Future stability of the 1  +  3 Milne model for the Einstein–Klein–Gordon system

Abstract

We study the small perturbations of the 1 + 3-dimensional Milne model for the Einstein-Klein-Gordon (EKG) system. We prove the nonlinear future stability, and show that the perturbed spacetimes are future causally geodesically complete. For the proof, we work within the constant mean curvature (CMC) gauge and focus on the 1 + 3 splitting of the Bianchi-Klein-Gordon equations. Moreover, we treat the Bianchi-Klein-Gordon equations as evolution equations and establish the energy scheme in the sense that we only commute the Bianchi-Klein-Gordon equations with the spatially covariant derivatives while the normal derivative is not allowed. Contents1 2 J. WANG 7.1. Closing the bootstrap argument 53 Appendix A. Local in time development of CMC data 58 Appendix B. ODE estimate 61 Appendix C. Basic geometric conventions and identities 61 References 68 the future direction is locally isometric to the k = −1 vacuum Friedmann-Lemîatre-Robertson-Walker (FLRW) model. Andersson and Moncrief [6] first consider the nonlinear stability of this model (called hyperbolic cone spacetimes there), and show that for the constant mean curvature (CMC) Cauchy data for the vacuum Einstein equations close to the standard data for the hyperbolic cone spacetimes, the maximal future Cauchy development is globally foliated by the CMC Cauchy surfaces and causally geodesically complete to the future. This also motivates the current project. We refer to [1, 2, 3, 17] and references therein for more backgrounds on the existence of CMC foliations. There are several results concerning the decay of the KG field [19, 20, 30, 31]. In contrast to the wave equation, the KG equation is not conformal invariant and hence does not commute well with the symmetry of scaling. Due to this fact, Klainerman [20] employs the hyperbloid foliations which respect the Lorentz invariance of the KG operator and derives the asymptotic behavior of the KG field by the energy method alone. Furthermore, Katayama [19] somehow overcomes the incompatibility between the wave and KG field. Other method including Fourier analysis for the wave-Klein-Gordon system is established by Ionescu-Pausader [18]. Global solutions for the semilinear KG equations in the FLRW spacetimes are studied by Galstian-Yagdjian [16]. Let us review some nonlinear stability results of the Minkowski spacetime. Christodoulou and Klainerman [13] initiate a covariant proof based on the Bel-Robinson energy. Their proof relies upon the geometric foliations of spacetime, including maximal t = const slices and a family outgoing null cones, and null condition hidden inside the Bianchi equations. Lindblad and Rodnianski [21,22] devise another proof based on the wave coordinate gauge, under which the Einstein equations satisfy the weak null condition. This method is extended to the case of massive Einstein-Vlasov system by Lindblad and Taylor [23]. All of these works require the full symmetries of the Minkowski spacetime, including the conformal symmetries, suggesting that these methods can not apply to the EKG system st...

Attractors of the Einstein-Klein-Gordon system

“…The system emerges as a projection of the Einstein equations on massive modes of the Fourier expansion of five-dimensional Kaluza-Klein metrics [19]. Independently, the system poses an interesting mathematical model as it consists of a system of quasilinear massless and massive wave equations.…”

“…The present paper considers the stability problem of the Milne model in the expanding direction as a solution to the Einstein-Klein-Gordon system. The EKGS has been studied intensely in recent years with an emphasis on the nonlinear stability problem [17,19,20] (cf. also [21] where the decay of the KG field on a Kerr-AdS background is investigated).…”

“…Recently, the stability of the Milne model has been generalized to the presence of a variety of matter models such as collisionless matter [12] by Andersson and the first author, fields emanating from generalized Kaluza-Klein spacetimes, in particular electromagnetic fields [8] by Branding, the first author and Kr€ oncke and also Klein-Gordon fields [19] by Wang. While the former two works use the CMCSH gauge to control the evolution of perturbations, in [19] a CMC-vanishing-shift gauge and Bel-Robinson-type energies (cf. [23]) are used.…”

“…to the Milne model and Minkowski space, several results appeared recently that generalize these works to the non-vacuum setting. The matter models that have been considered in these generalizations include Maxwell fields [7][8][9][10][11], collisionless matter [12][13][14][15][16] and scalar fields [17][18][19][20], in particular Klein-Gordon fields, which are the subject of this paper.…”

On the oscillations and future asymptotics of locally rotationally symmetric Bianchi type III cosmologies with a massive scalar field*

“…apply due to oscillations entering the system via the Klein-Gordon equation. Nevertheless there have been successful enquiries going this route, such as the recent works on various nonlinear stability problems for the Einstein-Klein-Gordon system concerning Minkowski spacetime [12,14,28] and the Milne model [10,27]. While those results concern vacuum dominated future asymptotics, in the present work we consider a class of models where the future is matter dominated.…”