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Linear Stability of Slowly Rotating Kerr Black Holes

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Linear Stability of Slowly Rotating Kerr Black Holes LINEAR STABILITY OF SLOWLY ROTATING KERR BLACK HOLES DIETRICH HAFNER,¨ PETER HINTZ, AND ANDRAS´ VASY Abstract. We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equations: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in a natural wave map/DeTurck gauge and show that the pure gauge term can be taken to lie in a fixed 7-dimensional space with a simple geometric interpretation. Our proof rests on a robust general framework, based on recent advances in microlocal analysis and non-elliptic Fredholm theory, for the analysis of resolvents of operators on asymptotically flat spaces. With the mode stability of the Schwarzschild metric as well as of certain scalar and 1-form wave operators on the Schwarzschild spacetime as an input, we establish the linear stability of slowly rotating Kerr black holes using perturbative arguments; in particular, our proof does not make any use of special algebraic properties of the Kerr metric. The heart of the paper is a detailed description of the resolvent of the linearization of a suitable hyperbolic gauge-fixed Einstein operator at low energies. As in previous work by the second and third authors on the nonlinear stability of cosmological black holes, constraint damping plays an important role. Here, it eliminates certain pathological generalized zero energy states; it also ensures that solutions of our hyperbolic formulation of the linearized Einstein equations have the stated asymptotics and decay for general initial data and forcing terms, which is a useful feature in nonlinear and numerical applications. Contents 1. Introduction ........................................................... 3 1.1. Ingredients of the proof .............................................. 8 1.1.1. Mode stability ................................................... 9 1.1.2. Zero energy modes; resolvent near zero ........................... 10 1.1.3. Perturbation to Kerr metrics ..................................... 12 1.2. Further related work ................................................. 13 1.3. Future directions .................................................... 14 1.4. Outline of the paper ................................................. 14 Acknowledgments ......................................................... 15 2. b- and scattering structures .......................................... 15 3. The spacetime manifold and the Kerr family of metrics . 20 3.1. The Schwarzschild metric ............................................ 20 3.2. The Kerr family ..................................................... 21 Date: August 30, 2020. Original version: June 3, 2019. 2010 Mathematics Subject Classification. Primary 83C05, 58J50, Secondary 83C57, 35B40, 83C35. Key words and phrases. Einstein's equation, black hole stability, constraint damping, low energy resolvent. 1 2 DIETRICH HAFNER,¨ PETER HINTZ, AND ANDRAS´ VASY 3.3. Stationarity, vector bundles, and geometric operators ................. 24 3.4. Properties of the null-geodesic flow ................................... 30 4. The gauge-fixed Einstein operator .................................. 32 4.1. The unmodified gauge-fixed Einstein operator ........................ 32 4.2. Constraint damping and the modified gauge-fixed Einstein operator . 38 5. Spherical harmonic decompositions ................................. 39 5.1. Spherical harmonics: functions, 1-forms, symmetric 2-tensors ......... 39 5.2. Decompositions on spacetime ........................................ 39 6. Mode analysis of the scalar wave operator ......................... 40 6.1. Growing zero modes ................................................. 43 7. Mode analysis of the 1-form wave operator ........................ 45 7.1. Absence of modes in Im σ > 0 ........................................ 48 7.2. Absence of real non-zero frequency modes ............................ 48 7.3. Description of zero modes ............................................ 50 7.4. Growing and generalized zero modes ................................. 53 8. Mode stability of the Schwarzschild metric ......................... 59 8.1. Scalar type perturbations ............................................ 61 8.1.1. Modes with l ≥ 2 ................................................ 62 8.1.2. Modes with l = 1 ................................................ 65 8.1.3. Spherically symmetric modes (l = 0) ............................. 67 8.2. Vector type perturbations ............................................ 68 8.2.1. Modes with l ≥ 2 ................................................ 68 8.2.2. Modes with l = 1 ................................................ 70 9. Modes of the unmodified linearized gauge-fixed Einstein operator 71 9.1. Modes in Im σ ≥ 0 ................................................... 71 9.2. Generalized stationary modes: linear growth ......................... 75 9.3. Generalized stationary modes: quadratic growth ...................... 79 10. Constraint damping (CD) .......................................... 81 10.1. Zero frequency improvements ....................................... 82 10.2. Mode stability of the gauge propagation operator in Im σ ≥ 0 . 87 11. Structure of the resolvent of the linearized modified gauge-fixed Einstein operator ......................................................... 93 11.1. Existence of the resolvent; rough description near zero energy . 94 11.2. Precise structure of the resolvent near zero energy ................... 101 12. Regularity of the resolvent in the spectral parameter . 108 12.1. Regularity at low frequencies ........................................ 108 12.2. Regularity at intermediate and high frequencies ..................... 116 12.3. Conormal regularity of the resolvent near zero ....................... 116 LINEAR STABILITY OF KERR BLACK HOLES 3 13. Decay estimates ..................................................... 119 14. Proof of linear stability ............................................. 123 References ................................................................. 129 1. Introduction We continue our investigation of stability problems in general relativity from a system- atic microlocal and spectral theoretic point of view. In previous work [HV18b, Hin18a], the second and third authors proved the full nonlinear stability of slowly rotating Kerr{de Sit- ter (KdS), resp. Kerr{Newman{de Sitter (KNdS) black holes as solutions of the Einstein vacuum equations, resp. Einstein{Maxwell equations, with positive cosmological constant Λ > 0. The proofs of these results rest on the completion of two main tasks: (1) control of asymptotics and decay of tensor-valued linear waves on exact slowly ro- tating KdS spacetimes via spectral theory/resonance analysis|we were in fact able to deduce the structure of resonances as well as mode stability of slowly rotating KdS black holes from that of spherically symmetric Schwarzschild{de Sitter (SdS) spacetimes; (2) robust control of the regularity of linear waves on asymptotically KdS spacetimes via microlocal analysis on the spacetime. (Combined with the spectral theoretic results on exact KdS spacetimes, this gives precise regularity and decay results for waves on asymptotically KdS spacetimes.) The present paper completes the first task on slowly rotating Kerr spacetimes: we show that solutions of the linearization of the Einstein vacuum equations around a slowly rotating Kerr solution decay at an inverse polynomial rate to a linearized Kerr metric, plus a pure gauge solution which, in a linearized wave map gauge, lies in an (almost) explicit 7-dimensional vector space. More precisely, recall that the metric of a Schwarzschild black hole with mass m > 0 is given in static coordinates by 2m 2m−1 g = 1 − dt2 − 1 − dr2 − r2g;= t 2 ; r 2 (2m; 1); (m;0) r r R 2 where g= is the standard metric on S [Sch16]. The more general Kerr family of metrics 3 g(m;a) [Ker63] depends in addition on the angular momentum a 2 R . These metrics are solutions of the Einstein vacuum equations1 Ric(g) = 0: (1.1) 4 Fix a mass parameter m0 > 0 and set b0 = (m0; 0) 2 R . Restricting to Kerr black hole 4 parameters b = (m; a) 2 R close to b0, we can regard gb as a smooth family of stationary 1As written, this is a single equation for a symmetric 2-tensor g. However, we follow standard terminology and refer to this as the Einstein equations in light of the fact that (1.1) is a coupled system of 10 nonlinear equations for the coefficients of g expressed in local coordinates. 4 DIETRICH HAFNER,¨ PETER HINTZ, AND ANDRAS´ VASY (time-independent) Lorentzian metrics on a fixed 4-dimensional manifold ◦ 2 M := Rt × [r−; 1) × S ; where r− < 2m0. The level sets of t here are equal to those of t in r > 4m0, i.e. far away from the black hole, and are regular and transversal to the future event horizon H+, which for b = b0 is located at the Schwarzschild radius r = 2m0. Linearizing equation (1.1) for g = gb in the parameters b, we see that the linearized Kerr metrics d _ _ 4 g_b(b) := gb+sb_ ; b 2 R ; ds s=0 _ are solutions of the linear equation Dgb Ric g_b(b) = 0. Our main result concerns the long-time behavior of general solutions of the linearized Einstein vacuum equations Dgb Ric(h) = 0: (1.2) To describe it, recall that the non-linear equation (1.1) admits a formulation as a Cauchy problem [CB52, CBG69]: fix a Cauchy surface ◦ −1 ◦ Σ0 = t (0) ⊂ M : ◦ Then the initial data are a Riemannian metric γ and a symmetric 2-tensor
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