[Paper Review] The global nonlinear stability of Minkowski space for self-gravitating massive fields. The wave-Klein-Gordon model

Philippe G. LeFloch, Yue Ma|arXiv (Cornell University)|Jul 4, 2015

Advanced Mathematical Physics Problems36 references80 citations

TL;DR

This paper establishes the global nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields by extending the Hyperboloidal Foliation Method to a wave-Klein-Gordon system on a curved background. The authors prove global existence of small-data solutions using Lorentz-invariant energy norms, refined sup-norm estimates, and a bootstrap argument that overcomes the loss of scaling symmetry due to the mass term, achieving sharp time decay and uniform energy bounds for the coupled system.

ABSTRACT

The Hyperboloidal Foliation Method (introduced by the authors in 2014) is extended here and applied to the Einstein equations of general relativity. Specifically, we establish the nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields, while existing methods only apply to massless scalar fields. First of all, by analyzing the structure of the Einstein equations in wave coordinates, we exhibit a nonlinear wave-Klein-Gordon model defined on a curved background, which is the focus of the present paper. For this model, we prove here the existence of global-in-time solutions to the Cauchy problem, when the initial data have sufficiently small Sobolev norms. A major difficulty comes from the fact that the class of conformal Killing fields of Minkowski space is significantly reduced in presence of a massive scalar field, since the scaling vector field is not conformal Killing for the Klein-Gordon operator. Our method relies on the foliation (of the interior of the light cone) of Minkowski spacetime by hyperboloidal hypersurfaces and uses Lorentz-invariant energy norms. We introduce a frame of vector fields adapted to the hyperboloidal foliation and we establish several key properties: Sobolev and Hardy-type inequalities on hyperboloids, as well as sup-norm estimates which correspond to the sharp time decay for the wave and the Klein-Gordon equations. These estimates allow us to control interaction terms associated with the curved geometry and the massive field, by distinguishing between two levels of regularity and energy growth and by a successive use of our key estimates in order to close a bootstrap argument.

Motivation & Objective

• Establish the global nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields, a problem previously open due to the lack of scaling symmetry.
• Extend the Hyperboloidal Foliation Method to handle the coupled wave-Klein-Gordon system arising from the Einstein equations in wave coordinates.
• Overcome the breakdown of the scaling vector field's conformal Killing property in the massive case, which invalidates classical vector field methods.
• Develop sharp pointwise decay estimates and Sobolev-type inequalities adapted to hyperboloidal foliations for both wave and Klein-Gordon equations.
• Close a bootstrap argument using refined energy and sup-norm estimates to control nonlinear interactions in the curved geometry.

Proposed method

• Apply the Hyperboloidal Foliation Method to foliate Minkowski spacetime by hyperboloids, which are asymptotic to the same light cone and preserve Lorentz invariance.
• Introduce a frame of vector fields adapted to the hyperboloidal foliation to analyze the structure of the wave-Klein-Gordon system on a curved background.
• Establish sharp sup-norm estimates for solutions to the wave and Klein-Gordon equations, corresponding to optimal time decay rates.
• Derive Sobolev and Hardy-type inequalities on hyperboloids to control derivatives and low-frequency components in the energy estimates.
• Use a two-level energy growth strategy, distinguishing between low and high regularity regimes, to manage nonlinear interaction terms.
• Implement a bootstrap argument relying on refined estimates for source terms, including nonlinearities like $ P^{\alpha\beta} \partial_\alpha v \partial_\beta v $ and $ R v^2 $, to close the energy bounds.

Experimental results

Research questions

• RQ1 Can the Hyperboloidal Foliation Method be extended to handle the wave-Klein-Gordon system arising from self-gravitating massive fields in general relativity?
• RQ2 What are the sharp pointwise decay rates for solutions to the wave and Klein-Gordon equations on hyperboloidal hypersurfaces in Minkowski spacetime?
• RQ3 How can energy estimates be controlled in the absence of the scaling vector field, which is no longer conformal for the massive Klein-Gordon operator?
• RQ4 What modifications are needed in the classical vector field method to accommodate the reduced conformal symmetry in the massive case?
• RQ5 How can nonlinear interaction terms involving the curved geometry and the massive field be controlled globally in time using refined sup-norm and energy estimates?

Key findings

• The paper proves the global existence of small-data solutions to the wave-Klein-Gordon system on a curved background derived from the Einstein equations in wave coordinates.
• Sharp time decay rates of $ s^{-1} $ for the wave and $ s^{-1/2} $ for the Klein-Gordon equation are established via refined sup-norm estimates on hyperboloids.
• Uniform energy bounds are achieved for all derivatives up to order $ N $, with energy growth controlled by $ s^{k\delta} $ for $ k $-th order derivatives.
• Nonlinear interaction terms such as $ P^{\alpha\beta} \partial_\alpha v \partial_\beta v $ and $ R v^2 $ are shown to be bounded in $ L^2 $ norm by $ (C_1 \varepsilon)^2 s^{-1 + k\delta} $, ensuring integrability.
• The bootstrap argument closes under the assumptions $ C_1 \geq 4CC_0 $ and $ \varepsilon \leq (4CC_1)^{-1} $, yielding global existence and uniform decay.
• By avoiding the scaling vector field, the method successfully handles the massive case where classical Klainerman-type methods fail, marking a significant extension of the nonlinear stability theory.