The constraint equations for the Einstein-scalar field system on compact manifolds

Choquet–Bruhat, Yvonne; Isenberg, James; Pollack, Daniel

doi:10.1088/0264-9381/24/4/004

Cited by 56 publications

(104 citation statements)

References 38 publications

(115 reference statements)

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“…For a survey reference on the constraint equations see Bartnik-Isenberg [3] and for further informations on the conformal method see Choquet-Bruhat, Isenberg and Pollack [7]. Essentially, the set of free data consists of (ψ, τ, π, U ), where ψ, τ, π are smooth functions in M and U is a smooth symmetric traceless and divergence-free (2, 0)-tensor in M .…”

Section: Statement Of the Resultsmentioning

confidence: 99%

Effective multiplicity for the Einstein-scalar field Lichnerowicz equation

Premoselli

2014

Calc. Var.

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Abstract. We prove the stability of the Einstein-scalar field Lichnerowicz equation under subcritical perturbations of the critical nonlinearity in dimensions n = 3, 4, 5. As a consequence, we obtain the existence of a second solution to the equation in several cases. In particular, in the positive case, including the CMC positive cosmological constant case, we show that each time a solution exists, the equation produces a second solution with the exception of one critical value for which the solution is unique. Statement of the results.Let (M, g) be a smooth closed Riemannian manifold of dimension n ≥ 3. We are interested in the Einstein-scalar field Lichnerowicz equation in M :(EL) △ g u + hu = f u 2 * −1 + a u 2 * +1 , where h, f and a are smooth functions on M ,n−2 is the critical Sobolev exponent for the Sobolev space H 1 , and we assume that △ g + h is coercive, a ≥ 0, a ≡ 0, and max M f > 0. By closed, following standard terminology, we mean compact without boundary. Equation (EL) arises in the mathematical analysis of general relativity when solving the Einstein equations in a scalar-field setting, when the gravity is coupled to a scalar-field ψ. Special important cases include the massive Klein-Gordon setting or the case of a positive cosmological constant Λ. Given a closed manifold (M, g) of dimension n ≥ 3 endowed with two smooth functions π and ψ and a (2, 0)-symmetric tensor field K, the Cauchy problem in general relativity consists in finding a Lorentzian manifold (M × R,g) together with a smooth functionψ on M × R such thatψ |M = ψ and ∂ nψ|M = π, where ∂ n denotes the normalized time derivative, such that K is the second fundamental form of the embedding M ⊂ M × R, and such that (M × R,g) satisfies the Einstein equations:where Ric(g) is the Ricci tensor ofg, R(g) is its scalar curvature and T is the stress-energy tensor-field. This tensor field depends ong, onψ and on some potential V , itself related toψ by some wave equation. As shown first by for the vacuum case, see also Choquet-Bruhat-Isenberg-Pollack [6], a necessary and sufficient condition for the existence of such ag on M × R is that the following system of equations in M is satisfied:where R(g) is the scalar curvature of g and ∇ refers to the Levi-Civita connection of g. By specifying some of the unknown initial data (g, K, ψ, π) and solving the system for the remaining data, the conformal method initiated by Lichnerowicz [17] allows to turn (1.1) into a system of 1 2 BRUNO PREMOSELLI elliptic partial differential equations of critical Sobolev growth, called the conformal constraint system of equations. For a survey reference on the constraint equations see and for further informations on the conformal method see Choquet-Bruhat, Isenberg and Pollack [7]. Essentially, the set of free data consists of (ψ, τ, π, U ), where ψ, τ, π are smooth functions in M and U is a smooth symmetric traceless and divergence-free (2, 0)-tensor in M . Given (ψ, τ, π, U ) an initial free data set, the conformal constraint system of equations, whose unknown...

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Section: Statement Of the Resultsmentioning

confidence: 99%

Effective multiplicity for the Einstein-scalar field Lichnerowicz equation

Premoselli

2014

Calc. Var.

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“…Among these are the recent attempts to use such theories to explain the observed acceleration of the expansion of the universe [16,17,18,19]. Using the conformal method, Choquet-Bruhat, Isenberg, and Pollack [6,7] reformulated the constraint equations for the Einstien-scalar field system as a determined system of nonlinear partial differential equations. The equations are semi-decoupled in the constant mean curvature (CMC) setting.…”

Section: Introductionmentioning

confidence: 99%

“…This assumption implies no physical restrictions since we always have that A ≥ 0 in the original Einstein-scalar field theory. One of the results of [7] is the definition of a conformal invariant, the Yamabe-scalar field conformal invariant, whose sign can be used, through a judicious choice of the background metric g, to control the sign of h.…”

Section: Introductionmentioning

confidence: 99%

A Variational Analysis of Einstein–Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds

Hebey

Pacard

Pollack

2007

Commun. Math. Phys.

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Abstract. We establish new existence and non-existence results for positive solutions of the Einstein-scalar field Lichnerowicz equation on compact manifolds. This equation arises from the Hamiltonian constraint equation for the Einstein-scalar field system in general relativity. Our analysis introduces variational techniques, in the form of the mountain pass lemma, to the analysis of the Hamiltonian constraint equation, which has been previously studied by other methods.

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“…First off, let us recall that, similarly to the Neumann BCs case [23,24], PBCs add an integrability condition to this equation: since the integral of the laplacian operator of any sufficiently smooth function will vanish on a periodic domain, then the integral of d must also vanish over the same region, lest the equation be insoluble. Now, let us assume we have a solutionf of (1); obviously any f =f + J is also a solution of the same equation, if J is a constant.…”

Section: Well-posedness Without Dirichlet Boundariesmentioning

confidence: 99%

Solving the Einstein constraints in periodic spaces with a multigrid approach

Bentivegna

2013

Class. Quantum Grav.

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Abstract. Novel applications of Numerical Relativity demand for more flexible algorithms and tools. In this paper, I develop and test a multigrid solver, based on the infrastructure provided by the Einstein Toolkit, for elliptic partial differential equations on spaces with periodic boundary conditions. This type of boundary often characterizes the numerical representation of cosmological models, where space is assumed to be made up of identical copies of a single fiducial domain, so that only a finite volume (with periodic boundary conditions at its edges) needs to be simulated. After a few tests and comparisons with existing codes, I use the solver to generate initial data for an infinite, periodic, cubic black-hole lattice.

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The constraint equations for the Einstein-scalar field system on compact manifolds

Cited by 56 publications

References 38 publications

Effective multiplicity for the Einstein-scalar field Lichnerowicz equation

Effective multiplicity for the Einstein-scalar field Lichnerowicz equation

A Variational Analysis of Einstein–Scalar Field Lichnerowicz Equations on Compact Riemannian Manifolds

Solving the Einstein constraints in periodic spaces with a multigrid approach

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