Stable fixed points of the Einstein flow with positive cosmological constant

Fajman, David; Kroencke, Klaus

doi:10.48550/arxiv.1504.00687

Cited by 7 publications

(14 citation statements)

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“…Thus the smallness of R 0ij + 2g 0ij H 3 (Σ,g 0 ) implies smallness for g 0ij − γ ij H 5 (Σ,γ) and hence the rescaled version of (1.3) entails that g 0ij −γ ij H 5 (Σ,γ) + k 0ij +γ ij H 4 (Σ,γ) ε. Now we can prove along the lines of the corresponding argument presented in [29,15] to show the existence of a CMC surface in the non vacuum setting. Therefore, the data in Theorem 1.1 can be generalized to be non-CMC.…”

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confidence: 88%

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

Wang

2019

Class. Quantum Grav.

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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...

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confidence: 88%

“…(1.4)- (1.5) for the rescaling). To match the condition in [15], one needs the smallness for g 0 − γ H s (Σ,γ) + k 0 + γ H s−1 (Σ,γ) , s > 3/2 + 1, where γ is the hyperbolic metric. We take s = 5 and choose the harmonic coordinates with respect to g 0 and γ.…”

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confidence: 99%

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

Wang

2019

Class. Quantum Grav.

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“…Considering sufficiently small initial data which is not necessarily CMC, the maximal globally hyperbolic development under the Einstein-Vlasov system is, locally in time, as close to the background geometry as desired in a suitable regularity [Ri]. The existence of a CMC surface in such a spacetime can be shown along the lines of the corresponding argument in the vacuum case presented for instance in [FK15]. 10.3.…”

Section: Global Existence and Completenessmentioning

confidence: 99%

Nonlinear Stability of the Milne Model with Matter

Andersson

Fajman

2020

Commun. Math. Phys.

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We show that any 3+1-dimensional Milne model is future nonlinearly, asymptotically stable in the set of solutions to the Einstein-Vlasov system. For the analysis of the Einstein equations we use the constantmean-curvature-spatial-harmonic gauge. For the distribution function the proof makes use of geometric L 2 -estimates based on the Sasaki-metric. The resulting estimates on the energy momentum tensor are then upgraded by employing the natural continuity equation for the energy density.

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“…With the spacetime topology of R × M , one has the freedom to choose the spatial slice as long as it is diffeomorphic to a Cauchy hypersurface. Let M be a Cauchy hypersurface with an induced metric g which together with (k, N, X) satisfies the Einstein evolution and constraint equations (7)(8)(9)(10). Now let φ be a an element of the identity component of the diffeomorphism group (D 0 ) of M .…”

Section: Field Equations and Gauge Fixingmentioning

confidence: 99%

“…In the case of '2+1' Einstein flow on R × S genus with genaus > 1, Teichmüller space plays the role of the Einstein moduli space and special techniques [3,12] are used to study the global existence (by utilizing the properness of the Dirichlet energy functional defined on the Teichmüller space). Recently [9] studied the Lyapunov stability of these background solutions including a positive cosmological constant. However, in order to establish the 'attractor' property of the background solutions, it is necessary to prove the asymptotic stability.…”

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confidence: 99%

Attractors of the `n+1' dimensional Einstein-$Λ$ flow

Mondal

2019

Preprint

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Here we prove a global existence theorem for sufficiently small however fully nonlinear perturbations of a family of background solutions of 'n + 1' vacuum Einstein equations in the presence of a positive cosmological constant Λ. The future stability of vacuum solutions in the small data and zero cosmological constant limit has been studied previously for both '3 + 1' and higher dimensional spacetimes. However, with the advent of dark energy driven accelerated expansion of the universe, it is of fundamental importance in mathematical cosmology to include a positive cosmological constant, the simplest form of the dark energy in the vacuum Einstein equations. Such Einsteinian evolution is entitled as the 'Einstein-Λ' flow. We study the background solutions of this 'Einstein-Λ' flow in 'n + 1' dimensional spacetimes in constant mean curvature spatial harmonic gauge, n ≥ 3 and establish both linear and non-linear stability of such solutions. We implement shadow gauge condition [7] to handle the nontrivial Einstein moduli spaces in n > 3 spatial dimensional case, where the background solution is allowed to vary with time. Such time dependence of the suitably re-scaled background spatial metric introduces modifications to the elliptic equations obtained through imposing gauge conditions. In addition to such modifications, the autonomous character of the suitably re-scaled Einstein flow breaks down as a consequence of including Λ(> 0). We construct a Lyapunov function (controlling suitable norm of the small data) similar to a wave equation type energy for the non-linear non-autonomous evolution of the small data and prove its decay in the direction of cosmological expansion. Our results demonstrate the future stability and geodesic completeness of the perturbed spacetimes, and show that the scale-free geometry converges to an element of the Einstein moduli space (a point for n = 3 and a finite dimensional space for n > 3), which has significant consequence on the cosmic topology while restricting to the case of n = 3.

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Stable fixed points of the Einstein flow with positive cosmological constant

Cited by 7 publications

References 14 publications

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

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

Nonlinear Stability of the Milne Model with Matter

Attractors of the `n+1' dimensional Einstein-$Λ$ flow

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