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![Depiction of the Schwarzschild manifold M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {M}}$$\end{document}. Also depicted is the region Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document} in which we apply the divergence theorem (2.11)](publication/361150384/figure/fig6/AS:11431281106522814@1670728275368/Depiction-of-the-Schwarzschild-manifold-Mdocumentclass12ptminimal.png)
Depiction of the Schwarzschild manifold M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {M}}$$\end{document}. Also depicted is the region Δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta $$\end{document} in which we apply the divergence theorem (2.11)
Source publication
In this paper, we derive the early-time asymptotics for fixed-frequency solutions $$\phi _\ell $$ ϕ ℓ to the wave equation $$\Box _g \phi _\ell =0$$ □ g ϕ ℓ = 0 on a fixed Schwarzschild background ( $$M>0$$ M > 0 ) arising from the no incoming radiation condition on $${\mathscr {I}}^-$$ I - and polynomially decaying data, $$r\phi _\ell \sim t^{-1}$...
Citations
... (4.18) of Thorne [61]). Of course, one might then wonder whether it is already too much to assume the validity of the framework all the way until the null cone C, rather than, say, just assuming it to hold up until some timelike cylinder T along which |t| ∼ r as in [27], see figure 3. Let us here just mention that the propagation of radiation from such a timelike cylinder to the null cone C was studied at the level of scalar waves in [30,66] on Schwarzschild: the result of these studies is that the leading-order decay towards u → −∞ is the same along the cylinder and along C. Second, the system that we consider, growing linearly in time, has the feature that all higherorder corrections within the post-Newtonian expansion (see §V.D of Thorne [61]), even though they are of smaller magnitude in v/c, feature the same decay towards I − , in just the same way as all electric angular modes of Ψ 0 or Ψ 4 feature the same decay towards I − . Of course, one could interpret the approximate statements (2.3) as the leading-order result for metric perturbations on fixed angular frequency. ...
This paper is the fourth in a series dedicated to the mathematically rigorous asymptotic analysis of gravitational radiation under astrophysically realistic set-ups. It provides an overview of the physical ideas involved in setting up the mathematical problem, the mathematical challenges that need to be overcome once the problem is posed, as well as the main new results we will obtain in upcoming work. From the physical perspective, this includes a discussion of how post-Newtonian theory provides a prediction on the gravitational radiation emitted by N infalling masses from the infinite past in the intermediate zone, i.e. up to some finite advanced time. From the mathematical perspective, we then take this prediction, together with the condition that there be no incoming radiation from I−, as a starting point to set up a scattering problem for the linearized Einstein vacuum equations around Schwarzschild and near spacelike infinity, and we outline how to solve this scattering problem and obtain the asymptotic properties of the scattering solution near i0 and I+. The full mathematical details will be presented in the sequel to this paper.
This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.
... At the time of writing, the general consensus appears to be that the smoothness assumptions in the classical definition of asymptotic simplicity are in some instances too restrictive to describe realistic physical scenarios. Kehrberger [36][37][38] has argued strongly that many systems of physical interest, e.g. the problem of N gravitating bodies coming in from infinity, violate not only peeling, but also the weaker decay assumptions and conclusions of Christodoulou-Klainerman [13,39]. There is some suggestion, however, that the weak assumptions of Bieri [15] are sufficient to capture these situations: see the chapter by Kehrberger [40] in this issue. ...
This is an introductory article for the proceedings associated with the Royal Society Hooke discussion meeting of the same title which took place in London in May 2023. We review the history of Penrose’s conformal compactification, null infinity and a number of related fundamental developments in mathematical general relativity from the last 60 years.
This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.
... It should be noted that it is expected that generic physically interesting Cauchy data do not satisfy peeling properties. See, for instance, the recent works of Kehrberger [53][54][55] in which the author considered the precise structure of gravitational radiation near infinity for the scalar field on Schwarzschild. ...
... The globally sharp v −1 τ −2 pointwise decay is first proven by Angelopoulos-Aretakis-Gajic [9,10] and the precise late-time asymptotic profile is calculated therein; Hintz [46] computed the v −1 τ −2 leading order term on both Schwarzschild and subextreme Kerr spacetimes and further obtained v −1 τ −2 −2 sharp asymptotics for ≥ modes in a compact region on Schwarzschild; Luk-Oh [64] derived sharp decay for the scalar field on a Reissner-Nordström background and used it to obtain linear instability of the Reissner-Nordström Cauchy horizon (see also their works [65,66] on a generalization to a nonlinear setting); Angelopoulos-Aretakis-Gajic based on their own earlier works and re-derived in [12] v −1 τ −2 −2 late time asymptotics for ≥ 0 modes in a finite radius region on Schwarzschild, and they further computed in [11] the asymptotic profiles of the = 0, = 1, and ≥ 2 modes in a subextreme Kerr spacetime; we [71] independently computed the global v −1 τ −2 −2 late time asymptotics for ≥ modes in a Schwarzschild spacetime. Additionally, Kehrberger [53][54][55] considered the precise structure of gravitational radiation near infinity for the scalar field on Schwarzschild. ...
In this work, we derive the global sharp decay, as both a lower and an upper bounds, for the spin ±s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pm {\mathfrak {s}}$$\end{document} components, which are solutions to the Teukolsky equation, in the black hole exterior and on the event horizon of a slowly rotating Kerr spacetime. These estimates are generalized to any subextreme Kerr background under an integrated local energy decay estimate. Our results apply to the scalar field (s=0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathfrak {s}}=0)$$\end{document}, the Maxwell field (s=1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathfrak {s}}=1)$$\end{document} and the linearized gravity (s=2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({\mathfrak {s}}=2)$$\end{document} and confirm the Price’s law decay that is conjectured to be sharp. Our analyses rely on a novel global conservation law for the Teukolsky equation, and this new approach can be applied to derive the precise asymptotics for solutions to semilinear wave equations.
... When Λ = 0 we identify the evolution equations relating the logarithmic terms, and also explain how the presence of these terms modifies the peeling behavior of the Weyl tensor by producing overleading non-smooth terms in Ψ 0 and Ψ 1 . Such violations of the peeling and arguments in favor of logarithmic terms have been discussed in the literature on numerous occasions [127,138,[141][142][143][144][145][146][147][148][149][150], but there is so far no agreement as to the type of realistic physical situations in which this would occur. Under some technical assumptions it has indeed been shown that compact sources with no incoming radiation preserve a smooth peeling [151][152][153]. ...
We present a detailed analysis of gravity in a partial Bondi gauge, where only the three conditions g_{rr}=0=g_{rA} g r r = 0 = g r A are fixed. We relax in particular the so-called determinant condition on the transverse metric, which is only assumed to admit a polyhomogeneous radial expansion. This is sufficient in order to build the solution space, which here includes a cosmological constant, time-dependent sources in the boundary metric, logarithmic branches, and an extra trace mode at subleading order in the transverse metric. The evolution equations are studied using the Newman–Penrose formalism in terms of covariant functionals identified from the Weyl scalars, and we build the explicit dictionary between this formalism and the tensorial Einstein equations. This provides in particular a new derivation of the (A)dS mass loss formula. We then study the holographic renormalisation of the symplectic potential, and the transformation laws under residual asymptotic symmetries. The advantage of the partial Bondi gauge is that it allows to contrast and treat in a unified manner the Bondi–Sachs and Newman–Unti gauges, which can each be reached upon imposing a further specific gauge condition. The differential determinant condition leads to the \Lambda Λ -BMSW gauge, while a differential condition on g_{ur} g u r leads to a generalized Newman–Unti gauge. This latter gives access to a new asymptotic symmetry which acts on the asymptotic shear and further extends the \Lambda Λ -BMSW group by an extra abelian radial translation. This generalizes results which we have recently obtained in three dimensions.
... However, in the full generality of the data considered in [10], the higher order regularity assumptions of [3], [35] do not hold. For a discussion of the physical significance of the lack of higher order regularity, see [26], [24], [25]. ...
... In particular, the work [9] suggested that physically relevant spacetimes do not admit peeling estimates or conformal smoothness. A recent series of papers has given mathematical evidence towards this view, see [19], [24], [25], [26]. (See also the upcoming [23] for the case of the linearised system considered here). ...
... We also devise another gauge fixing scheme for the purpose of studying the linear memory effect, as the laws of gravitational radiation are best exhibited in a gauge that is Bondi-normalised at I + and I − simultaneously. Thus we denote by the global scattering gauge the gauge that is Bondi-normalised at both I + , I − , that satisfies the conditions (1. 25), and which moreover satisfies ...
We construct a scattering theory for the linearised Einstein equations on a Schwarzschild background in a double null gauge. We build on the results of Part I \cite{Mas20}, where we used the energy conservation enjoyed by the Regge--Wheeler equation associated with the stationarity of the Schwarzschild background to construct a scattering theory for the Teukolsky equations of spin $\pm2$. We now extend the scattering theory of Part I to the full system of linearised Einstein equations by treating it as a system of transport equations which is sourced by solutions to the Teukolsky equations, leading to Hilbert space-isomorphisms between spaces of finite energy initial data and corresponding spaces of scattering states under suitably chosen gauge conditions on initial and scattering data. As a corollary, we show that for a solution which is Bondi-normalised at both past and future null infinity, past and future linear memories are related by an antipodal map.
... ], we will see in the present paper that other choices of f are equally important; see also [Kro00,Kro01,Keh21b,Keh22]. ...
... Then, by (4.20), it follows that when splitting ψ = ψ + ψ +1 , the ψ +1 contributes at higher order in τ −1 and u −1 , so the late-time asymptotics of ψ agree with the late-time asymptotics of ψ , which are given by (4.3) and (4.4), or (4.18) and (4.19). However, initial data on Σ 0 for which higher modes do not have higher regularity at infinity may arise naturally in scattering problems, see section 1.3.3 of [Keh22]. There, it is conjectured that compactly supported scattering data lead to solutions where all modes contribute to the late-time asymptotics at the same order. ...
... As explained in the introduction, instead of assuming, say, vanishing or 'peeling' asymptotics near I + , we will dynamically derive the asymptotics towards I + from a scattering data setup that (a) has no incoming radiation from I − and (b) resembles-in some sense-a system of N infalling masses following unbound Keplerian orbits near the infinite past. This section follows the works [Keh21a, Keh21b,Keh22]. ...
The last few years have seen considerable mathematical progress concerning the asymptotic structure of gravitational radiation in dynamical, astrophysical spacetimes. In this paper, we distil some of the key ideas from recent works and assemble them in a new way in order to make them more accessible to the wider general relativity community. In the process, we also discuss new physical findings. First, we introduce the conserved f(r)-modified Newman–Penrose charges on asymptotically flat spacetimes, and we show that these charges provide a dictionary that relates asymptotics of massless, general spin fields in different regions: asymptotic behaviour near i ⁺ (‘late-time tails’) can be read off from asymptotic behaviour towards I+ , and, similarly, asymptotic behaviour towards I+ can be read off from asymptotic behaviour near i ⁻ or I− . Using this dictionary, we then explain how: (I) the quadrupole approximation for a system of N infalling masses from i ⁻ causes the ‘peeling property towards I+ ’ to be violated, and (II) this failure of peeling results in deviations from the usual predictions for tails in the late-time behaviour of gravitational radiation: instead of the Price’s law rate rΨ[4]|I+∼u−6 as u → ∞, we predict that rΨ[4]|I+∼u−4 , with the coefficient of this latter decay rate being a multiple of the monopole and quadrupole moments of the matter distribution in the infinite past.
... The debate around the peeling vs non-peeling nature of null infinity is summarised in [51]. Here we point out that the assumption of smooth null infinity excludes the phenomenology of infalling matter from past timelike infinity [52][53][54][55][56]. We will also discuss logarithmic terms in u in section 7. ...
We derive the antipodal matching relations used to demonstrate the equivalence between soft graviton theorems and BMS charge conservation across spatial infinity. To this end we provide a precise map between Bondi data at null infinity $\mathscr{I}$ and Beig—Schmidt data at spatial infinity $i^0$ in a context appropriate to the gravitational scattering problem and celestial holography. In addition, we explicitly match the various proposals of BMS charges at $\mathscr{I}$ found in the literature with the conserved charges at $i^0$.
A bstract
We perform a complete and systematic analysis of the solution space of six-dimensional Einstein gravity. We show that a particular subclass of solutions — those that are analytic near $$ \mathcal{I} $$ I ⁺ — admit a non-trivial action of the generalised Bondi-Metzner-van der Burg-Sachs (GBMS) group which contains infinite-dimensional supertranslations and superrotations. The latter consists of all smooth volume-preserving Diff×Weyl transformations of the celestial S ⁴ . Using the covariant phase space formalism and a new technique which we develop in this paper (phase space renormalization), we are able to renormalize the symplectic potential using counterterms which are local and covariant . The Hamiltonian charges corresponding to GBMS diffeomorphisms are non-integrable. We show that the integrable part of these charges faithfully represent the GBMS algebra and in doing so, settle a long-standing open question regarding the existence of infinite-dimensional asymptotic symmetries in higher even dimensional non-linear gravity. Finally, we show that the semi-classical Ward identities for supertranslations and superrotations are precisely the leading and subleading soft-graviton theorems respectively.
