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.
Another approach to capturing the blue-shift instability at the Cauchy horizon @math is to identify a condition on the solution which ensures blow up at @math . Such a condition was given by Luk--Oh @cite_32 in the Reissner--Nordstr " o m case, who moreover showed that their condition indeed holds for solutions arising from compactly supported data posed on an asymptotically flat, hypersurface. (For some partial results concerning self-gravitating spherically symmetric scalar fields see @cite_18 @cite_27 .) In addition, the work @cite_32 gives an explicit characterization of the genericity assumption in terms of the asymptotics along future null infinity @math . In parallel with the present paper, Luk--Sbierski @cite_15 have obtained a Kerr analogue of the result of @cite_32 relating a polynomial lower bound along @math to infinite local energy at @math . Obtaining a characterization of spacelike initial data for which this lower bound holds remains an open problem. In broad terms, one expects that the strategy of @cite_15 will be applicable in the study of the instability properties of the Cauchy horizon in the full non-linear theory governed by the Einstein vacuum equations (cf. @cite_20 ).
@cite_18 @cite_32 @cite_27 @cite_15 @cite_20
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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. 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 We consider a spherically symmetric, double characteristic initial value problem for the (real) Einstein-Maxwell-scalar field equations. On the initial outgoing characteristic, the data is assumed to satisfy the Price law decay widely believed to hold on an event horizon arising from the collapse of an asymptotically flat Cauchy surface. We establish that the heuristic mass inflation scenario put forth by Israel and Poisson is mathematically correct in the context of this initial value problem. In particular, the maximal future development has a future boundary over which the space-time is extendible as a C0 metric but along which the Hawking mass blows up identically; thus, the space-time is inextendible as a C1 metric. In view of recent results of the author in collaboration with I. Rodnianski, which rigorously establish the validity of Price's law as an upper bound for the decay of scalar field hair, the C0 extendibility result applies to the collapse of complete, asymptotically flat, spacelike initial data where the scalar field is compactly supported. This shows that under Christodoulou's C0 formulation, the strong cosmic censorship conjecture is false for this system. © 2005 Wiley Periodicals, Inc. Abstract We study the Cauchy problem for the wave equation □ g ψ = 0 on extreme Kerr backgrounds. Specifically, we consider regular axisymmetric initial data prescribed on a Cauchy hypersurface Σ 0 which connects the future event horizon with spacelike or null infinity, and we solve the linear wave equation on the domain of dependence of Σ 0 . We show that the spacetime integral of an energy-type density is bounded by the initial conserved flux corresponding to the stationary Killing field T , and we derive boundedness of the non-degenerate energy flux corresponding to a globally timelike vector field N . Finally, we prove uniform pointwise boundedness and power-law decay for ψ up to and including the event horizon H + .