Phase Space for the Einstein Equations

Bartnik, Robert

doi:10.4310/cag.2005.v13.n5.a1

Cited by 56 publications

(129 citation statements)

References 31 publications

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“…Let M It will be shown in Theorem 3.2 that if g is a metric, defined on some open set, which satisfies (3) with some function λ, then g necessarily has constant scalar curvature. Furthermore, if Ω is indeed a closed manifold and g is a metric on Ω with negative scalar curvature for which (3) holds with some function λ, then g is an Einstein metric.…”

Section: Introductionmentioning

confidence: 99%

On the volume functional of compact manifolds with boundary with constant scalar curvature

2009

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Abstract. We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space forms, on which the standard metrics are critical points, are geodesic balls. In the zero scalar curvature case, assuming the boundary can be isometrically embedded in the Euclidean space as a compact strictly convex hypersurface, we show that the volume of a critical point is always no less than the Euclidean volume bounded by the isometric embedding of the boundary, and the two volumes are equal if and only if the critical point is isometric to a standard Euclidean ball. We also derive a second variation formula and apply it to show that, on Euclidean balls and "small" hyperbolic and spherical balls in dimensions 3 ≤ n ≤ 5, the standard space form metrics are indeed saddle points for the volume functional.

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Section: Introductionmentioning

confidence: 99%

On the volume functional of compact manifolds with boundary with constant scalar curvature

2009

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“…In a recent paper, Bartnik [1] shows that much of the finite-dimensional picture can be taken over to the Einstein system on asymptotically flat manifolds in the formulation given by Fischer and Marsden [6]. In particular, he shows that the constraint map Φ is smooth and surjective and that all its level sets, in particular the constraint manifold C, are smooth Hilbert submanifolds of the phase space of GR defined by the first and second fundamental forms (g, π ) of a suitable 3-dimensional manifold.…”

Section: General Description Of the Methodsmentioning

confidence: 99%

Algebraic stability analysis of constraint propagation

Frauendiener

Vogel

2005

Class. Quantum Grav.

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ABSTRACT. The divergence of the constraint quantities is a major problem in computational gravity today. Apparently, there are two sources for constraint violations. The use of boundary conditions which are not compatible with the constraint equations inadvertently leads to 'constraint violating modes' propagating into the computational domain from the boundary. The other source for constraint violation is intrinsic. It is already present in the initial value problem, i.e. even when no boundary conditions have to be specified. Its origin is due to the instability of the constraint surface in the phase space of initial conditions for the time evolution equations. In this paper, we present a technique to study in detail how this instability depends on gauge parameters. We demonstrate this for the influence of the choice of the time foliation in context of the Weyl system. This system is the essential hyperbolic part in various formulations of the Einstein equations.

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“…We remark that the approach is motivated by a constrained minimization scheme proposed by R. Bartnik [3] for his quasi-local mass program. The connection between the constrained minimization and mass rigidity was recently employed by D. Lee and the first named author in their proof to the rigidity conjecture of the spacetime positive mass theorem [14].…”

Section: Introductionmentioning

confidence: 99%