The Global Existence Problem in General Relativity

Andersson, Lars

doi:10.1007/978-3-0348-7953-8_3

Cited by 93 publications

(108 citation statements)

References 148 publications

(141 reference statements)

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“…We conjecture that this static solution, as it is for the classical Ricci flow equation (Ye [44]), the Einstein equations (Andersson-Moncrief [2]), and the reduced Einstein equations (Fischer-Moncrief [20]) is asymptotically stable under conformal Ricci flow; i.e., given any neighborhood U g h ⊂ M −1 of g h there exists a capture neighborhood U ′ g h ⊆ U g h such that if g 0 ∈ U ′ g h and g is the solution of the conformal Ricci system with initial value g 0 , then (i) g : [0, ∞) → M −1 is non-singular, (ii) for t ∈ [0, ∞), g t ∈ U g h , and (iii) g t → g h as t → ∞.…”

Section: Corollary 82 Letmentioning

confidence: 94%

An introduction to conformal Ricci flow

Fischer¹

2004

Class. Quantum Grav.

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Abstract. We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the conformal Ricci flow equations because of the role that conformal geometry plays in constraining the scalar curvature and because these equations are the vector field sum of a conformal flow equation and a Ricci flow equation. These new equations are given byfor a dynamically evolving metric g and a scalar non-dynamical field p. The conformal Ricci flow equations are analogous to the Navier-Stokes equations of fluid mechanics,Because of this analogy, the time-dependent scalar field p is called a conformal pressure and, as for the real physical pressure in fluid mechanics that serves to maintain the incompressibility of the fluid, the conformal pressure serves as a Lagrange multiplier to conformally deform the metric flow so as to maintain the scalar curvature constraint. The equilibrium points of the conformal Ricci flow equations are Einstein metrics with Einstein constant − 1 n . Thus the term −2(Ric(g)+ 1 n g) measures the deviation of the flow from an equilibrium point and acts as a nonlinear restoring force. The conformal pressure p ≥ 0 is zero at an equilibrium point and positive otherwise. The constraint force −pg acts pointwise orthogonally to the nonlinear restoring force −2(Ric(g) + 1 n g) and conformally deforms g so that the scalar curvature is preserved.A variety of properties of the conformal Ricci flow are discussed, including the reduced conformal Ricci flow, local existence and uniqueness, a variational formulation using a quasi-gradient of the Yamabe functional, strictly monotonically decreasing global and local volume results, and applications to 3-manifold geometry. The geometry of the conformal Ricci flow is discussed as well as the remarkable analytic fact that the constraint force does not lose derivatives and thus analytically the conformal Ricci equation is a bounded perturbation of the classical unnormalized Ricci equation. Lastly, we discuss potential applications to Perelman's proposed implementation of Hamilton's program to prove Thurston's 3-manifold geometrization conjectures.

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Section: Corollary 82 Letmentioning

confidence: 94%

An introduction to conformal Ricci flow

Fischer¹

2004

Class. Quantum Grav.

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“…As it follows from standard Kaluza-Klein lines, the solutions of the latter with polarized U(1) symmetry are equivalent to the Einstein-scalar field system in D = 3 (see e.g. [40] and [41], section 5). Hence the result of this section implies that we have constructed the most general known class of singular solutions of the Einstein vacuum equations in four spacetime dimensions.…”

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confidence: 99%

Kasner-Like Behaviour for Subcritical Einstein-Matter Systems

Damour¹,

Henneaux²,

Rendall³

et al. 2002

Annales Henri Poincare

146

227

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Confirming previous heuristic analysesà la Belinskii-Khalatnikov-Lifshitz, it is rigorously proven that certain "subcritical" Einstein-matter systems exhibit a monotone, generalized Kasner behaviour in the vicinity of a spacelike singularity. The D−dimensional coupled Einstein-dilaton-p-form system is subcritical if the dilaton couplings of the p-forms belong to some dimension dependent open neighbourhood of zero [1], while pure gravity is subcritical if D ≥ 11 [13]. Our proof relies, like the recent theorem [15] dealing with the (always subcritical [14]) Einstein-dilaton system, on the use of Fuchsian techniques, which enable one to construct local, analytic solutions to the full set of equations of motion. The solutions constructed are "general" in the sense that they depend on the maximal expected number of free functions.

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“…With a slight abuse of terminology the latter spacetime will also be referred to here as the Milne model. The stability of the latter model has been proved by Andersson and Moncrief (see [8, 11]). The result is that, given data for the Milne model on a manifold obtained by compactifying a hyperboloid in Minkowski space, the maximal Cauchy developments of nearby data are geodesically complete in the future.…”

Section: Global Existence For Small Datamentioning

confidence: 99%

Theorems on Existence and Global Dynamics for the Einstein Equations

Rendall

2005

Living Rev. Relativ.

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This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.

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The Global Existence Problem in General Relativity

Cited by 93 publications

References 148 publications

An introduction to conformal Ricci flow

An introduction to conformal Ricci flow

Kasner-Like Behaviour for Subcritical Einstein-Matter Systems

Theorems on Existence and Global Dynamics for the Einstein Equations

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