Flat Kasner Research Articles

Averaging generalized scalar-field cosmologies III: Kantowski\u2013Sachs and closed Friedmann\u2013Lema\xeetre\u2013Robertson\u2013Walker models

Scalar-field cosmologies with a generalized harmonic potential and matter with energy density rho _m, pressure p_m, and barotropic equation of state (EoS) p_m=(gamma -1)rho _m, ; gamma in [0,2] in Kantowski–Sachs (KS) and closed Friedmann–Lemaître–Robertson–Walker (FLRW) metrics are investigated. We use methods from non-linear dynamical systems theory and averaging theory considering a time-dependent perturbation function D. We define a regular dynamical system over a compact phase space, obtaining global results. That is, for KS metric the global late-time attractors of full and time-averaged systems are two anisotropic contracting solutions, which are non-flat locally rotationally symmetric (LRS) Kasner and Taub (flat LRS Kasner) for 0le gamma le 2, and flat FLRW matter-dominated universe if 0le gamma le frac{2}{3}. For closed FLRW metric late-time attractors of full and averaged systems are a flat matter-dominated FLRW universe for 0le gamma le frac{2}{3} as in KS and Einstein–de Sitter solution for 0le gamma <1. Therefore, a time-averaged system determines future asymptotics of the full system. Also, oscillations entering the system through Klein–Gordon (KG) equation can be controlled and smoothed out when D goes monotonically to zero, and incidentally for the whole D-range for KS and closed FLRW (if 0le gamma < 1) too. However, for gamma ge 1 closed FLRW solutions of the full system depart from the solutions of the averaged system as D is large. Our results are supported by numerical simulations.

A classification theorem for compact Cauchy horizons in vacuum spacetimes

We establish a complete classification theorem for the topology and for the orbital type of the null generators of compact non-degenerate Cauchy horizons of time orientable smooth vacuum \(3+1\)-spacetimes. We show that one and only one of the following must hold: (i) all generators are closed, (ii) only two generators are closed and any other densely fills a two-torus, (iii) every generator densely fills a two-torus, or (iv) every generator densely fills the horizon. We then show that, respectively to (i)–(iv), the horizon’s manifold is either: (i’) a Seifert manifold, (ii’) a lens space, (iii’) a two-torus bundle over a circle, or, (iv’) a three-torus. All the four possibilities are known to arise in examples. In the last case, (iv), (iv’), we show in addition that the spacetime is indeed flat Kasner, thus settling a problem posed by Isenberg and Moncrief for ergodic horizons. The results of this article open the door for a full parameterization of the metrics of all vacuum spacetimes with a compact Cauchy horizon. The method of proof permits direct generalizations to higher dimensions.

Symmetries of Cosmological Cauchy Horizons with Non-Closed Orbits

We consider analytic, vacuum spacetimes that admit compact, non-degenerate Cauchy horizons. Many years ago we proved that, if the null geodesic generators of such a horizon were all \textit{closed} curves, then the enveloping spacetime would necessarily admit a non-trivial, horizon-generating Killing vector field. Using a slightly extended version of the Cauchy-Kowaleski theorem one could establish the existence of infinite dimensional, analytic families of such `generalized Taub-NUT' spacetimes and show that, generically, they admitted \textit{only} the single (horizon-generating) Killing field alluded to above. In this article we relax the closure assumption and analyze vacuum spacetimes in which the generic horizon generating null geodesic densely fills a 2-torus lying in the horizon. In particular we show that, aside from some highly exceptional cases that we refer to as `ergodic', the non-closed generators always have this (densely 2-torus-filling) geometrical property in the analytic setting. By extending arguments we gave previously for the characterization of the Killing symmetries of higher dimensional, stationary black holes we prove that analytic, 4-dimensional, vacuum spacetimes with such (non-ergodic) compact Cauchy horizons always admit (at least) two independent, commuting Killing vector fields of which a special linear combination is horizon generating. We also discuss the \textit{conjectures} that every such spacetime with an \textit{ergodic} horizon is trivially constructable from the flat Kasner solution by making certain `irrational' toroidal compactifications and that degenerate compact Cauchy horizons do not exist in the analytic case.

Asymptotically free mimetic gravity

The idea of “asymptotically free” gravity is implemented using a constrained mimetic scalar field. The effective gravitational constant is assumed to vanish at some limiting curvature. As a result singularities in spatially flat Friedmann and Kasner universes are avoided. Instead, the solutions in both cases approach a de Sitter metric with limiting curvature. We show that quantum metric fluctuations vanish when this limiting curvature is approached.

A Unified Approach to the Klein\u2013Gordon Equation on Bianchi Backgrounds

In this paper, we study solutions to the Klein–Gordon equation on Bianchi backgrounds. In particular, we are interested in the asymptotic behaviour of solutions in the direction of silent singularities. The main conclusion is that, for a given solution u to the Klein–Gordon equation, there are smooth functions ui, i = 0,1, on the Lie group under consideration, such that {u_{sigma}(cdot,sigma)-u_{1}} and {u(cdot,sigma)-u_{1}sigma-u_{0}} asymptotically converge to zero in the direction of the singularity (where {sigma} is a geometrically defined time coordinate such that the singularity corresponds to {sigmarightarrow-infty}). Here ui, i = 0, 1, should be thought of as data on the singularity. Interestingly, it is possible to prove that the asymptotics are of this form for a large class of Bianchi spacetimes. Moreover, the conclusion applies for singularities that are matter dominated; singularities that are vacuum dominated; and even when the asymptotics of the underlying Bianchi spacetime are oscillatory. To summarise, there seems to be a universality as far as the asymptotics in the direction of silent singularities are concerned. In fact, it is tempting to conjecture that as long as the singularity of the underlying Bianchi spacetime is silent, then the asymptotics of solutions are as described above. In order to contrast the above asymptotics with the non-silent setting, we, by appealing to known results, provide a complete asymptotic characterisation of solutions to the Klein–Gordon equation on a flat Kasner background. In that setting, {u_{sigma}} does, generically, not converge.

The Mode Solution of the Wave Equation in Kasner Spacetimes and Redshift

We study the mode solution to the Cauchy problem of the scalar wave equation $\Box \varphi = 0$ in Kasner spacetimes. As a first result, we give the explicit mode solution in axisymmetric Kasner spacetimes, of which flat Kasner spacetimes are special cases. Furthermore, we give the small and large time asymptotics of the modes in general Kasner spacetimes. Generically, the modes in non-flat Kasner spacetimes grow logarithmically for small times, while the modes in flat Kasner spacetimes stay bounded for small times. For large times, however, the modes in general Kasner spacetimes oscillate with a polynomially decreasing amplitude. This gives a notion of large time frequency of the modes, which we use to model the wavelength of light rays in Kasner spacetimes. We show that the redshift one obtains in this way actually coincides with the usual cosmological redshift.

Is classical flat Kasner spacetime flat in quantum gravity?

Quantum nature of classical flat Kasner spacetime is studied using effective spacetime description in loop quantum cosmology (LQC). We find that even though the spacetime curvature vanishes at the classical level, nontrivial quantum gravitational effects can arise. For the standard loop quantization of Bianchi-I spacetime, which uniquely yields universal bounds on expansion and shear scalars and results in a generic resolution of strong singularities, we find that a flat Kasner metric is not a physical solution of the effective spacetime description, except in a limit. The lack of a flat Kasner metric at the quantum level results from a novel feature of the loop quantum Bianchi-I spacetime: quantum geometry induces nonvanishing spacetime curvature components, making it not Ricci flat even when no matter is present. The noncurvature singularity of the classical flat Kasner spacetime is avoided, and the effective spacetime transits from a flat Kasner spacetime in asymptotic future, to a Minkowski spacetime in asymptotic past. Interestingly, for an alternate loop quantization which does not share some of the fine features of the standard quantization, flat Kasner spacetime with expected classical features exists. In this case, even with nontrivial quantum geometric effects, the spacetime curvature vanishes. These examples show that the character of even a flat classical vacuum spacetime can alter in a fundamental way in quantum gravity and is sensitive to the quantization procedure.

Conformal Parameterizations of Slices of Flat Kasner Spacetimes

The Kasner metrics are among the simplest solutions of the vacuum Einstein equations, and we use them here to examine the conformal method of finding solutions of the Einstein constraint equations. After describing the conformal method’s construction of constant mean curvature (CMC) slices of Kasner spacetimes, we turn our attention to non-CMC slices of the smaller family of flat Kasner spacetimes. In this restricted setting we obtain a full description of the construction of certain U n-1 symmetric slices, even in the far-from-CMC regime. Among the conformal data sets generating these slices we find that most data sets construct a single flat Kasner spacetime, but that there are also far-from-CMC data sets that construct one-parameter families of slices. Although these non-CMC families are analogues of well-known CMC one-parameter families, they differ in important ways. Most significantly, unlike the CMC case, the condition signaling the appearance of these non-CMC families is not naturally detected from the conformal data set itself. In light of this difficulty, we propose modifications of the conformal method that involve a conformally transforming mean curvature.

Global geometry of [formula omitted]-symmetric spacetimes with weak regularity

We define the class of weakly regular spacetimes with T 2 -symmetry, and investigate their global geometrical structure. We formulate the initial value problem for the Einstein vacuum equations with weak regularity, and establish the existence of a global foliation by the level sets of the area R of the orbits of symmetry, so that each leaf can be regarded as an initial hypersurface. Except for the flat Kasner spacetimes which are known explicitly, R takes all positive values. Our weak regularity assumptions only require that the gradient of R is continuous while the metric coefficients belong to the Sobolev space H 1 (or have even less regularity).

Future asymptotics of vacuum Bianchi type VI0 solutions

In this paper, we present a thorough analysis of the future asymptotic dynamics of spatially homogeneous cosmological models of Bianchi type VI0. Each of these models converges to a flat Kasner solution (Taub solution) for late times; we give detailed asymptotic expansions describing this convergence. In particular, we prove that the future asymptotics of Bianchi type VI0 solutions cannot be approximated in any way by Bianchi type II solutions, which is in contrast to Bianchi type VIII and IX models (in the direction toward the singularity). The paper contains an extensive introduction where we put the results into a broader context. The core of these considerations consists in the fact that there exist regions in the phase space of Bianchi type VIII models where solutions can be approximated, to a high degree of accuracy, by type VI0 solutions. The behavior of solutions in these regions is essential for the question of ‘locality’, i.e., whether particle horizons form or not. Since Bianchi type VIII models are conjectured to be important role models for generic cosmological singularities, our understanding of Bianchi type VI0 dynamics might thus be crucial to help to shed some light on the important question of whether to expect generic singularities to be local or not.

On the area of the symmetry orbits in T2 symmetric spacetimes

We obtain a global existence result for the Einstein equations. We show that in the maximal Cauchy development of vacuum T2 symmetric initial data with nonvanishing twist constant, except for the special case of flat Kasner initial data, the area of the T2 group orbits takes on all positive values. This result shows that the areal time coordinate R which covers these spacetimes runs from zero to infinity, with the singularity occurring at R = 0.

Electromagnetic fields in a Taub-Nut-type cosmology

The behavior of a test electromagnetic field on R 1×T 3 and R 3×S 1 flat Kasner universes is used to probe the nature of the quasiregular singularity present. The stress-energy tensor for the field is calculated in coordinate and parallel-propagated orthonormal (PPON) frames. Theorems about the convergence properties of the field at the singularity are proven. A conjecture is made concerning the stability of the singularity structure.

Cosmologies with quasiregular singularities. I. Spacetimes and test waves

The nature of spacetimes with quasiregular singularities is discussed. Such singularities are the end points of incomplete, inextendible geodesics at which the Riemann tensor and its derivatives remain at least bounded in all parallel-propagated orthonormal frames; observers approaching such a singularity would find that their world lines come to an end in a finite proper time, without encountering infinite tidal forces. Particular attention is paid to Taub-NUT-(Newman-Unti-Tamburino) type cosmologies, which are an interesting class of spacetimes containing quasiregular singularities. These cosmologies are characterized by incomplete geodesics which spiral infinitely around a topologically closed spatial dimension: They include the ${R}^{3}\ifmmode\times\else\texttimes\fi{}{S}^{1}$ and ${R}^{1}\ifmmode\times\else\texttimes\fi{}{T}^{3}$ flat Kasner universes, the two-parameter family of Taub-NUT universes, and an infinite-dimensional subclass of Einstein-Rosen-Gowdy spacetimes studied by Moncrief. The global structure of each of these spacetimes is described. The flat Kasner and Moncrief universes both possess a null hypersurface which is a Cauchy and a Killing horizon and which contains a quasiregular singularity; the Taub-NUT universes possess two such null hypersurfaces. Timelike geodesics exhibit two sorts of behavior in the vicinity of any one of these null hypersurfaces: They may pass right through the hypersurface or they may approach it asymptotically, spiraling around a closed spatial dimension. The behavior of scalar test fields in each of the Taub-NUT-type cosmologies is also very similar in the vicinity of a singularity-containing null hypersurface. In each case there are three types of wave modes: There are modes whose amplitude diverges logarithmically in the vicinity of the null hypersurface; there are other modes whose phase diverges logarithmically; and there are modes without any divergent behavior. It is shown that generic finite data on an initial Cauchy hypersurface lead to divergent test fields at the null hypersurface. The possibility of the existence of additional Taub-NUT-type cosmologies among spatially homogeneous spacetimes of the various Bianchi types and among inhomogeneous spacetimes is also discussed. The geodesic and scalar field behavior in these spacetimes is used in a subsequent paper to investigate the stability of Taub-NUT-type cosmologies.

Cosmologies with quasiregular singularities. II. Stability considerations.

The stability properties of a class of spacetimes with quasiregular singularities is discussed. Quasiregular singularities are the end points of incomplete, inextendible geodesics at which the Riemann tensor and its derivatives remain at least bounded in all parallel-propagated orthonormal (PPON) frames; observers approaching such a singularity would find that their world lines come to an end in a finite proper time. The Taub-NUT (Newman-Unti-Tamburino)-type cosmologies investigated are ${R}^{1}$\ifmmode\times\else\texttimes\fi{}${T}^{3}$ and ${R}^{3}$\ifmmode\times\else\texttimes\fi{}${S}^{1}$ flat Kasner spacetimes, the two-parameter family of spatially homogeneous but anisotropic Bianchi type-IX Taub-NUT spacetimes, and an infinite-dimensional family of Einstein-Rosen-Gowdy spacetimes studied by Moncrief. The behavior of matter near the quasiregular singularity in each of these spacetimes is explored through an examination of the behavior of the stress-energy tensors and scalars for conformally coupled and minimally coupled, massive and massless scalar waves as observed in both coordinate and PPON frames. A conjecture is postulated concerning the stability of the nature of the singularity in these spacetimes. The conjecture for a Taub-NUT-type background spacetime is that if a test-field stress-energy tensor evaluated in a PPON frame mimics the behavior of the Riemann tensor components which indicate a particular type of singularity (quasiregular, nonscalar curvature, or scalar curvature), then a complete nonlinear backreaction calculation, in which the fields are allowed to influence the geometry, would show that this type of singularity actually occurs. Evidence supporting the conjecture is presented for spacetimes whose symmetries are unchanged when fields with the same symmetries are added. The conjecture and the exact solutions which support it both indicate that most waves mimic scalar curvature singularities; only very special wave modes mimic nonscalar curvature or quasiregular singularities. Therefore if general fields are added to the idealized empty Taub-NUT-type cosmologies, one would expect the quasiregular singularities to be converted into scalar curvature singularities.

Quantum vacuum energy in Taub-NUT (Newman-Unti-Tamburino)-type cosmologies

The effects of vacuum polarization on the mildest possible sort of cosmological singularity, the Taub-NUT (Newman-Unti-Tamburino)-type singularities, are studied. Unlike stronger sorts of singularities where physical quantities (e.g., curvature, energy density) diverge, in these universes the only barrier is a pathological topology. Quantum effects, known to be important in regions of large spacetime curvature, are found to also be important in these universes, where the curvature may be arbitrarily small or even zero. The vacuum expectation value of the stress-energy tensor for a conformal scalar field is calculated on a flat archetype of the Taub-NUT-type universes, the Misner universe (flat Kasner spacetime with ${S}^{1}\ifmmode\times\else\texttimes\fi{}{R}^{3}$ topology). The vacuum stress energy diverges at the singularity and on its associated Cauchy horizons. This divergence, together with the "fixed" nature of the spacetime's topology, suggests that these boundaries will be replaced by curvature singularities in a better approximation to full quantum gravity.

“Taub-NUT-like” cosmologies

We investigate vacuum cosmologies with quasiregular singularities, in particular Taub-NUT, three-torus flat Kasner, and Moncrief's spatially inhomogeneous spacetimes. In each model the metric and topology, null geodesics, and massless scalar test fields show similar structure near a null hypersurface containing the singularity and its associated Cauchy horizons.

Comments on the coherent-state representation of a scalar field in the early universe

Previous work on the coherent states of a cosmological scalar field is extended to illustrate the effect of spacetime curvature on the coherent-state parametrization. Coherent states for the "flat Kasner" ("Rindler wedge") cosmology are constructed as an example. It is shown that the coherent-state representation is invariant under a conformal transformation which does not alter the definition of positive frequency.

Electromagnetic waves in a Bianchi type-I universe

A general formalism is developed to deal with electromagnetic waves in a metric of the form $\ensuremath{-}d{s}^{2}={g}_{\ensuremath{\mu}\ensuremath{\nu}}d{x}^{\ensuremath{\mu}}d{x}^{\ensuremath{\nu}}=\ensuremath{-}d{t}^{2}+\ensuremath{\Sigma}{i=1}^{3}{{A}_{i}}^{2}(t){(d{x}^{i})}^{2}$. The propagation problem is reduced to the integration of a second-order differential equation determining the time evolution of the ${F}_{\ensuremath{\mu}\ensuremath{\nu}}$ tensor. The observable electric and magnetic fields are obtained referring ${F}_{\ensuremath{\mu}\ensuremath{\nu}}$ to a suitable orthonormal tetrad and are shown to consist of a superposition of spectral components, each labeled by a set of parameters ${k}_{i}$ and transverse to a time-varying direction specified by $\frac{{k}_{i}}{{A}_{i}(t)}$. The ratio between energy and spin angular momentum for a single spectral component is found to be ${({k}_{i}{k}^{i})}^{\frac{1}{2}}$, which in the WKB limit is the angular frequency of the waves. The high-frequency solutions for the fields are used to discuss the possible effects of anisotropy in the expansion of the universe after decoupling on polarization and intensity distributions of the microwave background. Exact solutions are given for waves propagating along the coordinate axes of the general Kasner spacetimes and along any direction in the flat Kasner spacetime. Propagation out of the singularity is seen to alter considerably amplitude and phase relationships between the fields. Waveforms traveling along the coordinate axes without a backward tail are constructed.