PROPAGATION OF SINGULARITIES ON AdS SPACETIMES FOR GENERAL BOUNDARY CONDITIONS AND THE HOLOGRAPHIC HADAMARD CONDITION

Gannot, Oran; Wrochna, Michał

doi:10.1017/s147474802000002x

Cited by 14 publications

(80 citation statements)

References 47 publications

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“…Observe in particular that h x in (2) does not need to be an Einstein metric nor ∂M is required to be diffeomorphic to R×S n−2 . Since we prefer to make a close connection to both [26] and [16] we stick to their nomenclature.…”

Section: Remark 22mentioning

confidence: 99%

“…In this section we introduce the main analytic tools that play a key rôle in our investigation. We start by recollecting the main results from [16] which are, in turn, based on [26] and [40,41].…”

Section: Analytic Preliminariesmentioning

confidence: 99%

“…In this case the analysis of partial differential equations such as the Klein-Gordon one becomes more involved due to admissible class of backgrounds and, in particular, due to the lack of isometries of the metric. Noteworthy has been the recent analysis by Gannot and Wrochna, [26], in which, using techniques proper of b-calculus they have investigated the structural properties of the Klein-Gordon operator with Robin boundary conditions. In between the several results proven, we highlight in particular the theorem of propagation of singularities and the existence of advanced and retarded fundamental solutions.…”

Section: Introductionmentioning

confidence: 99%

“…Yet, as strongly advocated in [10], the class of boundary conditions which are of interest in concrete models is greater than the one considered in [26], a notable example in this direction being the so-called Wentzell boundary conditions, see e.g. [9,13,19,38,42].…”

Section: Introductionmentioning

confidence: 99%

“…[9,13,19,38,42]. For this reason in [16], we started an investigation aimed at generalizing the results of [26] proving a theorem of propagation of singularities for the Klein-Gordon operator on an asymptotically anti-de Sitter spacetime M such that the boundary condition is implemented by a b-pseudodifferential operator ∈ k b (∂M) with k 2, see Section 3.1 for the definitions.…”

Section: Introductionmentioning

confidence: 99%

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Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

Dappiaggi

Marta

2021

Math Phys Anal Geom

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We consider the Klein-Gordon operator on an n-dimensional asymptotically anti-de Sitter spacetime (M,g) together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on ∂M of order up to 2. Using techniques from b-calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-solutions of the underlying equations of motion with a prescribed form of their wavefront set and whose anti-symmetric part is proportional to the difference between the advanced and the retarded fundamental solutions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary.

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Section: Remark 22mentioning

confidence: 99%

Section: Analytic Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

Dappiaggi

Marta

2021

Math Phys Anal Geom

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A generalization of the propagation of singularities theorem on asymptotically anti‐de Sitter spacetimes

Dappiaggi

Marta

2022

Mathematische Nachrichten

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In a recent paper O. Gannot and M. Wrochna considered the Klein-Gordon equation on an asymptotically anti-de Sitter spacetime subject to Robin boundary conditions, proving in particular a propagation of singularities theorem. In this work we generalize their result considering a more general class of boundary conditions implemented on the conformal boundary via pseudodifferential operators of suitable order. Using techniques proper of 𝑏-calculus and of twisted Sobolev spaces, we prove also for the case in hand a propagation of singularity theorem along generalized broken bicharacteristics, highlighting the potential presence of a contribution due to the pseudodifferential operator encoding the boundary condition. K E Y W O R D S asymptotically anti-de Sitter spacetimes, boundary conditions, propagation of singularities M S C ( 2 0 2 0 ) 58J32, 58J45, 58J47

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Diophantine approximation as Cosmic Censor for Kerr–AdS black holes

Kehle

2021

Invent. math.

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The purpose of this paper is to show an unexpected connection between Diophantine approximation and the behavior of waves on black hole interiors with negative cosmological constant $$\Lambda <0$$ Λ < 0 and explore the consequences of this for the Strong Cosmic Censorship conjecture in general relativity. We study linear scalar perturbations $$\psi $$ ψ of Kerr–AdS solving $$\Box _g\psi -\frac{2}{3}\Lambda \psi =0$$ □ g ψ - 2 3 Λ ψ = 0 with reflecting boundary conditions imposed at infinity. Understanding the behavior of $$\psi $$ ψ at the Cauchy horizon corresponds to a linear analog of the problem of Strong Cosmic Censorship. Our main result shows that if the dimensionless black hole parameters mass $${\mathfrak {m}} = M \sqrt{-\Lambda }$$ m = M - Λ and angular momentum $${\mathfrak {a}} = a \sqrt{-\Lambda }$$ a = a - Λ satisfy a certain non-Diophantine condition, then perturbations $$\psi $$ ψ arising from generic smooth initial data blow up $$|\psi |\rightarrow +\infty $$ | ψ | → + ∞ at the Cauchy horizon. The proof crucially relies on a novel resonance phenomenon between stable trapping on the black hole exterior and the poles of the interior scattering operator that gives rise to a small divisors problem. Our result is in stark contrast to the result on Reissner–Nordström–AdS (Kehle in Commun Math Phys 376(1):145–200, 2020) as well as to previous work on the analogous problem for $$\Lambda \ge 0$$ Λ ≥ 0 —in both cases such linear scalar perturbations were shown to remain bounded. As a result of the non-Diophantine condition, the set of parameters $${\mathfrak {m}}, {\mathfrak {a}}$$ m , a for which we show blow-up forms a Baire-generic but Lebesgue-exceptional subset of all parameters below the Hawking–Reall bound. On the other hand, we conjecture that for a set of parameters $${\mathfrak {m}}, {\mathfrak {a}} $$ m , a which is Baire-exceptional but Lebesgue-generic, all linear scalar perturbations remain bounded at the Cauchy horizon $$|\psi |\le C$$ | ψ | ≤ C . This suggests that the validity of the $$C^0$$ C 0 -formulation of Strong Cosmic Censorship for $$\Lambda <0$$ Λ < 0 may change in a spectacular way according to the notion of genericity imposed.

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PROPAGATION OF SINGULARITIES ON AdS SPACETIMES FOR GENERAL BOUNDARY CONDITIONS AND THE HOLOGRAPHIC HADAMARD CONDITION

Cited by 14 publications

References 47 publications

Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

A generalization of the propagation of singularities theorem on asymptotically anti‐de Sitter spacetimes

Diophantine approximation as Cosmic Censor for Kerr–AdS black holes

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