Decay Properties of Klein‐Gordon Fields on Kerr‐AdS Spacetimes

Holzegel, Gustav; Smulevici, Jacques

doi:10.1002/cpa.21470

Cited by 94 publications

(202 citation statements)

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“…Note that this vanishes as u → ∞, i.e., it vanishes at the bifurcation surface B. 17 The resulting solution in region II is simply Φ out,+ . This vanishes as u → ∞ so it vanishes at H + R and hence matches continuously to the solution in region I.…”

Section: Non-smooth Perturbationsmentioning

confidence: 99%

“…This was first noticed by studying quasinormal modes of such black holes [16]. This slow decay arises from the stable trapping of null geodesics in such spacetimes [17]: there exist null geodesics which orbit the black hole, and such orbits are stable against perturbations. The very slow decay of perturbations outside the black hole is expected to strengthen the instability of the Cauchy horizon.…”

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confidence: 95%

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The BTZ black hole violates strong cosmic censorship

Dias

Reall

Santos

2019

J. High Energ. Phys.

139

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We investigate the stability of the inner horizon of a rotating BTZ black hole. We show that linear perturbations arising from smooth initial data are arbitrarily differentiable at the inner horizon if the black hole is sufficiently close to extremality. This is demonstrated for scalar fields, for massive Chern-Simons fields, for Proca fields, and for massive spin-2 fields. Thus the strong cosmic censorship conjecture is violated by a near-extremal BTZ black hole in a large class of theories. However, we show that a weaker "rough" version of the conjecture is respected. We calculate the renormalized energymomentum tensor of a scalar field in the Hartle-Hawking state in the BTZ geometry. We show that the result is finite at the inner horizon of a near-extremal black hole. Hence the backreaction of vacuum polarization does not enforce strong cosmic censorship.

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Section: Non-smooth Perturbationsmentioning

confidence: 99%

mentioning

confidence: 95%

The BTZ black hole violates strong cosmic censorship

Dias

Reall

Santos

2019

J. High Energ. Phys.

139

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“…(i) Taking into account the proof of Lemma 6.1, we just have to notice that ∂ u ̟, ∂ v ̟ and ∂ u κ are continuous according to (22), (23) and (26). (ii) We just provide a sketch of the proof.…”

Section: Proofmentioning

confidence: 99%

On the global uniqueness for the Einstein–Maxwell-scalar field system with a cosmological constant: I. Well posedness and breakdown criterion

Costa

Girão

Natário

et al. 2014

Class. Quantum Grav.

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This paper is the first part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the EinsteinMaxwell-scalar field system with a cosmological constant Λ, with the data on the outgoing initial null hypersurface given by a subextremal ReissnerNordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development (MGHD) as a "suitably regular" Lorentzian manifold.In this first part we establish well posedness of the Einstein equations for characteristic data satisfying the minimal regularity conditions leading to classical solutions. We also identify the appropriate notion of maximal solution, from which the construction of the corresponding MGHD follows, and determine breakdown criteria. This is the unavoidable starting point of the analysis; our main results will depend on the detailed understanding of these fundamentals.In the second part of this series [12] we study the stability of the radius function at the Cauchy horizon. In the third and final paper [13] we show that, depending on the decay rate of the initial data, mass inflation may or may not occur; in fact, it is even possible to have (non-isometric) extensions of the spacetime across the Cauchy horizon as classical solutions of the Einstein equations.

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“…The metric in this case exhibits strong (elliptic) trapping, which suggests that the decay of linear waves is very slow because of the high frequency contributions. A logarithmic upper bound was proved in [HoSm11], and existence of resonances exponentially close to the real axis and a logarithmic lower bound were established in [Ga,HoSm13].…”

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confidence: 99%

Asymptotics of Linear Waves and Resonances with Applications to Black Holes

Dyatlov

2015

Commun. Math. Phys.

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Abstract. We apply the results of [Dy13] to describe asymptotic behavior of linear waves on stationary Lorentzian metrics with r-normally hyperbolic trapped sets, in particular Kerr and Kerr-de Sitter metrics with |a| < M and M Λa 1. We prove that if the initial data is localized at frequencies ∼ λ 1, then the energy norm of the solution is bounded by O(λ 1/2 e −(νmin−ε)t/2 ) + O(λ −∞ ), for t ≤ C log λ, where ν min is a natural dynamical quantity. The key tool is a microlocal projector splitting the solution into a component with controlled rate of exponential decay and an O(λe −(νmin−ε)t ) + O(λ −∞ ) remainder; this splitting can be viewed as an analog of resonance expansion. Moreover, for the Kerr-de Sitter case we study quasi-normal modes; under a dynamical pinching condition, a Weyl law in a band holds.The subject of this paper are decay properties of solutions to the wave equation for the rotating Kerr (cosmological constant Λ = 0) and Kerr-de Sitter (Λ > 0) black holes, as well as for their stationary perturbations. In the recent decade, there has been a lot of progress in understanding the upper bounds on these solutions, producing a polynomial decay rate O(t −3 ) for Kerr and an exponential decay rate O(e −νt ) for Kerrde Sitter (the latter is modulo constant functions). The weaker decay for Λ = 0 is explained by the presence of an asymptotically Euclidean infinite end; however, this polynomial decay comes from low frequency contributions.We instead concentrate on the decay of solutions with initial data localized at high frequencies ∼ λ 1; it is related to the geometry of the trapped set K, consisting of lightlike geodesics that never escape to the spatial infinity or through the event horizons. The trapped set for both Kerr and Kerr-de Sitter metrics is r-normally hyperbolic, and this dynamical property is stable under stationary perturbations of the metric -see §3.6. The key quantities associated to such trapping are the minimal and maximal transversal expansion rates 0 < ν min ≤ ν max , see (2.9), (2.10). Using our recent work [Dy13], we show the exponential decay rate O(λ 1/2 e −(ν min −ε)t/2 ) + O(λ −∞ ), valid for t = O(log λ) (Theorem 1). This bound is new for the Kerr case, complementing Price's law.Our methods give a more precise microlocal description of long time propagation of high frequency solutions. In Theorem 2, we split a solution u(t) into two approximate solutions to the wave equation, u Π (t) and u R (t), with the rate of decay of u Π (t) between e −(νmax+ε)t/2 and e −(ν min −ε)t/2 and u R (t) bounded from above by λe −(ν min −ε)t , all 1 arXiv:1305.1723v1 [gr-qc]

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Decay Properties of Klein‐Gordon Fields on Kerr‐AdS Spacetimes

Abstract: This paper investigates the decay properties of solutions to the massive linear wave equation left□gψ+αl2ψ=0 for g being the metric of a Kerr‐AdS spacetime satisfying | a |l Show more

Cited by 94 publications

References 59 publications

The BTZ black hole violates strong cosmic censorship

The BTZ black hole violates strong cosmic censorship

On the global uniqueness for the Einstein–Maxwell-scalar field system with a cosmological constant: I. Well posedness and breakdown criterion

Asymptotics of Linear Waves and Resonances with Applications to Black Holes

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