Previous and related work

The aforementioned papers [140], [75], [76]—on which the analysis of the present paper directly builds—and our general philosophy to the study of waves on black hole

space-times, mostly with Λ>0, build on a host of previous works.

On Schwarzschild–de Sitter space, Bachelot [6] set up the functional analytic scat-tering theory, and S´a Barreto–Zworski [123] and Bony–H¨afner [17] studied resonances and exponential wave decay away from the event horizon; Melrose, S´a Barreto and Vasy [108] proved exponential decay to constants across the horizons.

The precise study of waves on rotating Kerr–de Sitter spacetimes requires an anal-ysis at normally hyperbolically trapped sets, which was first accomplished in the break-through work of Wunsch–Zworski [147]. This was later extended and simplified by Nonnenmacher–Zworski [115] and Dyatlov [56]; see also [74]. This enabled Dyatlov to obtain full asymptotic expansions for linear waves on exact, slowly rotating Kerr–

de Sitter spaces into quasinormal modes (resonances) [53], following his earlier work on exponential energy decay [51], [52]; see also the more recent [54].

Using rather different, physical space, techniques, Dafermos–Rodnianski [43] proved super-polynomial energy decay on Schwarzschild–de Sitter spacetimes. Such techniques were also used by Schlue in his analysis of linear waves in thecosmological part of Kerr–

de Sitter spacetimes [127], and by Keller for the Maxwell equation [88]. We furthermore mention Warnick’s physical space approach to the study of resonances [145].

Regarding work on spacetimeswithout black holes, but in the microlocal spirit, we mention specifically the works [9]–[12].

The general microlocal analytic and geometric framework underlying our global study of asymptotically Kerr–de Sitter-type spaces by compactifying them to manifolds with boundary, which are then naturally equipped with b-metrics, is Melrose’s b-analysis [110]. The considerable flexibility and power of a microlocal point of view is exploited throughout the present paper, especially in§5, §8 and §9. We specifically mention the ease with which bundle-valued equations can be treated, as first noted in [140], and shown concretely in [72], [78], where the authors prove decay to stationary states for Maxwell’s (and more general) equations. We also point out that a stronger notion of normal hyperbolicity, calledr-normal hyperbolicity—which is stable under perturbations [79]—was proved for Kerr and Kerr–de Sitter spacetimes in [147], [54], and allows for global results for (non-)linear waves under very general assumptions [140], [76]. Since, as we show, solutions to Einstein’s equations near Kerr–de Sitter always decay to anexact Kerr–de Sitter solution (up to exponentially decaying tails), the flexibility afforded by r-normal hyperbolicity is not used here.

Linear and non-linear wave equations on black hole spacetimes with Λ=0, specifi-cally Kerr and Schwarzschild, have received more attention. They do not directly fit into the general frameworks mentioned above; a fundamental difference is that waves decay at most at a fixed polynomial rate on general asymptotically flat (Λ=0) spacetimes, which

is in stark contrast to the exponential decay rate on spacetimes with asymptotically hy-perbolic (Λ>0) ends. Directly related to the topic of the present paper is the recent proof of the linear stability of the Schwarzschild spacetime under gravitational perturbations without symmetry assumptions on the data [37], which we already discussed above; a less quantitative version of this was obtained by simpler means by Hung, Keller, and Wang [83]. After pioneering work by Wald [143] and Kay–Wald [87], Dafermos, Rodnianski and Shlapentokh–Rothman [42], [44] recently proved polynomial decay for the scalar wave equation on all (exact) subextremal Kerr spacetimes; Tataru and Tohaneanu [131], [132]

proved Price’s law, i.e. precise polynomial decay rates, for slowly rotating Kerr space-times, and Marzuola, Metcalfe, Tataru and Tohaneanu obtained Strichartz estimates [104], [137]. There is also work by Donninger, Schlag and Soffer [49] onL^∞ estimates on Schwarzschild black holes, followingL^∞ estimates of Dafermos and Rodnianski [40], and of Blue and Soffer [16] on non-rotating charged black holes giving L⁶ estimates.

Apart from [37], bundle-valued (or coupled systems of) equations were studied in par-ticular in the contexts of Maxwell’s equations by Andersson and Blue [15], [4], [5] and Sterbenz–Tataru [130] (see also [84], [50]), and for Dirac equations by Finster, Kamran, Smoller and Yau [58]. Non-linear problems on exterior Λ=0 black hole spacetimes were studied by Dafermos, Holzegel and Rodnianski [38], who constructedbackward solutions of the Einstein vacuum equations settling down to Kerr exponentially fast (regarding this point, see also [45]); for forward problems, Dafermos [35], [36] studied the non-linear Einstein–Maxwell–scalar field system under the assumption of spherical symmetry. We also mention Luk’s work [101] on semi-linear equations on Kerr, as well as the steps towards understanding a model problem related to Kerr stability under the assumption of axial symmetry [85].

A fundamental driving force behind a large number of these works is Klainerman’s vector field method [90]; subsequent works by Klainerman and Christodoulou [91], [24]

introduce the ‘null condition’ which plays a major role in the analysis of non-linear in-teractions near the light cone, in particular in (3+1)-dimensional asymptotically flat spacetimes—in the asymptotically hyperbolic case which we study here, there is no ana-logue of this condition.

Using these and related techniques, a number of works prove the global non-linear stability of Minkowski space as a solution to Einstein’s field equations coupled to various matter models; we mention the works by Speck [129] for the Einstein–Maxwell system, LeFloch–Ma [96] for the Einstein equation coupled to a massive scalar field, Taylor [133]

for Einstein–Vlasov, and references therein. There is also a large amount of literature studying stability questions under symmetry assumptions on the spacetime: we only mention the work by Choquet-Bruhat and Moncrief [22], in which they in particular

solve for a (time-dependent) finite-dimensional (Teichm¨uller) parameter, but we point out that this is unrelated to the finite-dimensional gauge issues discussed in§1.1.

There is also ongoing work by Dafermos–Luk [39] on the stability of the interior (‘Cauchy’) horizon of Kerr black holes; note that the black hole interior is largely un-affected by the presence of a cosmological constant, but the a-priori decay assumptions along the event horizon, which determine regularity properties at the Cauchy horizon, are vastly different: the merely polynomial decay rates on asymptotically flat (Λ=0) space-times, as compared to the exponential decay rate on asymptotically hyperbolic (Λ>0) spacetimes, is a low frequency effect, related to the very delicate behavior of the resolvent near zero energy on asymptotically flat spaces. A precise study in the spirit of [140] and [46] is currently in progress [67].

In the physics community, hole perturbation theory, i.e. the study oflinearized per-turbations of black hole spacetimes, has a long history. For us, the most convenient formulation, which we use heavily in§7, is due to Ishibashi, Kodama and Seto [93], [92], [86], building on earlier work by Kodama–Sasaki [94]. The study was initiated in the seminal paper by Regge–Wheeler [118], with extensions by Vishveshwara [142] and Zerilli [149], analyzing metric perturbations of the Schwarzschild spacetime; a gauge-invariant formalism was introduced by Moncrief [113], later extended to allow for coupling with matter models by Gerlach–Sengupta [62] and Martel–Poisson [103]. A different approach to the study of gravitational perturbations, relying on the Newman–Penrose formalism [114], was pursued by Bardeen–Press [7] and Teukolsky [136], who discovered that certain curvature components satisfy decoupled wave equations; their mode stability was proved by Whiting [146]. We refer to Chandrasekhar’s monograph [19] for a more detailed account.

For surveys of numerical investigations of quasinormal modes, often with the goal of quantifying the phenomenon of ringdown discussed in §1.2, we refer the reader to the articles [95] and [13], and the references therein. We also mention the paper by Dyatlov–Zworski [57] connecting recent mathematical advances in particular related to quasinormal modes with the physics literature.