Analytic description of singularities in Gowdy spacetimes

Kichenassamy, Satyanad; Rendall, Alan D.

doi:10.1088/0264-9381/15/5/016

Cited by 163 publications

(307 citation statements)

References 12 publications

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“…The general form of a Fuchsian system ( [31,17,35] and references therein) for a vectorvalued unknown function u(t, x) = (u 1 (t, x), ..., u k (t, x)), defined on an open subset of R×R n with values in R k , is…”

Section: B1 Fuchs Theoremmentioning

confidence: 99%

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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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Section: B1 Fuchs Theoremmentioning

confidence: 99%

“…The solutions of this asymptotic system are precisely given by Kasner-like metrics. Fuchsian methods [17] can then be used to prove that, given a solution of the velocity dominated system, there exists a (geometrically unique) solution of…”

Section: Introductionmentioning

confidence: 99%

Describing general cosmological singularities in Iwasawa variables

2008

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“…The terms on the right hand side not written out are obtained from the terms on the right hand side of equations (11) and (12) if the indices m, p and q are replaced by the pairs (m, l), (p, l 1 ) and (q, l 2 ), summing over l 1 + l 2 = l. The recursion relations implied by the constraints are identical except for the addition of an extra index l. In a similar way, the consistency conditions lead to…”

Section: Perturbative Solutionsmentioning

confidence: 99%

Asymptotics of Solutions of the Einstein Equations with Positive Cosmological Constant

Rendall

2004

Ann. Henri Poincaré

135

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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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“…The condition which makes the expansion possible is that a polynomial u 0 can be found which satisfies the equation up to an error of sufficiently high order in t. The only obstruction to this is if the shifted eigenvalue becomes zero at some stage in the process and this is prevented by the assumption made on the eigenvalues of the original matrix. In cases such as the system arising in the analysis of Gowdy spacetimes in [11] the solutions cannot be expanded in integral powers of t. Instead non-integral (and even x-dependent) powers of t and logarithms occur. For this reason the method of [6] does not apply directly.…”

Section: T ∂U ∂T + N(x)u = T α F (T X U D X U)mentioning

confidence: 99%

“…Another line of development of the applications of Fuchsian equations in general relativity starts with the work of Kichenassamy and Rendall [11] on singularities in analytic Gowdy spacetimes. It builds on previous work of Kichenassamy outside general relativity (see [10] and references therein).…”

Section: T ∂U ∂T + N(x)u = T α F (T X U D X U)mentioning

confidence: 99%

Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity

Rendall

2000

Class. Quantum Grav.

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114

133

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Abstract. Fuchsian equations provide a way of constructing large classes of spacetimes whose singularities can be described in detail. In some of the applications of this technique only the analytic case could be handled up to now. This paper develops a method of removing the undesirable hypothesis of analyticity. This is applied to the specific case of the Gowdy spacetimes in order to show that analogues of the results known in the analytic case hold in the smooth case. As far as possible the likely strengths and weaknesses of the method, as applied to more general problems, are displayed.

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Analytic description of singularities in Gowdy spacetimes

Cited by 163 publications

References 12 publications

Describing general cosmological singularities in Iwasawa variables

Describing general cosmological singularities in Iwasawa variables

Asymptotics of Solutions of the Einstein Equations with Positive Cosmological Constant

Fuchsian analysis of singularities in Gowdy spacetimes beyond analyticity

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