Long time existence from interior gluing

Chruściel, Piotr T.

doi:10.1088/1361-6382/aa769d

“…Let us explain this terminology: One can now restrict these data to a region, let's call it the exterior region, sufficiently close to spacelike infinity in a way so that the data in this exterior region are vacuum and have arbitrarily small || • || CK -norm. By the gluing results [24,25], one can then extend these exterior data to interior data whose || • || CK -norm can also be chosen sufficiently small so that the resulting glued data are C-K small. Therefore, the results of [15] apply to the (C-K small) glued data, and thus, by the domain of dependence property, they apply to the domain of dependence of the exterior part of the (C-K compatible) original data, i.e.…”

Section: Christodoulou's Argument Against Smooth Null Infinitymentioning

confidence: 99%

The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

Kehrberger

¹

2021

Ann. Henri Poincaré

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This paper initiates a series of works dedicated to the rigorous study of the precise structure of gravitational radiation near infinity. We begin with a brief review of an argument due to Christodoulou (in: The Ninth Marcel Grossmann Meeting, World Scientific Publishing Company, Singapore, 2002) stating that Penrose’s proposal of smooth conformal compactification of spacetime (or smooth null infinity) fails to accurately capture the structure of gravitational radiation emitted by N infalling masses coming from past timelike infinity $$i^-$$ i - . Modelling gravitational radiation by scalar radiation, we then take a first step towards a dynamical understanding of the non-smoothness of null infinity by constructing solutions to the spherically symmetric Einstein–Scalar field equations that arise from polynomially decaying boundary data, $$r\phi \sim t^{-1}$$ r ϕ ∼ t - 1 as $$t\rightarrow -\infty $$ t → - ∞ , on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, $$r\partial _v\phi =0$$ r ∂ v ϕ = 0 , on past null infinity. We show that if the initial Hawking mass at $$i^-$$ i - is nonzero, then, in accordance with the non-smoothness of $${\mathcal {I}}^+$$ I + , the asymptotic expansion of $$\partial _v(r\phi )$$ ∂ v ( r ϕ ) near $${\mathcal {I}}^+$$ I + reads $$\partial _v(r\phi )=Cr^{-3}\log r+{\mathcal {O}}(r^{-3})$$ ∂ v ( r ϕ ) = C r - 3 log r + O ( r - 3 ) for some non-vanishing constant C. In fact, the same logarithmic terms appear already in the linear theory, i.e. when considering the spherically symmetric linear wave equation on a fixed Schwarzschild background. As a corollary, we can apply our results to the scattering problem on Schwarzschild: Putting compactly supported scattering data for the linear (or coupled) wave equation on $${\mathcal {I}}^-$$ I - and on $${\mathcal {H}}^-$$ H - , we find that the asymptotic expansion of $$\partial _v(r\phi )$$ ∂ v ( r ϕ ) near $${\mathcal {I}}^+$$ I + generically contains logarithmic terms at second order, i.e. at order $$r^{-4}\log r$$ r - 4 log r .

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“…Let us explain this terminology: One can now restrict these data to a region, let's call it the exterior region, sufficiently close to spacelike infinity in a way so that the data in this exterior region are vacuum and have arbitrarily small || • || CK -norm. By the gluing results [24,25], one can then extend these exterior data to interior data whose || • || CK -norm can also be chosen sufficiently small so that the resulting glued data are C-K small. Therefore, the results of [16] apply to the (C-K small) glued data, and, thus, by the domain of dependence property, they apply to the domain of dependence of the exterior part of the (C-K compatible) original data, i.e.…”

Section: Christodoulou's Argument Against Smooth Null Infinitymentioning

confidence: 99%

The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

Kehrberger

2021

Preprint

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This paper initiates a series of works dedicated to the rigorous study of the precise structure of gravitational radiation near infinity.We begin with a brief review of an argument due to Christodoulou [1] stating that Penrose's proposal of smooth conformal compactification of spacetime (or smooth null infinity) fails to accurately capture the structure of gravitational radiation emitted by N infalling masses coming from past timelike infinity i − .Modelling gravitational radiation by scalar radiation, we then take a first step towards a rigorous, fully general relativistic understanding of the non-smoothness of null infinity by constructing solutions to the spherically symmetric Einstein-Scalar field equations. Our constructions are motivated by Christodoulou's argument: They arise dynamically from polynomially decaying boundary data, rφ ∼ |t| −p as t → −∞, on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, r∂ v φ = 0, on past null infinity. We show that if the initial Hawking mass at past timelike infinity i − is non-zero, then there exists a constant C = 0 such that, in the case p = 1, we obtain the following asymptotic expansion near I + , precisely in accordance with the non-smoothness ofSimilarly, if p > 1, we find constant coefficient logarithmic terms appearing at higher orders in the expansion of ∂ v (rφ).Even though these results are obtained in the non-linear setting, we show that the same logarithmic terms appear already in the linear theory, i.e. when considering the spherically symmetric linear wave equation on a fixed Schwarzschild background.As a corollary, we can apply our results to the scattering problem on Schwarzschild: Putting smooth compactly supported scattering data for the linear (or coupled) wave equation on I − and on H − , we find that the asymptotic expansion of ∂ v (rφ) near I + generically contains logarithmic terms at second order, i.e. at order r −4 log r.

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“…for the initial data constructed by Carlotto-Schoen [CaSch16], which are non-trivial only in conic wedges.) Bieri and Chruściel [BiCh16,Ch17] construct a piece of I + for the data considered in [BiZi09] but without a smallness assumption. Further works on the stability of Minkowski space for the Einstein equations coupled to other fields include those by Speck [Sp14] on (a generalization of) the Einstein-Maxwell system, Taylor [Ta16], Lindblad-Taylor [LiTa17], and Fajman-Joudioux-Smulevici [FaJoSm17] for both the massless and the massive Einstein-Vlasov system.…”

Section: Introductionmentioning

confidence: 99%

Stability of Minkowski space and polyhomogeneity of the metric

Hintz,

Vasy

2017

Preprint

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We first give a new proof of the non-linear stability of the (3 + 1)-dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. We then show that the metric admits a full asymptotic expansion at infinity, more precisely at the boundary hypersurfaces (corresponding to spacelike, null, and timelike infinity) of a suitable compactification of R 4 adapted to the bending of outgoing light cones. We work in a wave map/DeTurck gauge closely related to the standard wave coordinate gauge. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the non-linear equation using an iteration scheme in which we solve a linearized equation globally at each step; to fix the geometry of null infinity throughout the iteration scheme, we devise a hyperbolic formulation of Einstein's equation which ensures constraint damping. The weak null condition of Lindblad and Rodnianski arises naturally as a nilpotent coupling of certain metric components in a linear model operator at null infinity. Furthermore, we relate the Bondi mass to a logarithmic term in the expansion of the metric at null infinity and prove the Bondi mass loss formula.

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Long time existence from interior gluing

Cited by 5 publications

References 24 publications

The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

Stability of Minkowski space and polyhomogeneity of the metric

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