The non-linear stability of the Schwarzschild family of black holes

Abstract

We prove the non-linear asymptotic stability of the Schwarzschild family as solutions to the Einstein vacuum equations in the exterior of the black hole region: general vacuum initial data-with no symmetry assumedsufficiently close to Schwarzschild data evolve to a vacuum spacetime which (i) possesses a complete future null infinity I + (whose past J − (I + ) is moreover bounded by a regular future complete event horizon H + ),(ii) remains close to Schwarzschild in its exterior, and(iii) asymptotes back to a member of the Schwarzschild family as an appropriate notion of time goes to infinity, provided that the data are themselves constrained to lie on a teleologically constructed codimension-3 "submanifold" of moduli space. This is the full nonlinear asymptotic stability of Schwarzschild since solutions not arising from data lying on this submanifold should by dimensional considerations approach a Kerr spacetime with rotation parameter a = 0, i.e. such solutions cannot satisfy (iii). The statement is effective, providing quantitative bounds from explicit initial data quantities, and the global nearness to Schwarzschild at top order can be measured with respect to the same quantity as initial data, i.e. without loss of derivatives. The proof employs teleologically normalised double null gauges, is expressed entirely in physical space and makes essential use of the analysis in our previous study of the linear stability of the Kerr family around Schwarzschild [DHR], as well as techniques developed over the years to control the non-linearities of the Einstein equations, in particular in the difficult radiation zone associated to subtle non-linear effects like Christodoulou memory. The present work, however, is entirely self-contained. In view of the recent [DHR19, TdCSR20], our approach can be applied to the full non-linear asymptotic stability of the subextremal Kerr family.cone which is a level surface of the exterior optical function. This is still compatible with decay of all quantities for fixed r since the "interior" region was covered by a different optical functions whose null cones were not related to the null cones of the exterior optical function. In the exterior region of [CK93], on the other hand, the r decay is sufficient to compensate for the lack of decay for some fully rescaled quantities in u, so as for all nonlinearities to still be controllable.We begin with the point p ∈ S and v ∈ T p S. By a well known result of Klingenberg [Kli59], if in (4.4.1) is sufficiently small, there exists a geodesic polar neighbourhood S p, 7 8 π ⊂ S around p of, say, radius 7 8 π. Let us denote the chart by the diffeomorphism ϕ p, 7 8 π : S p, 7 8 π → D 7 8 π .Associated to this chart are geodesic polar coordinates which we shall denote as (θ, φ). These are uniquely determined by declaring that φ = 0 corresponds to the direction ∂ v and the frame ∂ θ , ∂ φ is compatible with the orientation where defined. Let γ(s) denote the geodesic in S with initial condition γ(0) = p, γ (0) = v. This is parametrised by arc length. ...

Instability of the Kerr Cauchy horizon under linearised gravitational perturbations

“…This difference impacts a priori on relating the coefficients in the separated picture to the Teukolsky transform of the characteristic initial data (more about this later) and also on the regularity of the transmission and reflection coefficients: as before we can write Recall that the structure of the blow-up argument only requires information on the frequency regime near 15 Recall that ψ degenerates (vanishes) at H l . 16 Here, λ rss ml pωq denotes the eigenvalue associated to the eigenfunction Y rss ml p¨; ωq of the spin 2-weighted spheroidal Laplacian, see Section 5.1.…”

“…See[40],[16],[41] for recent results on the black hole stability problem 2. See also[45] for the linearised case.…”

Linear Stability of Black Holes and Naked Singularities

“…We should recall at this point that such a proof only exists for: (i) de Sitter spacetime (proved by H. Friedrich in 1986 [26,27]), (ii) Minkowski spacetime (proved by Christdoulou and Klainerman in 1993 [16]), (iii) Schwarzschild de Sitter BH, (proved by Hintz and Vasy in 2016 [32]). A preprint is available since a few months ago with a proof of non-linear stability of the Λ = 0 Schwarzschild BH (see [18]), the proof takes five hundred pages. For the Kerr BH, at the moment, we must content ourselves with a proof of linear stability.…”

“…The analysis of the stability of black hole outer regions, having passed decades ago the test of modal linear stability, has had, after a long period of little activity, a remarkable progress in the last few years. An incomplete list of recent advances related to the stability of the Schwarschild and Kerr black holes follow: (i) for Λ ≥ 0 Schwarzschild black holes the nonmodal linear stability was established in [24,25]; (ii) in the Λ = 0 case, the decay in time of generic linear perturbations of the Schwarzschild black hole, leaving a "slowly" rotating Kerr black hole was proved in [17]; (iii) the conditional stability of the Λ < 0 Schwarzschild black hole, and the breaking of the even/odd symmetry mediated by the Chandrasekhar operators (93) and (94) was studied in [6]; (iv) the non-linear stability of the Schwarzschild de Sitter black hole was proved in [32]; (v) a preprint is now available with a proof of the non-linear stability of the Λ = 0 Schwarzschild black hole [18]; (vi) pointwise decay estimates for solutions of the linearized Einstein's equations on the outer region of a Kerr black hole were obtained in [1]; (vii) the role of hidden symmetries (see the review [28]) in type D spacetimes, and the reconstruction of (gravitational, Maxwell and spinor) perturbation fields from "Debye potentials" (first introduced in [37,47]), was studied in depth and made clear in the series of papers [2][3][4][5].…”

Nonlinear stability of the slowly-rotating Kerr-de Sitter family

“…The stability of Minkowski was first shown in the breakthrough result of Christodoulou-Klainerman in [5]. Following substantial developments in the field, nonlinear stability of Schwarzschild was then shown by Klainerman-Szeftel in [15], and by Dafermos-Holzegel-Rodnianski in [6]. Most recently, Klainerman-Szeftel were able to obtain a nonlinear stability result for the slowly-rotating Kerr family [16].…”