In this paper, we provide an elementary, unified treatment of two distinct blue-shift instabilities for the scalar wave equation on a fixed Kerr black hole background: the celebrated blue-shift at the Cauchy horizon (familiar from the strong cosmic censorship conjecture) and the time-reversed red-shift at the event horizon (relevant in classical scattering theory). Our first theorem concerns the latter and constructs solutions to the wave equation on Kerr spacetimes such that the radiation field along the future event horizon vanishes and the radiation field along future null infinity decays at an arbitrarily fast polynomial rate, yet, the local energy of the solution is infinite near any point on the future event horizon. Our second theorem constructs solutions to the wave equation on rotating Kerr spacetimes such that the radiation field along the past event horizon (extended into the black hole) vanishes and the radiation field along past null infinity decays at an arbitrarily fast polynomial rate, yet, the local energy of the solution is infinite near any point on the Cauchy horizon. The results make essential use of the scattering theory developed in Dafermos, Rodnianski and Shlapentokh-Rothman (A scattering theory for the wave equation on Kerr black hole exteriors (2014). arXiv:1412.8379) and exploit directly the time-translation invariance of the scattering map and the non-triviality of the transmission map.
Let us also note several other classical attempts in the physics literature to understand the blue-shift instability at the Cauchy horizon @cite_34 @cite_4 @cite_9 . In the case of Reissner--Nordstr " o m or Kerr, the local red-shift along @math and the local blue-shift along @math both vanish. This leads to fundamentally different expectations for the qualitative behavior of waves in the black hole exterior and interior, see @cite_35 @cite_10 @cite_28 @cite_6 @cite_0 @cite_12 for the current state of the art. We note, however, that even the question of boundedness for general solutions to on extremal Kerr exteriors remains an open problem (see the discussion in @cite_25 ).
@cite_35 @cite_4 @cite_28 @cite_9 @cite_6 @cite_0 @cite_34 @cite_10 @cite_25 @cite_12
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The stability of the inner Reissner-Nordstroem geometry is studied with test massless integer-spin fields. In contrast to previous mathematical treatments we present physical arguments for the processes involved and show that ray tracing and simple first-order scattering suffice to elucidate most of the results. Monochromatic waves which are of small amplitude and ingoing near the outer horizon develop infinite energy densities near the inner Cauchy horizon (as measured by a freely falling observer). Previous work has shown that certain derivatives of the field in a general (nonmonochromatic) disturbance must fall off exponentially near the inner (Cauchy) horizon (r = r sub - ) if energy densities are to remain finite. Thus the solution is unstable to physically reasonable perturbations which arise outside the black hole because such perturbations, if localized near past null infinity (I sup - ), cannot be localized near r sub + , the outer horizon. The mass-energy of an infalling disturbance would generate multipole moments on the black hole. Price, Sibgatullin, and Alekseev have shown that such moments are radiated away as ''tails'' which travel outward and are rescattered inward yielding a wave field with a time dependence t sup -p , p > 0. This decay in time is sufficiently slow that themore » tails yield infinite energy densities on the Cauchy horizon. (The amplification of the low-frequency tails upon interacting with the time-dependent potential between the horizons is an important feature guaranteeing the infinite energy density.) The interior structure of the analytically extended solution is thus disrupted by finite external disturbances. have further shown that even perturbations which are localized as they cross the outer horizon produce singularities at the inner horizon. It is shown that this singularity arises when the incoming radiation is first scattered just inside the outer horizon« less These lecture notes, based on a course given at the Zurich Clay Summer School (June 23-July 18, 2008), review our current mathematical understanding of the global behaviour of waves on black hole exterior backgrounds. Interest in this problem stems from its relationship to the non-linear stability of the black hole spacetimes themselves as solutions to the Einstein equations, one of the central open problems of general relativity. After an introductory discussion of the Schwarzschild geometry and the black hole concept, the classical theorem of Kay and Wald on the boundedness of scalar waves on the exterior region of Schwarzschild is reviewed. The original proof is presented, followed by a new more robust proof of a stronger boundedness statement. The problem of decay of scalar waves on Schwarzschild is then addressed, and a theorem proving quantitative decay is stated and its proof sketched. This decay statement is carefully contrasted with the type of statements derived heuristically in the physics literature for the asymptotic tails of individual spherical harmonics. Following this, our recent proof of the boundedness of solutions to the wave equation on axisymmetric stationary backgrounds (including slowly-rotating Kerr and Kerr-Newman) is reviewed and a new decay result for slowly-rotating Kerr spacetimes is stated and proved. This last result was announced at the summer school and appears in print here for the first time. A discussion of the analogue of these problems for spacetimes with a positive cosmological constant follows. Finally, a general framework is given for capturing the red-shift effect for non-extremal black holes. This unifies and extends some of the analysis of the previous sections. The notes end with a collection of open problems. We consider solutions to the linear wave equation in the interior region of extremal Reissner–Nordstrom black holes. We show that, under suitable assumptions on the initial data, the solutions can be extended continuously beyond the Cauchy horizon and, moreover, that their local energy is finite. This result is in contrast with previously established results for subextremal Reissner–Nordstrom black holes, where the local energy was shown to generically blow up at the Cauchy horizon. Abstract We prove that a large class of smooth solutions ψ to the linear wave equation □ g ψ = 0 on subextremal rotating Kerr spacetimes which are regular and decaying along the event horizon become singular at the Cauchy horizon. More precisely, we show that assuming appropriate upper and lower bounds on the energy along the event horizon, the solution has infinite (non-degenerate) energy on any spacelike hypersurfaces intersecting the Cauchy horizon transversally. Extrapolating from known results in the Reissner–Nordstrom case, the assumed upper and lower bounds required for our theorem are conjectured to hold for solutions arising from generic smooth and compactly supported initial data on a Cauchy hypersurface. This result is motivated by the strong cosmic censorship conjecture in general relativity.