein's equations with variableG and A scalars are considered for a Bianchi type I metric. A solution is found in which the cosmological term varies inversely with the square of time. As in the case of the flat Friedmann-Lemaître-Robertson-Walker (flrw) models discussed recently, there is then no dimensional constant associated with A. However, it is shown that the time behaviour of Bianchi type I inflationary solutions cannot be of the pure de Sitter type. This shows that if the flat FLRW inflationary solutions previously considered are perturbed by the introduction of Bianchi type I anisotropy, then the time evolution may be perturbed from the pure exponential form.     

Frequency doubling of laser light in an isotropic medium with electrically destroyed centre of inversion

The non-zero non-linear susceptibility tensor components ² (E ⁰)which account for frequency doubling of laser light in a naturally isotropic medium immersed in a DC efectric field (E ⁰) are discussed. The conditions for extremal second harmonic generation (SHG) are derived, which depend onE ⁰ and on the incident laser intensity but primarily on the microstructure of the medium. With growingE ⁰, the susceptibilities ² (E ⁰) increase for cigar-like microsystems but decrease for disc-shaped ones, according to whether the induced dipole helps or hinders the permanent dipole in reorienting the microsystem along (E ⁰). Non-linear electronic distortion alone is insufficient for explaining the anomalous experimental results.Generally, upon the electrically induced anisotropy, an anisotropy self-induced by the laser beam is superimposed, with optical axis along the propagation direction if the beam is circularly polarised or unpolarised, or along the oscillation direction of the light vector if it is linearly polarised.Extremal anisotropy of the medium occurs at saturation of electric or optical reorientation. This is experimentally inachievable in molecular substances even with very strong fields but is easy to obtain in solutions of macromolecules or colloid particles, where yet other opto-electronic processes intervene significantly.     

Nonscalar singularities in spatially homogeneous cosmologies

Singularities in vacuum spatially homogeneous cosmological models are investigated. It is shown that in general the curvature scalarR _* ^abcd R^*abcddiverge and that the only solutions which have curvature singularities at which this scalar does not diverge describe certain plane-wave space-times. It is argued that with matter present these nonscalar singularities are even less likely to occur. The exceptional case of Bianchi type VI_–1/9 is not considered.