Quiescent Cosmological Singularities

Andersson, Lars; Rendall, Alan D.

doi:10.1007/s002200100406

Cited by 199 publications

(547 citation statements)

References 25 publications

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“…3 that though the x direction is initially expanding, eventually all three directions contract. This is what one would expect if the metrics of [10] represent the generic behavior near the singularity since these metrics have all three directions contracting in a neighborhood of the singularity.…”

Section: Resultsmentioning

confidence: 73%

Harmonic coordinate method for simulating generic singularities

2002

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This paper presents both a numerical method for general relativity and an application of that method. The method involves the use of harmonic coordinates in a 3+1 code to evolve the Einstein equations with scalar field matter. In such coordinates, the terms in Einstein's equations with the highest number of derivatives take a form similar to that of the wave equation. The application is an exploration of the generic approach to the singularity for this type of matter. The preliminary results indicate that the dynamics as one approaches the singularity is locally the dynamics of the Kasner spacetimes.

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Section: Resultsmentioning

confidence: 73%

Harmonic coordinate method for simulating generic singularities

2002

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“…If we assume that N (x) is bounded then by choosing M large enough it can be ensured that the matrix N (x)+M I is positive definite. Assuming that f and N are regular in the analytic sense the existence theorem of [1] implies the existence of a unique regular solution v vanishing at t = 0. Expressing u in terms of v gives a solution of the original equation which has the given asymptotic expansion up to order M .…”

Section: Fuchsian Analysismentioning

confidence: 99%

Asymptotics of Solutions of the Einstein Equations with Positive Cosmological Constant

Rendall

2004

Ann. Henri Poincaré

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129

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A positive cosmological constant simplifies the asymptotics of forever expanding cosmological solutions of the Einstein equations. In this paper a general mathematical analysis on the level of formal power series is carried out for vacuum spacetimes of any dimension and perfect fluid spacetimes with linear equation of state in spacetime dimension four. For equations of state stiffer than radiation evidence for development of large gradients, analogous to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution satisfying minimal asymptotic conditions has a full asymptotic expansion given by the formal series. In four spacetime dimensions, and for spatially homogeneous spacetimes of any dimension, these minimal conditions can be derived for appropriate initial data. Using Fuchsian methods the existence of vacuum spacetimes with the given formal asymptotics depending on the maximal number of free functions is shown without symmetry assumptions.

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“…On the other hand, the p-form fields A have limits as t → 0. Theories exhibiting this asymptotic behavior are, for instance, pure gravity in D ≥ 11 [6,7,13], and gravity coupled to a scalar field in any dimensions [4,14].…”

Section: Introductionmentioning

confidence: 99%

“…These Fuchsian methods have been used to mathematically describe cosmological singularities in various simplified contexts: Gowdy spacetimes [17], plane symmetric spacetimes with a massless scalar field [18], polarized and half-polarized U (1) symmetric vacuum spacetimes [19], spacetimes with collisionless matter and spherical, plane or hyperbolic symmetry [20], and a particular subset of general Gowdy spacetimes [21]. It has also been possible to use Fuchsian methods to mathematically describe singularities without any symmetries: notably for the Einsteinscalar system [14], and for many Einstein-matter models including pure gravity in D ≥ 11 dimensions [13].…”

Section: Introductionmentioning

confidence: 99%

Describing general cosmological singularities in Iwasawa variables

2008

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Belinskii, Khalatnikov, and Lifshitz (BKL) conjectured that the description of the asymptotic behavior of a generic solution of Einstein equations near a spacelike singularity could be drastically simplified by considering that the time derivatives of the metric asymptotically dominate (except at a sequence of instants, in the 'chaotic case') over the spatial derivatives. We present a precise formulation of the BKL conjecture (in the chaotic case) that consists of basically three elements: (i) we parametrize the spatial metric g ij by means of Iwasawa variables (β a , N a i ); (ii) we define, at each spatial point, a (chaotic) asymptotic evolution system made of ordinary differential equations for the Iwasawa variables; and (iii) we characterize the exact Einstein solutions β, N whose asymptotic behavior is described by a solution β [0] , N [0] of the previous evolution system by means of a 'generalized Fuchsian system' for the differenced variablesβ = β − β [0] ,N = N − N [0] , and by requiring thatβ andN tend to zero on the singularity. We also show that, in spite of the apparently chaotic infinite succession of 'Kasner epochs' near the singularity, there exists a well-defined asymptotic geometrical structure on the singularity : it is described by a partially framed flag. Our treatment encompasses Einstein-matter systems (comprising scalar and p-forms), and also shows how the use of Iwasawa variables can simplify the usual ('asymptotically velocity term dominated') description of non-chaotic systems.

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Quiescent Cosmological Singularities

Cited by 199 publications

References 25 publications

Harmonic coordinate method for simulating generic singularities

Harmonic coordinate method for simulating generic singularities

Asymptotics of Solutions of the Einstein Equations with Positive Cosmological Constant

Describing general cosmological singularities in Iwasawa variables

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