Review: Aspects of Solution-generating Techniques for Space-times with Two Commuting Killing Vectors

Certain aspects of solution-generatingtechniques for spacetimes with two commuting Killingvectors are reviewed. A brief historical introduction tostationary axisymmetric systems is given. The importance of the Homogeneous Hilbert problem associatedwith the equations, unifying the group-theoretic withthe soliton-theoretic approaches, is emphasized. Theformalism of generating functions is introduced, both for vacuum and electrovacuum.Sibgatullin's technique for electrovacuum solutions isrelated to the Hauser Ernst variables and a method byErnst is briefly discussed. The solitonic methods ofBelinsky-Zakharov and Alekseev are reviewed. Their relation isemphasized by an explicit proof, at the level ofgenerating techniques, that the BZ two soliton with twocomplex conjugate poles is isomorphic to the Alekseev one-soliton (restricted to vacuum) with trivialgauge. The Alekseev non-soliton technique is discussed.Some recent developments are brieflydiscussed.     

Hypersurface-Orthogonal Generators of an Orthogonally Transitive G2I, Topological Identifications, and Axially and Cylindrically Symmetric Spacetimes

A criterion given by Castejón-Amenedo and MacCallum for the existence of (locally) hypersurface-orthogonal generators of an orthogonallytransitive two-parameter Abelian group of motions (a G₂I) in spacetime is re-expressed as a test for linear dependence with constant coefficients between the three components of the metric in the orbits in canonical coordinates. In general, it is shown that such a relation implies that the metric is locally diagonalizable in canonical coordinates, or has a null Killing vector, or can locally be written in a generalized form of the windmill solutions characterized by McIntosh. If the orbits of the G₂I have cylindrical or toroidal topology and a periodic coordinate is used, these metric forms cannot in general be realized globally as they would conflict with the topological identification. The geometry then has additional essential parameters, which specify the topological identification. The physical significance of these parameters is shown by their appearance in global holonomy and by examples of exterior solutions where they have been related to characteristics of physical sources. These results lead to some remarks about the definition of cylindrical symmetry.     

Lax Pair Tensors and Integrable Spacetimes

The use of Lax pair tensors as a unifying framework for Killing tensors of arbitrary rank is discussed. Some properties of the tensorial Lax pair formulation are stated. A mechanical system with a well-known Lax representation—the three-particle open Toda lattice—is geometrized by a suitable canonical transformation. In this way the Toda lattice is realized as the geodesic system of a certain Riemannian geometry. By using different canonical transformations we obtain two inequivalent geometries which both represent the original system. Adding a timelike dimension gives four-dimensional spacetimes which admit two Killing vector fields and are completely integrable.     

String Cosmological Models in Five-Dimensional Spacetimes

We present some classes of solutions for dust matter coupled to the string cloud in five-dimensional Kaluza-Klein spacetimes. The solutions have one or two distinct singularities depending upon the sign of the constant of integration. Some of the classes of models exhibit inflation in the initial stage. The behaviour of associated parameters has been discussed in detail.     

Unbounded Entropy in Spacetimes with Positive Cosmological Constant

In theories of gravity with a positive cosmological constant, we consider product solutions with flux, of the form (A)dS _p ×S ^q . Most solutions are shown to be perturbatively unstable, including all uncharged dS _p ×S ^q spacetimes. For dimensions greater than four, the stable class includes universes whose entropy exceeds that of de Sitter space, in violation of the conjectured N-bound. Hence, if quantum gravity theories with finite-dimensional Hilbert space exist, the specification of a positive cosmological constant will not suffice to characterize the class of spacetimes they describe.     

An Anisotropic Cosmological Model in Self-creation Cosmology

The paper presents an exact solution of spatially homogeneous and anisotropic cylindrically symmetric cosmological model in Barber’s second self-creation theory of gravitation in the presence of perfect fluid with pressure equal to energy density. Some physical properties of this model are also discussed.     

Some Anisotropic Homogeneous Cosmological Models in Self-creation Cosmology

The paper presents an exact solution of cylindrically symmetric cosmological models which are of Petrov type-I or Petov type-D in Barber’s second self-creation theory of gravitation. Some physical and geometrical properties of these models are also discussed.     

Phantom Appearance of Non-phantom Matter in Anisotropic Cosmological Model

Two cosmological models with non-phantom matter having the same expansion of the universe as phantom cosmologies are constructed in anisotropic Bianchi type-V universe. The exact solutions to the corresponding Einstein field equations have been obtained. The statefinder diagnostic pair i.e. {r,s} parameters have been obtained for disordered radiation i.e. γ=1/3. We have also discussed the well-known astrophysical phenomena, namely the look-back time, luminosity distance and event horizon with redshift.     

Bianchi Type-III Cosmic Strings and Domain Walls Cosmological Model in General Relativity

Bianchi type-III space time is considered in the presence of cosmic strings and thick domain walls source in the frame work of general relativity. Exact cosmological models using various cases of ρ=α λ and p=γ ρ are presented. It is observed that the behavior of these models (with cosmic strings and domain walls), based on their physical and kinematical properties, is found to be identical.     

Anisotropic Bulk Viscous Fluid Cosmological Model with Zero-Rest-Mass Scalar Field and Time-Dependent Cosmological Term

In this paper, we have investigated Bianchi type-III cosmological model in the presence of a bulk viscous fluid together with zero-rest-mass scalar field and time-dependent cosmological term. We have shown that the field equations are solvable for any arbitrary cosmic scale function. Exact solutions of Einstein’s field equations are obtained which represent an expanding, shearing, non-rotating and decelerating model of the universe. Some physical and geometrical behaviours of the cosmological model are discussed.     

Anisotropic Dark Energy Bianchi Type-II Cosmological Models in Lyra Geometry

The spatially homogeneous and totally anisotropic Bianchi type-II cosmological model has been discussed in general relativity in the presence of a hypothetical anisotropic dark energy fluid with constant deceleration parameter within the frame work of Lyra’s manifold with uniform and time varying displacement field vector. With the help of special law of variation for Hubble’s parameter proposed by Bermann (Nuovo Cimento 74B:182, 1983) a dark energy cosmological model is obtained in this theory. We use the power law relation between average Hubble parameter H and average scale factor R to find the solution. The assumption of constant deceleration parameter leads to two models of universe, i.e. power law model and exponential model. Some physical and kinematical properties of the model are also discussed.     

An Anisotropic Homogeneous Cosmological Model 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 spatially homogeneous and anisotropic cosmological model corresponding to Bianchi type-I solution. The physical and geometrical properties of this model has been discussed. Finally this model has been transformed to the original form (1961) of Brans-Dicke theory (including a variable cosmological term).     

The Role of Magnetic Field in Anisotropic String Cosmological Model

In this paper we consider a spatially homogenous and anisotropic Bianchi type-V space-time model to investigate the effects of a magnetic field in string cosmology. We assume that the string’s direction and magnetic field are along x-axis. The field equations are solved by using the equation of state for a cloud of strings and variable magnetic permeability. We derive exact solutions for three types of strings: (i) Nambu strings, (ii) string model where the sum of energy density and string tension density is zero and (iii) Takabayasi strings. We examine the behaviour of scale factors and other physical parameters with and without magnetic field and it is found that the magnetic field effects the dynamics of the universe at early time. During late time the universe becomes isotropic even in the presence of magnetic field. The universe expands with decelerated rate during early stages of the evolution of the universe but it goes to marginal inflation at late times.     

Generalized Chaplygin Gas Dominated Anisotropic Bianchi Type-I Cosmological Models

We present Bianchi type-I cosmological models in the presence of generalized Chaplygin gas and perfect fluid for early and late time epochs. Exact solutions of Einstein’s field equations for this model are obtained. The general solutions of gravitational field equations are expressed in an exact parametric form, with average scale factor as parameter. In the limiting cases of small and large values of the average scale factor, the solutions of the field equations are expressed in exact analytic forms. Moreover, this model predicts that the expansion of Universe is accelerating for the late times. The physical and geometrical properties of the corresponding cosmological models are discussed.     

Anisotropic Cosmological Models with Perfect Fluid and Dark Energy Reexamined

We consider a self consistent system of Bianchi type-I (BI) gravitational field and a binary mixture of perfect fluid and dark energy. The perfect fluid is taken to be the one obeying the usual equation of state, i.e., p = ξε, with ζ∉[0, 1] whereas, the dark energy is considered to be obeying a quintessence-like equation of state. The modification of the ordinary quintessence lies in the fact that its pressure becomes positive if the (dark) energy density exceeds some critical value. Exact solutions to the corresponding Einstein equations are obtained. The model in consideration gives rise to a Universe which is spatially finite. Depending on the choice of problem parameters the Universe is either close with a space-time singularity, or an open one which is oscillatory, regular and infinite in time. PACS numbers: 04.20.Ha, 03.65.Pm, 04.20.Jb     

Troubles with Quantum Anisotropic Cosmological Models: Loss of Unitarity

The anisotropic Bianchi I cosmological model coupled with perfect fluid is quantized in the minisuperspace. The perfect fluid is described by using the Schutz formalism which allows to attribute dynamical degrees of freedom to matter. A Schrödinger-type equation is obtained where the matter variables play the role of time. However, the signature of the kinetic term is hyperbolic. This Schrödinger-like equation is solved and a wave packet is constructed. The norm of the resulting wave function comes out to be time dependent, indicating the loss of unitarity in this model. The loss of unitarity is due to the fact that the effective Hamiltonian is hermitian but not self-adjoint. The expectation value and the bohmian trajectories are evaluated leading to different cosmological scenarios, what is a consequence of the absence of a unitary quantum structure. The consistency of this quantum model is discussed as well as the generality of the absence of unitarity in anisotropic quantum models.     

Anisotropic Cosmological Models with Variable G and Decaying Vacuum Energy

We present Bianchi type-I cosmological models with a perfect fluid source and time-dependent gravitational and cosmological constants based on new exact solutions of Einstein’s equations. The perfect fluid is chosen to obey a barotropic equation of state. The models obtained represent a radiation dominated phase and a dust era. In some of the models the expansion changes from a decelerating phase to an accelerating one and these models asymptotically tend to the de Sitter universe. The paper is dedicated to late Prof. S.R. Roy, Ex-Head, Department of Mathematics, Banaras Hindu University, Varanasi, India.     

Anisotropic Viscous Fluid Cosmological Models from Deceleration to Acceleration in String Cosmology

In the present study, anisotropic locally rotationally symmetric (LRS) Bianchi type II models representing massive strings have been presented with variable mean deceleration parameter. The field equations in scalar-tensor theory with energy momentum tensor proposed by Letelier (Phys. Rev. D 28: 2414, 1983), have been solved for the bulk viscous fluid and perfect fluid under the following physically relevant assumptions: (a) scale factor varies with time as $a(t) = (\sinh(\alpha t))^{\frac{1}{n}}$ , (b) Expansion (θ) in the model is proportional to shear (σ), (c) p=γρ, where γ (0≤γ≤1) is a constant, (d) ξ(t)=ξ ₀ ρ ^η . It has been observed that the presented universe has a phase transition from the early decelerating phase to the accelerating phase at present epoch which is in good agreement with the recent astronomical observations. Moreover, some physical and geometric properties of the models have been discussed in detail.