Kasner Solution Research Articles

Chaos of Yang-Mills field in class A Bianchi spacetimes

Studying the Yang-Mills field and the gravitational field in class A Bianchi spacetimes, we find that chaotic behavior appears in the late phase (the asymptotic future). In this phase, the Yang-Mills field behaves as that in Minkowski spacetime, in which we can understand it by a potential picture, except for types VIII and IX. At the same time, in the initial phase (near the initial singularity), we numerically find that the behavior seems to approach the Kasner solution. However, we show that the Kasner circle is unstable and the Kasner solution is not an attractor. From an analysis of stability and numerical simulation, we find a Mixmaster-like behavior in Bianchi I spacetime. Although this result may provide a counterexample to the Belinskii, Khalatnikov, and Lifshitz (BKL) conjecture, we show that the BKL conjecture is still valid in Bianchi IX spacetime. We also analyze a multiplicative effect of two types of chaos, that is, chaos with the Yang-Mills field and that in vacuum Bianchi IX spacetime. Two types of chaos seem to coexist in the initial phase. However, the effect due to the Yang-Mills field is much smaller than that of the curvature term.

Intersection rules, dynamics and symmetries

We consider theories containing gravity, at most one dilaton and form field strengths. We show that the existence of particular BPS solutions of intersecting extremal closed branes select the theories, which upon dimensional reduction to three dimensions possess a simple simply laced Lie group symmetry G. Furthermore these theories can be fully reconstructed from the dynamics of such branes and of their openings. Amongst such theories are the effective actions of the bosonic sector of M-theory and of the bosonic string. The BPS intersecting brane solutions form representations of a subgroup of the group of Weyl reflections and outer automorphisms of the triple Kac-Moody extension G+++ of the G algebra, which cannot be embedded in the overextended Kac-Moody subalgebra G++ characterising the cosmological Kasner solutions.

Kasner–AdS spacetime and anisotropic brane-world cosmology

Anisotropic generalization of Randall and Sundrum brane-world model is considered. A new class of exact solutions for brane and bulk geometry is found; it is related to anisotropic Kasner solution. In view of this, the old question of isotropy of initial conditions in cosmology rises once again in the brane-world context.

The Asymptotic Regimes of Tilted Bianchi II Cosmologies

In this paper we give, for the first time, a complete description of the dynamics of tilted spatially homogeneous cosmologies of Bianchi type II. The source is assumed to be a perfect fluid with equation of state $p = (\gamma -1) \mu$, where $\gamma$ is a constant. We show that unless the perfect fluid is stiff, the tilt destabilizes the Kasner solutions, leading to a Mixmaster-like initial singularity, with the tilt being dynamically significant. At late times the tilt becomes dynamically negligible unless the equation of state parameter satisfies $\gamma > {10/7}$. We also find that the tilt does not destabilize the flat FL model, with the result that the presence of tilt increases the likelihood of intermediate isotropization.

The Bianchi type Iminisuperspace model

The minisuperspace model of a Bianchi type I universe with compactspatial sections is investigated. The classical solutions are broughtinto a form where the difference between compact and infinite spatialsections is manifest. One of the features of the compact case is that ithas a non-trivial moduli space. The solution space of the compact Bianchitype I universe is 10-dimensional, whereas the Kasner solutions onlyhave a one-dimensional solution space. We also include the classicalsolutions with a dustand a cosmological constant. Solutions to the Wheeler-DeWitt equationare obtained in the light of the tunnelling boundary proposal by Vilenkin.Backreaction effects from a simple scalar field are also investigated.

D-dimensionalp-brane cosmological models associated with a Lie algebra of the typeA m

We study a D-dimensional cosmological model on the manifold M=R×M0×R×Mn describing an evolution of n+1 Einstein factor spaces Mi in a theory with several dilatonic scalar fields and differential forms admitting an interpretation in terms of intersecting p-branes. The equations of motion of the model are reduced to the Euler-Lagrange equations for the so-called pseudo-Euclidean Toda-like system. Assuming that the characteristic vectors related to the configuration of p-branes and their couplings to the dilatonic scalar fields can be interpreted as the root vectors of a Lie algebra of the type Am≡sl(m+1C), we reduce the model to an open Toda chain, which is integrable by the customary methods. The resulting metric has the form of the Kasner solution. We single out the particular model describing the Friedmanlike evolution of the three-dimensional external factor space M0 (in the Einsteinian conformal gauge) and the contraction of the internal factor spaces M1,...,Mn.

Rapid asymmetric inflation and early cosmology in theories with sub-millimeter dimensions

It was recently pointed out that the fundamental Planck mass could be close to the TeV scale with the observed weakness of gravity at long distances being due the existence of new sub-millimeter spatial dimensions. In this picture the standard model fields are localized to a (3+1)-dimensional wall or “3-brane”. We show that in such theories there exist attractive models of inflation that occur while the size of the new dimensions are still small. We show that it is easy to produce the required number of efoldings, and further that the density perturbations δρ/ρ as measured by COBE can be easily reproduced, both in overall magnitude and in their approximately scale-invariant spectrum. In the minimal approach, the inflaton field is just the moduli describing the size of the internal dimensions, the role of the inflationary potential being played by the stabilizing potential of the internal space. We show that under quite general conditions, the inflationary era is followed by an epoch of contraction of our world on the brane, while the internal dimensions slowly expand to their stabilization radius. We find a set of exact solutions which describe this behavior, generalizing the well-known Kasner solutions. During this phase, the production of bulk gravitons remains suppressed. The period of contraction is terminated by the blue-shifting of Hawking radiation left on our wall at the end of the inflationary de Sitter phase. The temperature to which this is reheated is consistent with the normalcy bounds. We give a precise definition of the radion moduli problem.

Dualities versus singularities

We show that a subgroup of the modular group of M-theory compactified on a ten torus, implies the Lorentzian structure of the moduli space, that is usually associated with naive discussions of quantum cosmology based on the low energy Einstein action. This structure implies a natural division of the asymptotic domains of the moduli space into regions which can/cannot be mapped to Type II string theory or 11D Supergravity (SUGRA) with large radii. We call these the safe and unsafe domains. The safe domain is the interior of the future light cone in the moduli space while the unsafe domain contains the spacelike region and the past light cone. Within the safe domain, apparent cosmological singularities can be resolved by duality transformations and we briefly provide a physical picture of how this occurs. The unsafe domains represent true singularities where all field theoretic description of the physics breaks down. They violate the holographic principle. We argue that this structure provides a natural arrow of time for cosmology. All of the Kasner solutions, of the compactified SUGRA theory interpolate between the past and future light cones of the moduli space. We describe tentative generalizations of this analysis to moduli spaces with less SUSY.

Einstein and Brans–Dicke Frames in Multidimensional Cosmology

Inhomogeneous multidimensional cosmological models with a higher-dimensional space-time manifold \(M = \overline {M_0 }\) 0 i=1 n Mi (n ≥ 1) are in stigated under dimensional reduction to a D 0-dimensional effective non-minimally coupled σ-model which generalizes the familiar Brans–Dicke model. The general form of the Einstein frame representation of multidimensional solutions known in the Brans–Dicke frame is given with respect to cosmic synchronous time. As an example, the transformation is demonstrated explicitly for the generalized Kasner solutions where it is shown that solutions in the Einstein frame show no inflation of the external space although they can undergo deflation after the cosmic synchronous time inversion.

Kasner‐like, inflationary and steady‐state solutions in multi‐dimensional cosmology

AbstractMulti‐dimensional cosmological models with n(n > 1) Ricci‐flat spaces and a scalar field are discussed classically and with respect to canonical quantization. These models are integrable. Two classes of solutions are obtained. One class of solutions generalizes the Kasner solutions. For special additional conditions we find another solution describing extended inflation connected with a constant volume of the whole multidimensional space (steady‐state).

Semiclassical corrections to Kasner cosmologies

We consider the Einstein equations for a Bianchi type I geometry, modified by first-order semiclassical quantum corrections. Using reduction techniques developed by Parker and Simon, we simplify these equations, obtaining reduced forms containing only first and second derivatives. We then find analytical solutions for both the vacuum case and for the case of a perfect fluid with a stiff equation of state. In the vacuum case we find that the Kasner solution maintains the same form in both the classical and semiclassical regimes. In the matter-filled case we observe, however, that a qualitatively different behavior emerges in the semiclassical era. We comment on the nature of these differences.

Kasner solutions and the causality principle in the relativistic theory of gravitation

Kasner solutions and the causality principle in the relativistic theory of gravitation

Anisotropic solutions in the 5D space-time-mass theory

In this paper we present anisotropic families of cosmological solutions in the 5-dimensional space-time-mass theory of gravity. In particular, the 5D analogue of the Kasner solution of General Relativity is obtained. Some comments are given on the solutions.

Anisotropic, time-dependent solutions in maximally Gauss-Bonnet extended gravity

In an arbitrary number of dimensions, we find the full exact anisotropic, time-dependent, diagonal-metric solutions to maximally Gauss-Bonnet extended gravity theory. This class of theories, for which the lagrangian is an arbitrary linear combination of dimensionally extended Euler forms, is the most general gravitational theory in which the field equations contain no more than second derivatives of the metric. We show that the space-time exponentially approaches an asymptotic state of constant, anisotropic curvature and prove three theorems concerning two generic types of singularities. The first theorem gives conditions for the existence of Kasner-like curvature singularities. For these the metric diverges as t Pi where ∑ p i = 2 k max- 1 and k max is the highest power of the curvature in the lagrangian. Other critical point singularities can arise from the polynomial nature of the theory. The remaining theorems demonstrate that the generic solution is extendible at all of these other critical points and that the generic critical points occur at moments of external volume density of space-time. We give an explicit coordinate transformation which produces a smooth extension through the critical point. The space-time may therefore alternately expand and contract for many cycles before expanding forever or contracting to a singularity. Many particular cases are treated in detail including several power series solutions, the generalized Kasner solution to general relativity with or without cosmological constant, the perturbative solution for quadratic string gravity, and five-dimensional extended gravity.

Gravitational field of static and thick domain walls

The behavior of static and thick domain walls with a scalar field is studied. If there are asymptotically vacuum regions on both sides of a domain wall, their regions can be expressed by the static Kasner solutions and the relation between the Kasner parameters on both sides is the same as that in the case of thin domain walls.

On the approach to the cosmological singularity in quadratic theories of gravity: The Kasner regimes

We study the Kasner-type solutions of the vacuum field equations of D-dimensional theories of gravity which derive from lagrangians that are non-linear in the curvature. We treat both the generic quadratic case which yields field equations of fourth order in the metric and the Lovelock theory whose field equations remain second order. We show that in four-dimensional space-times the standard Kasner solution is always a solution is always a solution (and often the only one) whereas in D > 4 space times all spatial dimensions can always contract as one goes backward in time, contrary to what happens in einsteinian gravity. The bearing of these results on the conjecture that the chaotic approach to the Big Bang singularity is specific to four-dimensional space-times is briefly discussed.

Are Kaluza-Klein models of the universe chaotic?

The generic behavior of vacuum inhomogeneous and spatially homogeneous Kaluza-Klein models is studied in the vicinity of the cosmological singularity. It is shown that, in space-time dimensions ≥ 11, the generalized Kasner solution, with monotonic power-law behavior of the spatial distances, becomes a general solution of the Einstein vacuum field equations and that, moreover, the chaotic oscillatory behavior disappears. On the other hand, the chaotic oscillatory behavior, absent in diagonal spatially homogeneous cosmological models in space-time dimensions between 5 and 10, can be reestablished when off-diagonal terms are included.

Generation of LRS Bianchi type I universes filled with perfect fluids

A generating technique is presented which converts known LRS Bianchi type I models into new models of the same type. Starting from the general Kasner solutions new classes of models are obtained which add to the rare perfect-fluid solutions not satisfying the equation of state. The physical and kinematical properties of cosmological models are studied.

Structure of the singularities produced by colliding plane waves.

When gravitational plane waves propagating and colliding in an otherwise flat background interact, they produce spacetime singularities. If the colliding waves have parallel (linear) polarizations, the mathematical analysis of the field equations in the interaction region is especially simple. Using the formulation of these field equations previously given by Szekeres, we analyze the asymptotic structure of a general colliding parallel-polarized plane-wave solution near the singularity. We show that the metric is asymptotic to an inhomogeneous Kasner solution as the singularity is approached. We give explicit expressions which relate the asymptotic Kasner exponents along the singularity to the initial data posed along the wave fronts of the incoming, colliding plane waves. It becomes clear from these expressions that for specific choices of initial data the curvature singularity created by the colliding waves degenerates to a coordinate singularity, and that a nonsingular Killing-Cauchy horizon is thereby obtained. Our equations prove that these horizons are unstable in the full nonlinear theory against small but generic perturbations of the initial data, and that in a very precise sense, ``generic'' initial data always produce all-embracing, spacelike curvature singularities without Killing-Cauchy horizons. We give several examples of exact solutions which illustrate some of the asymptotic singularity structures that are discussed in the paper. In particular, we construct a new family of exact colliding parallel-polarized plane-wave solutions, which create Killing-Cauchy horizons instead of a spacelike curvature singularity. The maximal analytic extension of one of these solutions across its Killing-Cauchy horizon results in a colliding plane-wave spacetime, in which a Schwarzschild black hole is created out of the collision between two plane-symmetric sandwich waves propagating in a cylindrical universe.

Vacuum solutions admitting a geodesic null congruence with shear proportional to expansion

Algebraically general, nontwisting solutions for the vacuum to vacuum generalized Kerr–Schild (GKS) transformation are obtained. These solutions admit a geodesic null congruence with shear proportional to expansion. In the Newman–Penrose formalism, if lμ is chosen to be the null vector of the GKS transformation, this property is stated as σ=aρ and Da=0. It is assumed that a is a constant, and the background is chosen as a pp-wave solution. For generic values of a, the GKS metrics consist of the Kasner solutions. For a=±(1±(2)1/2), there are solutions with less symmetries including special cases of the Kóta–Perjés and Lukács solutions.