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The linear stability of the Schwarzschild solution to gravitational perturbations
Abstract
We prove in this paper the linear stability of the celebrated Schwarzschild family of black holes in general relativity: Solutions to the linearisation of the Einstein vacuum equations around a Schwarzschild metric arising from regular initial data remain globally bounded on the black hole exterior and in fact decay to a linearised Kerr metric. We express the equations in a suitable double null gauge. To obtain decay, one must in fact add a residual pure gauge solution which we prove to be itself quantitatively controlled from initial data. Our result a fortiori includes decay statements for general solutions of the Teukolsky equation (satisfied by gauge-invariant null-decomposed curvature components). These latter statements are in fact deduced in the course of the proof by exploiting associated quantities shown to satisfy the Regge--Wheeler equation, for which appropriate decay can be obtained easily by adapting previous work on the linear scalar wave equation. The bounds on the rate of decay to linearised Kerr are inverse polynomial, suggesting that dispersion is sufficient to control the non-linearities of the Einstein equations in a potential future proof of nonlinear stability. This paper is self-contained and includes a physical-space derivation of the equations of linearised gravity around Schwarzschild from the full non-linear Einstein vacuum equations expressed in a double null gauge.
... These are then analysed using techniques adapted from the study of wave equations on black hole backgrounds. This framework has proven extremely successful, particularly in recent years, in tackling linear or nonlinear stability questions for Schwarzschild [4][5][6], Kerr [7][8][9][10][11][12][13], Reissner-Nordström [14,15] and Kerr-Newman spacetimes [16]. 1 However, the use of the Teukolsky formalism, while powerful, comes at a cost: it requires the exploitation of the background's Petrov type and principal null directions. Moreover, the implementation of standard analytic tools available for the wave equation do not straightforwardly carry over to the decoupled Teukolsky equations. ...
... Despite the absence of conserved energies for the Teukolsky variables at the level of first derivatives-even in vacuum-substantial progress has been made in establishing stability properties of the Reissner-Nordström spacetime. Key breakthroughs were achieved by introducing derived quantities-higher-order combinations of the gauge-invariant fields-that satisfy Regge-Wheeler-type wave equations, first introduced in [4] in Schwarzschild. These derived equations allow to derive energy estimates, allowing for the application of the vector field method to obtain uniform energy boundedness, integrated decay, and pointwise control. ...
... In [1], this work was extended to show how one can ascend a hierarchy in the system of linearised gravity to control on the gauge-invariant Teukolsky quantities from the coercive estimates obtained from these conservation laws rather than the transformation theory to Regge-Wheeler quantities [4]. ...
We study the linearised Einstein--Maxwell equations on the Reissner--Nordstr\"om spacetime and derive the canonical energy conservation law in double null gauge. In the spirit of the work of Holzegel and the second author, we avoid any use of the hyperbolic nature of the Teukolsky equations and rely solely on the conservation law to establish control of energy fluxes for the gauge-invariant Teukolsky variables, previously identified by the third author, along all outgoing null hypersurfaces, for charge-to-mass ratio . This yields uniform boundedness for the Teukolsky variables in Reissner--Nordstr\"om.
... 22 The exterior optical function (ext) u is initialized on the last slice t = t * , by the construction of a foliation (inverse lapse foliation) initialized at space-like infinity. It is thus readjusted dynamically as t * → ∞. 23 Note however that even though the linearized system around Minkowski does not contain instabilities, the proof of the nonlinear stability of the Minkowski space in [20] takes into account (in a fundamental way!) general covariance. Indeed the presence of the ADM mass affects the causal structure of the far, asymptotic, region of the perturbed spacetime. ...
... All results on the linear stability of Kerr in the physics literature during the 10-15 years after Roy Kerr's 1963 discovery, often called the "Golden Age of Black Hole Physics", are based on mode decompositions. One makes use of the separability 31 of the linearized equations, more 29 Thus, for example, in their well known linear stability result around Schwarzschild [23], the authors derive satisfactory results (compatible with what is needed in nonlinear theory) for components of the curvature tensor, and some Ricci coefficients, but not all. Similar comments apply to [24,25]. ...
... A first quantitative proof of the linear stability of Schwarzschild spacetime was established 37 by Dafermos-Holzegel-Rodnianski (DHR) in [23]. Notable in their analysis is the treatment of the Teukolsky equation in a fixed Schwarzschild background. ...
This is a short survey on the black hole stability problem written in honor of Yvonne Choquet Bruhat’s 100th birthday, subject very dear to her and to which she has contributed greatly, in particular her foundational local existence result [1] and the maximal globally hyperbolic development of a given initial data set. The main focus is on the recent results on the stability of slowly rotating Kerr contained in the sequence of works [2, 3, 4, 5, 6].
... Recently, there has also been a substantial advance in the last step of this "linear program", i.e. the proof of linear stability of the Schwarzschild metric under gravitational perturbations. Indeed, a recent result of Dafermos, Holzegel and Rodnianski [12] shows that the Schwarzschild metric is stable under linearized gravitational perturbations. ...
... The third and main motivation for this work, though, lies in the connection with the aforementioned problem of linear stability of the Schwarzschild metric. It has become evident that a very similar approach to the one in [12] can be adopted to address the decay properties of the spin ±1 Teukolsky equations and of the Maxwell system on Schwarzschild. ...
... We briefly recall the strategy followed by the authors in [12]. Given a solution to the equation for the extreme curvature components (i.e., the spin ±2 Teukolsky equation), the authors find a second order differential transformation that performs the following: if we apply the transformation on the extreme curvature component, the resulting expression satisfies a "good" equation (i.e., an equation for which Morawetz and energy estimates can be proved.) ...
In this note we prove decay for the spin 1 Teukolsky Equations on the Schwarzschild spacetime. These equations are those satisfied by the extreme components ( and ) of the Maxwell field, when expressed with respect to a null frame. The subject has already been addressed in the literature, and the interest in the present approach lies in the connection with the recent work by Dafermos, Holzegel and Rodnianski on linearized gravity [M. Dafermos, G. Holzegel and I. Rodnianski, The linear stability of the Schwarzschild solution to gravitational perturbations, arXiv:1601.06467]. In analogy with the spin case, it seems difficult to directly prove Morawetz estimates for solutions to the spin Teukolsky Equations. By performing a differential transformation on the extreme components and , we obtain quantities which satisfy a Fackerell--Ipser Equation, which does admit a straightforward Morawetz estimate, and is the key to the decay estimates. This approach is exactly analogous to the strategy appearing in the aforementioned work on linearized gravity. We achieve inverse polynomial decay estimates by a streamlined version of the physical space method of Dafermos and Rodnianski. Furthermore, we are also able to prove decay for all the components of the Maxwell system. The transformation that we use is a physical space version of a fixed-frequency transformation which appeared in the work of Chandrasekhar. The present note is a version of the author's master thesis and also serves the "pedagogical" purpose to be as complete as possible in the presentation.
... has played a particularly prominent role. Being the simplest black hole solution to the vacuum Einstein equations, it has been the object of most of the works pioneering the mathematical study of both scalar and gravitational perturbations of black hole spacetimes [8][9][10]. Alongside its intrinsic interest, a solid understanding of the Schwarzschild solution and its stability properties has proven fundamental to address the more general, and complicated, Kerr geometry [1,7,11,13,14,20,21,24,25]. ...
... As we shall explain in the present introduction, the nature and significance of the simplifications concern the treatment of so-called gauge normalisations. After briefly discussing the meaning of gauge normalisations, we illustrate the new features of our analysis in relation to previous work [8]. Some more detailed (but still introductory) comments are deferred to the overview in Sect. 2. ...
... Geometric gauges governed by transport equations have been future normalised when adopted in linear stability problems for black hole spacetimes (e.g. [8,12]). This means that, given a general solution to the linearised system, the choice of gauge normalisation is made depending on the evaluation at (all) future times of certain (linearised) dynamical quantities associated to the solution. ...
We present a new proof of linear stability of the Schwarzschild solution to gravitational perturbations. Our approach employs the system of linearised gravity in the new geometric gauge of Benomio (A new gauge for gravitational perturbations of Kerr spacetimes I: the linearised theory, 2022, https://arxiv.org/abs/2211.00602), specialised to the case. The proof fundamentally relies on the novel structure of the transport equations in the system. Indeed, while exploiting the well-known decoupling of two gauge invariant linearised quantities into spin Teukolsky equations, we make enhanced use of the red-shifted transport equations and their stabilising properties to control the gauge dependent part of the system. As a result, an initial-data gauge normalisation suffices to establish both orbital and asymptotic stability for all the linearised quantities in the system. The absence of future gauge normalisations is a novel element in the linear stability analysis of black hole spacetimes in geometric gauges governed by transport equations. In particular, our approach simplifies the proof of Dafermos et al. (Acta Math 222:1–214, 2019), which requires a future normalised (double-null) gauge to establish asymptotic stability for the full system.
... The precise formulas are given in Section 2.3. Moreover, the transversal perturbations correspond to the linearised pure gauge solutions of [23]. ...
... In this section we derive the linearised formulas of null structure equations about the reference Minkowski sphere data m. We follow the linearisation procedure from [23]. Formally, for the C 3 -sphere data, we write X u,v =(Ω, g /, Ωtrχ, χ, Ωtrχ, χ, η, ω, Dω, D 2 ω, ω, Dω, D 2 ω, α, Dα, α, Dα) (2.13) and differentiate in ε at ε = 0. ...
... By equation (242) in [23] the linearisation of the Gauss curvature iṡ ...
In this paper, we investigate the -null gluing problem for the Einstein vacuum equations, that is, we consider the null gluing of up to and including third-order derivatives of the metric. In the regime where the characteristic data is close to Minkowski data, we show that this -null gluing problem is solvable up to a 20-dimensional space of obstructions. The obstructions correspond to 20 linearly conserved quantities: 10 of which are already present in the -null gluing problem analysed by Aretakis, Czimek and Rodnianski, and 10 are novel obstructions inherent to the -null gluing problem. The 10 novel obstructions are linearly conserved charges calculated from third-order derivatives of the metric.
... The equations were originally derived by Teukolsky [Teu72] in the physics literature and form one of the landmarks of the "golden age" of black hole physics. In the asymptotically flat case, there is now an extensive mathematical literature on the long time behaviour of general solutions to the Teukolsky equations: The works of [DHR19b,DHR19a,Ma20] developed a robust understanding of the boundedness and decay properties of solutions in the Schwarzschild and slowly rotating Kerr case, which forms a key ingredient in the recent proofs of the nonlinear stability of these spacetimes [DHRT21,GKS22]. More recently, a treatment of the full-subextremal range |a| < M was provided in [STDC23]. ...
... We review here the generalisation of the physical space transformation theory introduced for the Teukolsky equation in the asymptotically flat case in [DHR19b]. are now coupled to one another and to the lower order Teukolsky boundary values as follows: ...
... Finally, one obtains bounds for α [−2] by integrating from data in the outgoing direction (using the bounds just established on ψ [−2] ) and then -now using the corresponding boundary condition relating α [−2] to α [+2] -bounds for α [+2] by integrating from the boundary in the ingoing direction (using the bounds just established on ψ [+2] ). While this procedure, carried out in Section 4.2, encounters an obvious loss of derivatives, this loss can finally be recovered through elliptic and inhomogeneous Morawetz estimates similar to the arguments carried out in [DHR19b,DHRT21] but now with a careful treatment of the coupling of the boundary terms in these estimates. The details are collected in Sections 4.3-4.6. ...
We prove boundedness and inverse logarithmic decay in time of solutions to the Teukolsky equations on Schwarzschild-Anti-de Sitter backgrounds with standard boundary conditions originating from fixing the conformal class of the non-linear metric on the boundary. The proofs rely on (1) a physical space transformation theory between the Teukolsky equations and the Regge-Wheeler equations on Schwarzschild-Anti de Sitter backgrounds and (2) novel energy and Carleman estimates handling the coupling of the two Teukolsky equations through the boundary conditions thereby generalising earlier work of \cite{Hol.Smu13} for the covariant wave equation. Specifically, we also produce purely physical space Carleman estimates.
... Remark 1.1. The theorem should be directly compared with the result of [DHR19] in the Λ = 0 case. The main difference is that here only a logarithmic decay rate (as opposed to inverse polynomial in [DHR19]) can be concluded. ...
... The theorem should be directly compared with the result of [DHR19] in the Λ = 0 case. The main difference is that here only a logarithmic decay rate (as opposed to inverse polynomial in [DHR19]) can be concluded. This is characteristic of the reflective boundary conditions as explained above. ...
... main difference compared with the asymptotically flat case is that here all quantities can be shown to decay without adding a residual pure gauge solution. This is to be constrasted with Theorem 3 in [DHR19] which establishes boundedness and Theorem 4 in [DHR19] where decay is established after having added to the solution an appropriately future normalised (dynamically determined) pure gauge solution. ...
This is the main paper of a series establishing the linear stability of Schwarzschild-Anti-de Sitter (AdS) black holes to gravitational perturbations. Specifically, we prove that solutions to the linearisation of the Einstein equations with around a Schwarzschild-AdS metric arising from regular initial data and with standard Dirichlet-type boundary conditions imposed at the conformal boundary (inherited from fixing the conformal class of the non-linear metric) remain globally uniformly bounded on the black hole exterior and in fact decay inverse logarithmically in time to a linearised Kerr-AdS metric. The proof exploits a hierarchical structure of the equations of linearised gravity in double null gauge and crucially relies on boundedness and logarithmic decay results for the Teukolsky system, which are independent results proven in Part II of the series. Contrary to the asymptotically flat case, addition of a residual pure gauge solution to the original solution is not required to prove decay of all linearised null curvature and Ricci coefficients. One may however normalise the solution at the conformal boundary to be in standard AdS-form by adding such a pure gauge solution, which is constructed dynamically from the trace of the original solution at the conformal boundary and quantitatively controlled by initial data.
... Moreover, it is expected that vanishing of both α [+2] and α [−2] reduces the space of solutions of the full system of linearised Einstein equations to (the infinite dimensional family of) infinitesimal diffeomorphisms, frame changes and (the finite dimensional family of) linearised Kerr solutions [33]. These facts, together with the decoupled equation (1) that they satisfy, make quantitative bounds on α [±2] an essential ingredient in many approaches to the stability problem [1,4,5,[13][14][15]18,22,23,30]. ...
... See also [27] for precise asymptotic decay rates. Progress started with a complete treatment of the Schwarzschild case in [14], where the authors prove (amongst other things) boundedness and integrated decay estimates for the Teukolsky equation. The key insight was a physical space version of the transformation theory of Chandrasekhar [7]: by applying a second order differential operator, the Teukolsky equation is transformed into a Regge-Wheeler-type equation, which does originate from a Lagrangian and for which estimates are known -in particular, there is a conserved energy! ...
... The main idea is to add a pure gauge solution that achieves the normalisation (59) for the linearised quantities on a fixed late outgoing cone C u f thereby making the flux on C u f appearing in the conservation law coercive. Note that this normalisation is different from the one used in [14] (where the solution is normalised at the horizon itself) but converges to the latter in the limit as u f → ∞. Of course, in the context of [14] estimates on (1) α, (1) α had already been obtained from the transformation theory (independently of any gauge choice) while the premise of the present paper is precisely to prove such estimates on any cone C u f . ...
We consider the equations of linearised gravity on the Schwarzschild spacetime in a double null gauge. Applying suitably commuted versions of the conservation laws derived in earlier work of the second author we establish control on the gauge invariant Teukolsky quantities α[±2]\upalpha ^{[\pm 2]} without any reference to the decoupled Teukolsky wave equation satisfied by these quantities. More specifically, we uniformly bound the energy flux of all first derivatives of α[±2]\upalpha ^{[\pm 2]} along any outgoing cone from an initial data quantity at the level of first derivatives of the linearised curvature and second derivatives of the linearised connection components. Analogous control on the energy fluxes along any ingoing cone is established a posteriori directly from the Teukolsky equation using the outgoing bounds.
... Out of all the various proofs of stability of the Minkowski spacetime floating about in the literature (e.g. [1, [46][47][48][49][50][51] and references therein), only Bieri's class of perturbations [52] and that of [53] is large enough to allow for such slow decay-all other proofs assume in particular that g = (1 + (2M/r))δ + o(r −1 )! Somewhat surprisingly, even though we must thus conclude that most stability works starting from a Cauchy hypersurface make assumptions too strong to model a physically relevant class of perturbations, we will see that these very perturbations exhibit sufficient decay towards I + so as to still be compatible with most stability works that start from an asymptotically null initial data hypersurface [54][55][56] (see §8). 5 (c) Structure of the remainder of the paper In §2, we give an outline of the post-Newtonian analysis of the generation problem for a system of N infalling masses coming from the infinite past. ...
... The (double null) system of linearized gravity around Schwarzschild is obtained by considering a general one-parameter family of Einstein metrics g(ε) on M M such that g(0) = g M : Ω, etc. and only keeping terms of order ε) (see [54] for details). The result is a coupled system of 10 hyperbolic equations for the linearized curvature coefficients-the linearized Bianchi equations-together with a system of around 30 transport and elliptic equations for the linearized metric and connection coefficients. ...
... In view of the existence of these solutions, it is very helpful to extract quantities The quantitative stability of (LGS) was first shown in [54] (and provided the central ingredient to the more recent nonlinear stability of Schwarzschild proof [55]), and a global scattering theory to (LGS) was written down in [68]. ...
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’.
... Through this paper, we follow the notations which were used in [67,68] (see also [14,18]) on the round metric and projected covariant derivatives on the 2-sphere S 2 (t,r) . ...
... To prove the third commutator, we have (see section 4.3.2 in [18]): ...
... This work can be useful for the construction of conformal scattering theories on Kerr spacetime for the scalar wave, tensorial wave and Maxwell equations.• The peeling properties of the tensorial wave equations(18) and(19)(where their rescaled forms are the tensorial Fackerell-Ipser equations ...
In this paper, we establish the constructions of conformal scattering theories for the tensorial wave equation such as the tensorial Fackerell-Ipser and the spin Teukolsky equations on Schwarzschild spacetime. In our strategy, we construct the conformal scattering for the tensorial Fackerell-Ipser equations which are obtained from the Maxwell equation and spin Teukolsky equations. Our method combines Penrose's conformal compactification and the energy decay results of the tensorial fields satisfying the tensorial Fackerell-Ipser equation to prove the energy equality of the fields through the conformal boundary \mathfrak{H}^+\cup \scri^+ (resp. \mathfrak{H}^-\cup \scri^-) and the initial Cauchy hypersurface . We will prove the well-posedness of the Goursat problem by using a generalization of H"ormander's results for the tensorial wave equations. By using the results for the tensorial Fackerell-Ipser equations we will establish the construction of conformal scattering for the spin Teukolsky equations.
... Conservation laws are essential ingredients when trying to prove energy estimates for hyperbolic equations on black hole backgrounds. Such energy estimates then let one infer boundedness and decay properties of solutions to such equations (see, for example, [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]). In many cases, such estimates arise from applications of the divergence theorem to identities satisfied by energy currents. ...
... The generalisation of the symplectic current (5) to the linearised vacuum Einstein equation (3) is ...
... This is in no small part due to the fact one has to make sure that the initial data one is considering satisfies the aforementioned admissibility criteria. Indeed, even for the 'basic' case of the stability of the 4-dimensional Schwarzschild black hole, the original work of Hollands and Wald did not establish positivity (the Schwarzschild black hole is after all (non-)linearly stable [5,6]). This was partially rectified in the work of Prabu and Wald [24] where they show that the canonical energy of a metric perturbation arising from a 'Hertz potential' is positive. ...
In this paper, we study the canonical energy associated with solutions to the linearised vacuum Einstein equation on a stationary spacetime. The main result of this paper establishes, in the context of the 4-dimensional Schwarzschild exterior, a direct correspondence between the conservation law satisfied by the canonical energy and the conservation laws deduced by Holzegel for gravitational perturbations in double null gauge. Since the latter exhibit useful coercivity properties (leading to energy and pointwise boundedness statements) we obtain coercivity results for the canonical energy in the double null gauge as a corollary. More generally, the correspondence suggests a systematic way to uncover coercivity properties in the conservation laws for the canonical energy on Kerr.
... (5) The spacetime metric, which determines the principal part of the wave operators under consideration, roughly speaking only needs to settle down to a stationary metric at a rate t −δ * , δ > 0. The class of metrics we can allow in 3 + 1 dimensions includes those arising in nonlinear stability problems for asymptotically flat solutions of Einstein's field equations [CK93,KN03,Lin17,HV20,KS21a,DHRT21,KS21b]; see also [HV23b,Hin23a]. The non-stationary wave operators are subject to a linear version of a weak null condition at null infinity [LR03], [HV20,Remark 1.7]. ...
... The first quasilinear results in asymptotically flat black hole settings were obtained by Lindblad-Tohaneanu [LT18,LT20]; see [LT22,Loo22c,Loo22a,Loo22b] for further nonlinear results. More recently, Dafermos-Holzegel-Taylor-Rodnianski [DHRT21] proved the codimension 3 nonlinear stability of the Schwarzschild family of black holes as solutions of the Einstein vacuum equations; while this is a tensorial equation, [DHRT21] phrases it is a coupled system for a large number of unknowns and utilizes a delicate hierarchical structure of the equations in which decay estimates for carefully chosen scalar quantities play a key role. Klainerman-Szeftel [KS21b] similarly base their proof of the nonlinear stability of slowly rotating Kerr spacetimes on estimates for a scalar quantity (the Teukolsky scalar). ...
... The first quasilinear results in asymptotically flat black hole settings were obtained by Lindblad-Tohaneanu [LT18,LT20]; see [LT22,Loo22c,Loo22a,Loo22b] for further nonlinear results. More recently, Dafermos-Holzegel-Taylor-Rodnianski [DHRT21] proved the codimension 3 nonlinear stability of the Schwarzschild family of black holes as solutions of the Einstein vacuum equations; while this is a tensorial equation, [DHRT21] phrases it is a coupled system for a large number of unknowns and utilizes a delicate hierarchical structure of the equations in which decay estimates for carefully chosen scalar quantities play a key role. Klainerman-Szeftel [KS21b] similarly base their proof of the nonlinear stability of slowly rotating Kerr spacetimes on estimates for a scalar quantity (the Teukolsky scalar). ...
We introduce a novel framework for the analysis of linear wave equations on nonstationary asymptotically flat spacetimes, under the assumptions of mode stability and absence of zero energy resonances for a stationary model operator. Our methods apply in all spacetime dimensions and to tensorial equations, and they do not require any symmetry or almost-symmetry assumptions on the spacetime metrics or on the wave type operators. Moreover, we allow for the presence of terms which are asymptotically scaling critical at infinity, such as inverse square potentials. For simplicity of presentation, we do not allow for normally hyperbolic trapping or horizons. In the first part of the paper, we study stationary wave type equations, i.e. equations with time-translation symmetry, and prove pointwise upper bounds for their solutions. We establish a relationship between pointwise decay rates and weights related to the mapping properties of the zero energy operator. Under a nondegeneracy assumption, we prove that this relationship is sharp by extracting leading order asymptotic profiles at late times. The main tool is the analysis of the resolvent at low energies. In the second part, we consider a class of wave operators without time-translation symmetry which settle down to stationary operators at a rate as an appropriate hyperboloidal time function tends to infinity. The main result is a sharp solvability theory for forward problems on a scale of polynomially weighted spacetime -Sobolev spaces. The proof combines a regularity theory for the nonstationary operator with the invertibility of the stationary model established in the first part. The regularity theory is fully microlocal and utilizes edge-b-analysis near null infinity, as developed in joint work with Vasy, and 3b-analysis in the forward cone.
... Thus the main result of this work, namely the blow-up asymptotics of ψ +2 at CH + , is a linear curvature instability statement for the Kerr Cauchy horizon. The Teukolsky equations were originally introduced to study the stability of the exterior of black holes, for example in [9] for the linear stability of the exterior of Schwarzschild black holes, and in [2,27] for the linearized stability of Kerr black holes. In the non-linear setting, they were used in [10,19] to prove the non-linear stability of Content courtesy of Springer Nature, terms of use apply. ...
... Moreover, by a Cauchy-Schwarz inequality 9 , ...
Using a purely physical-space analysis, we prove the precise oscillatory blow-up asymptotics of the spin +2 Teukolsky field in the interior of a subextremal Kerr black hole. In particular, this work gives a new proof of the blueshift instability of the Kerr Cauchy horizon against linearized gravitational perturbations that was first shown by Sbierski (Ann PDE 9:Art. 7, 2023). In that sense, this work supports the Strong Cosmic Censorship conjecture in Kerr spacetimes. The proof is an extension to the Teukolsky equation of the work by Ma and Zhang (Trans Am Math Soc 376:7815–7856, 2023) that treats the scalar wave equation in the interior of Kerr. The analysis relies on the generic polynomial decay on the event horizon of solutions of the Teukolsky equation that arise from compactly supported initial data, as recently proved by Ma and Zhang (Comm Math Phys 401:433–434, 2023) and Millet (, 2023, arXiv:2302.06946) in subextremal Kerr.
... These black holes are stationary solutions to the vacuum Einstein equation, a (3+1)-dimensional quasilinear wave equation. Our problem is simpler in several ways, including the gauge choice (compare our choice described in Section 1.5 with [9,22,25]) and the analysis of the linearized problem (compare the discussion in Section 1.2 with [3,8,18,20]). Nevertheless, in this paper and [29] we satisfactorily resolve a key issue that is shared by many soliton stability problems, but not with the black hole stability problem -this is the issue of modulation of the translation and boost parameters. ...
... wherek(r) = 0 for r < 0 andk(r) := − ∞ r k(s)ds for r ≥ 0. To motivate the final orthogonality condition, we want to choose Ωφ so that ℘ satisfies 9 8 We could also use the same definition as in (4.9) and (4.10) but since the moment condition for some constant β > 0 and with ω to be determined below (see the discussion leading to equation (3.34) in [29] for the motivation for introducing ω). Comparing with (3.9), and defining ...
We prove that the 3-dimensional catenoid is asymptotically stable as a solution to the hyperbolic vanishing mean curvature equation in Minkowski space, modulo suitable translation and boost (i.e., modulation) and with respect to a codimension one set of initial data perturbations. The modulation and the codimension one restriction on the initial data are necessary (and optimal) in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. The 3-dimensional problem is more challenging than the higher (specifically, 5 and higher) dimensional case addressed in the previous work of the authors with J.~L\"uhrmann, due to slower temporal decay of waves and slower spatial decay of the catenoid. To overcome these issues, we introduce several innovations, such as a proof of Morawetz- (or local-energy-decay-) estimates for the linearized operator with slowly decaying kernel elements based on the Darboux transform, a new method to obtain Price's-law-type bounds for waves on a moving catenoid, as well as a refined profile construction designed to capture a crucial cancellation in the wave-catenoid interaction. In conjunction with our previous work on the higher dimensional case, this paper outlines a systematic approach for studying other soliton stability problems for (3+1)-dimensional quasilinear wave equations.
... The first nonlinear stability result for any family of black hole spacetimes without symmetry assumptions was proved for slowly rotating Kerr-de Sitter black holes (ƒ > 0) by Hintz-Vasy [115]. The results in the asymptotically flat Kerr setting are not quite yet complete, though stability under special symmetries [139] as well as the full codimensional stability of the Schwarzschild family [59] are known. See also [137,138,140] for progress in the slowly rotating case. ...
... See also [137,138,140] for progress in the slowly rotating case. Finally, we refer the readers to [59,§IV.2] for a discussion of (necessarily codimension restricted) nonlinear stability statements that one could attempt to prove in the extremal case. 8 The metric perturbation is trace-free (with respect to the background Kerr metric) and has vanishing contraction with the ingoing principal null vector field. ...
... Bieri [6] lowered the decay assumptions to 0 > − 1 2 and required only N = 3 derivatives on the initial data. (There is a vast literature on extensions and variants of the nonlinear stability problem on asymptotically flat spacetimes, including [1,3,10,[13][14][15]19,24,28,29,31,37,44,45,47].) ...
... where ϒ(g; g m ) = g(g m ) −1 δ g G g g m as in (2.1). We then define 13 13 The definition of P g,E C ,E ϒ is consistent with the motivational Definition 2.2 for g = g m , as follows from a brief calculation using Lemma 2.1. ...
We give a short proof of the existence of a small piece of null infinity for (3+1)(3+1)-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the standard wave coordinate gauge in which all non-physical metric degrees of freedom have strong decay at null infinity. Using a formulation of the gauge-fixed Einstein vacuum equations which implements constraint damping, we establish this strong decay regardless of the validity of the constraint equations. On a technical level, we use notions from geometric singular analysis to give a streamlined proof of semiglobal existence for the relevant quasilinear hyperbolic equation.
... The Chandrasekhar transformation. The first quantitative proof of the linear stability of Schwarzschild was obtained by Dafermos-Holzegel-Rodnianski [DHR19b], where they extended the Chandrasekhar transformation, a transformation relating curvature and metric perturbations previously known only for mode solutions, to general solutions in physical-space. By applying such transfor- [HHV21], using adaptations of the Newman-Penrose formalism and wave coordinates respectively. ...
... To this equation one can apply techniques developed for the standard wave equation and deduce boundedness and decay properties for solutions to the original Teukolsky equation. Once control of the gauge-invariant curvature components and is obtained, the remaining work in[DHR19b] is to derive decay for the other curvature components and linearized Ricci coefficients associated to the their choice of gauge, given by the double null foliation, by making carefully chosen gauge conditions.A similar method to the above can be found in the case of the Kerr spacetime. Ma[Ma20] and Dafermos-Holzegel-Rodnianski[DHR19a] obtained generalizations of the Chandrasekhar transformation to Kerr which takes the Teukolsky equations to a generalized version of the Regge-Wheeler (gRW) equation. ...
... This framework is the one of angularly regular spacetimes in double null gauge. Requiring no symmetry assumption, the double null gauge has proved very flexible over the years, see for instance the important works [90][91][92]. If S is a compact 2-surface and u * , u * > 0, the Lorentzian manifold [0, u * ] × [0, u * ] × S, g is said to be in double null gauge if ...
We present the literature on Burnett’s conjecture in general relativity, which relate weak limits of vacuum solutions to relativistic kinetic theory. A special care is put on relating these works with early Choquet-Bruhat’s results on high-frequency gravitational waves and geometric optics.
... Note that Andersson et al. [69] also obtained a linear stability result for the Kerr family, but their decay assumptions on the data are stronger, which eliminates the linearized Kerr solution. Earlier results by Dafermos, Holzegel and Rodnianski [70] as well as Hung, Keller and Wang [71] establish linear stability of the Schwarzschild solution. ...
We review some of the important ideas of spectral and microlocal analysis which have been applied to problems in Mathematical General Relativity over the last decades.
... Dafermos-Luk proved in [23] the stability of Kerr's Cauchy horizon with respect to vacuum perturbations relaxing to Kerr at a fast, integrable rate (consistent with the fast rates one would obtain in the exterior problem [20,21]). For the model (1.1)-(1.4), the relaxation is conjectured to occur at a slower rate if = 0 (see the heuristics/numerics from [10,47,48]), which is a serious obstruction to asymptotic stability, even in spherical symmetry. ...
We construct a new one-parameter family, indexed by ϵ, of two-ended, spatially-homogeneous black hole interiors solving the Einstein–Maxwell–Klein–Gordon equations with a (possibly zero) cosmological constant Λ and bifurcating off a Reissner–Nordström-(dS/AdS) interior (ϵ=0). For all small ϵ≠0, we prove that, although the black hole is charged, its terminal boundary is an everywhere-spacelike Kasner singularity foliated by spheres of zero radius r. Moreover, smaller perturbations (i.e. smaller |ϵ|) are more singular than larger ones, in the sense that the Hawking mass and the curvature blow up following a power law of the form r-O(ϵ-2) at the singularity {r=0}. This unusual property originates from a dynamical phenomenon—violent nonlinear collapse—caused by the almost formation of a Cauchy horizon to the past of the spacelike singularity {r=0}. This phenomenon was previously described numerically in the physics literature and referred to as “the collapse of the Einstein–Rosen bridge”. While we cover all values of Λ∈R, the case Λ<0 is of particular significance to the AdS/CFT correspondence. Our result can also be viewed in general as a first step towards the understanding of the interior of hairy black holes.
... In this paper, we derive the Morawetz estimates for the axially symmetric gRW equation through the use of a novel choice of multiplier, which depends on the charge parameter, but which is independent on any mode decomposition. This is to be compared with previous choices appeared in Schwarzschild, for example in [6][12] (whose proof makes use of the decomposition in spherical harmonics 3 ) and [8] [25], which is independent of mode decomposition. See also [26] for choices in various spherically symmetric spacetimes. ...
We establish boundedness and polynomial decay results for the Teukolsky system in the exterior spacetime of very slowly rotating and strongly charged sub-extremal Kerr-Newman black holes, with a focus on axially symmetric solutions. The key step in achieving these results is deriving a physical-space Morawetz estimate for the associated generalized Regge-Wheeler system, without relying on spherical harmonic decomposition.
... where M := 1 − 2M r and the area radius r M (u, v) is implicitly defined by (see (98) in [26]) ...
This is the third paper in a series of papers adressing the characteristic gluing problem for the Einstein vacuum equations. We provide full details of our characteristic gluing (including the 10 charges) of strongly asymptotically flat data to the data of a suitably chosen Kerr spacetime. The choice of the Kerr spacetime crucially relies on relating the 10 charges to the ADM energy, linear momentum, angular momentum and the center-of-mass. As a corollary, we obtain an alternative proof of the Corvino-Schoen spacelike gluing construction for strongly asymptotically flat spacelike initial data.
... The propagation of waves in the space-time of a single black hole and the partial differential equations describing them have been studied for quite a long time, and exhaustive answers to many interesting aspects of the problems such as the linear stability of Schwarzschild black holes, decay of small solutions, Price's law, the formal mode analysis of the linearized equations, black hole shadow, particle creation, "John problem", and the Strauss conjecture are known. (See, e.g., [1,2,4,5,7,8,9,10,11,12,18,20,22,23,25,27,30,31,32,33] and references therein.) In most publications on the partial differential equations in cosmological backgrounds, the black hole is assumed to be eternal, that is, the space-time and the Schwarzschild radius are assumed to be static. ...
In this paper we prove the existence of global in time small data solutions of semilinear Klein-Gordon equations in the space-time with a static Schwarzschild radius in the expanding universe.
... In the Schwarzschild context considered here, such Morawetz estimate dates back to the works [1], [10] and various different approaches have been developed to obtain such estimates for linear solutions in [2], [6], [9], [11], [17], [20], [21], [35]. Also see generalizations and extensions to the more complicate family of Kerr black hole spacetimes in [3], [4], [16], [23], [24], [31], [48], [51], [52] and to the asymptotic flat spacetimes in [5], [37], [38], [50]. ...
In this paper, we study the long time dynamics of solutions to the defocusing semilinear wave equation on the Schwarzschild black hole spacetimes. For and sufficiently smooth and localized initial data, we show that the solution decays like in the domain of outer communication. The proof relies on the r-weighted vector field method of Dafermos-Rodnianski together with the Strichartz estimates for linear waves by Marzuola-Metcalfe-Tataru-Tohaneanu.
... We will discuss this in more detail in the following. Let us make the perturbation by ingoing shells an iterative process: we represent an ingoing, polynomially decreasing flux of radiation (which is usually the source of mass inflation [2,32] stemming from the decay of perturbations on the geometry [110,156]) with a sequence of ingoing shells of progressively smaller mass, ...
It is the goal of this thesis to revisit and revise the problem of black hole formation and evolution in semiclassical gravity -- a theory in which spacetime is treated classically, while matter admits a quantum description, coupling to gravity through an expectation value of a stress-energy tensor operator. We examine the vacuum expectation value of this operator in a variety of spacetimes in which trapped regions form or are close to forming, drawing conclusions regarding the semiclassical dynamics of spacetime in these scenarios.
... The stability of the Kerr black hole was recently obtained in a series of works by Klainerman, Szeftel and Giorgi [21,33,34,35,36]. In the case of the Schwarzschild black hole, Dafermos, Holzegel, Rodnianski and Taylor [16,17,18] showed codimensional stability and the asymptotic stability. Finally, Hintz and Vasy [25] proved nonlinear stability of Kerr under de Sitter gravity. ...
We consider the vacuum Einstein field equations under the Belinski-Zakharov symmetries. Depending on the chosen signature of the metric, these spacetimes contain most of the well-known special solutions in General Relativity, including well-known black holes. In this paper, we prove global existence of small Belinski-Zakharov spacetimes under a natural nondegeneracy condition. We also construct new energies and virial functionals to provide a description of the energy decay of smooth global cosmological metrics inside the light cone. Finally, some applications are presented in the case of generalized Kasner solitons.
... It might even become repulsive as conjectured in the framework of some alternative gravitational models. Still the Schwartzield model is the object of very refined mathematical investigations, see for instance [1] and the references therein. In addition to that, a lot of recent activity has been devoted to another type of singular black hole, not spherically symmetric but axi-symmetric, called Kerr's model, cf. in particular the very difficult results obtained in the long paper [2] and the bibliography therein. ...
... We note that the scaling vector field played a fundamental role in that work, similar to this one. In addition, we mention the exciting recent work concerning black hole stability (see, for example, [12], [11], and [29] and the references therein). These works required extending the philosophy of using weighted vector fields as commutators and multipliers to nontrivial backgrounds. ...
In this paper, we initiate the study of global stability for anisotropic systems of quasilinear wave equations. Equations of this kind arise naturally in the study of crystal optics, and they exhibit birefringence. We introduce a physical space strategy based on bilinear energy estimates that allows us to prove decay for the nonlinear problem. This uses decay for the homogeneous wave equation as a black box. The proof also requires us to interface this strategy with the vector field method and take advantage of the scaling vector field. A careful analysis of the spacetime geometry of the interaction between waves is necessary in the proof.
We study the problem of stability of the catenoid, which is an asymptotically flat rotationally symmetric minimal surface in Euclidean space, viewed as a stationary solution to the hyperbolic vanishing mean curvature equation in Minkowski space. The latter is a quasilinear wave equation that constitutes the hyperbolic counterpart of the minimal surface equation in Euclidean space. Our main result is the nonlinear asymptotic stability, modulo suitable translation and boost (i.e., modulation), of the n-dimensional catenoid with respect to a codimension one set of initial data perturbations without any symmetry assumptions, for . The modulation and the codimension one restriction on the data are necessary and optimal in view of the kernel and the unique simple eigenvalue, respectively, of the stability operator of the catenoid. In a broader context, this paper fits in the long tradition of studies of soliton stability problems. From this viewpoint, our aim here is to tackle some new issues that arise due to the quasilinear nature of the underlying hyperbolic equation. Ideas introduced in this paper include a new profile construction and modulation analysis to track the evolution of the translation and boost parameters of the stationary solution, a new scheme for proving integrated local energy decay for the perturbation in the quasilinear and modulation-theoretic context, and an adaptation of the vectorfield method in the presence of dynamic translations and boosts of the stationary solution.
General relativity is an area at the interface of partial differential equations, differential geometry, global analysis, mathematical physics and dynamical systems. It interacts with astrophysics, cosmology, high energy physics, and numerical analysis. The field is rapidly expanding and has witnessed remarkable developments and interconnections with other fields in recent years.The workshop Mathematical Aspects of General Relativity was organised by Carla Cederbaum (Tübingen), Mihalis Dafermos (Cambridge/Princeton), Jim Isenberg (Eugene) and Hans Ringström (KTH Stockholm). There were 48 on-site and 4 online participants. There were 16 one hour talks, nine 30 minute talks and four 10 minute talks.
We study the Einstein-Yang-Mills system in both the Lorenz and harmonic gauges, where the Yang-Mills fields are valued in any arbitrary Lie algebra , associated to any compact Lie group G. This gives a system of hyperbolic partial partial differential that does not satisfy the null condition and that has new complications that are not present for the Einstein vacuum equations nor for the Einstein-Maxwell system. We prove the exterior stability of the Minkowski space-time, , governed by the fully coupled Einstein-Yang-Mills system in the Lorenz gauge, valued in any arbitrary Lie algebra , without any assumption of spherical symmetry. We start with an arbitrary sufficiently small initial data, defined in a suitable energy norm for the perturbations of the Yang-Mills potential and of the Minkowski space-time, and we show the well-posedness of the Cauchy development in the exterior, and we prove that this leads to solutions converging in the Lorenz gauge and in wave coordinates to the zero Yang-Mills fields and to the Minkowski space-time. This provides a first detailed proof of the exterior stability of Minkowski governed by the fully non-linear Einstein-Yang-Mills equations in the Lorenz gauge, by using a null frame decomposition that was first used by H. Lindblad and I. Rodnianski for the case of the Einstein vacuum equations. We note that in contrast to the much simpler case of the Einstein-Maxwell equations where one can omit the potential, in fact in the non-abelian case of the Einstein-Yang-Mills equations, the question of stability, or non-stability, is a purely gauge dependent statement and the partial differential equations depend on the gauge on the Yang-Mills potential that is needed to write up the equations.
In this last part of the series we prove that the slow (inverse logarithmic) decay in time of solutions to the linearised Einstein equations on Schwarzschild-Anti-de Sitter backgrounds obtained in~\cite{Gra.Hol24,Gra.Hol24a} is in fact optimal by constructing quasimode solutions for the Teukolsky system. The main difficulties compared with the case of the scalar wave equation treated in earlier works arise from the first order terms in the Teukolsky equation, the coupling of the Teukolsky quantities at the conformal boundary and ensuring that the relevant quasimode solutions satisfy the Teukolsky-Starobinsky relations. The proof invokes a quasimode construction for the corresponding Regge-Wheeler system (which can be fully decoupled at the expense of a higher order boundary condition) and a reverse Chandrasekhar transformation which generates solutions of the Teukolsky system from solutions of the Regge-Wheeler system. Finally, we provide a general discussion of the well-posedness theory for the higher order boundary conditions that typically appear in the process of decoupling.
In this paper, we prove the codimension-one nonlinear asymptotic stability of the extremal Reissner-Nordstr\"om family of black holes in the spherically symmetric Einstein-Maxwell-neutral scalar field model, up to and including the event horizon. More precisely, we show that there exists a teleologically defined, codimension-one "submanifold" of the moduli space of spherically symmetric characteristic data for the Einstein-Maxwell-scalar field system lying close to the extremal Reissner-Nordstr\"om family, such that any data in evolve into a solution with the following properties as time goes to infinity: (i) the metric decays to a member of the extremal Reissner-Nordstr\"om family uniformly up to the event horizon, (ii) the scalar field decays to zero pointwise and in an appropriate energy norm, (iii) the first translation-invariant ingoing null derivative of the scalar field is approximately constant on the event horizon , (iv) for "generic" data, the second translation-invariant ingoing null derivative of the scalar field grows linearly along the event horizon. Due to the coupling of the scalar field to the geometry via the Einstein equations, suitable components of the Ricci tensor exhibit non-decay and growth phenomena along the event horizon. Points (i) and (ii) above reflect the "stability" of the extremal Reissner-Nordstr\"om family and points (iii) and (iv) verify the presence of the celebrated "Aretakis instability" for the linear wave equation on extremal Reissner-Nordstr\"om black holes in the full nonlinear Einstein-Maxwell-scalar field model.
We prove global existence, boundedness and decay for small data solutions to a general class of quasilinear wave equations on Kerr black hole backgrounds in the full sub-extremal range . The method extends our previous [DHRT22], which considered such equations on a wide class of background spacetimes, including Kerr, but restricted in that case to the very slowly rotating regime (which may be treated simply as a perturbation of Schwarzschild a=0). To handle the general case, our present proof is based on two ingredients: (i) the linear inhomogeneous estimates on Kerr backgrounds proven in [DRSR16], further refined however in order to gain a derivative in elliptic frequency regimes, and (ii) the existence of appropriate physical space currents satisfying degenerate coercivity properties, but which now must be tailored to a finite number of wave packets defined by suitable frequency projection. The above ingredients can be thought of as adaptations of the two basic ingredients of [DHRT22], exploiting however the peculiarities of the Kerr geometry. The novel frequency decomposition in (ii), inspired by the boundedness arguments of [DR11, DRSR16], is defined using only azimuthal and stationary frequencies, and serves both to separate the superradiant and non-superradiant parts of the solution and to localise trapping to small regions of spacetime. The strengthened (i), on the other hand, allows us to relax the required coercivity properties of our wave-packet dependent currents, so as in particular to accept top order errors provided that they are localised to the elliptic frequency regime. These error terms are analysed with the help of the full Carter separation.
The late-time behaviour of the solutions of the Fackerell-Ipser equation (which is a wave equation for the spin-zero component of the electromagnetic field strength tensor) on the closure of the domain of outer communication of sub-extremal Kerr spacetime is studied numerically. Within the Kerr family, the case of Schwarzschild background is also considered. Horizon-penetrating compactified hyperboloidal coordinates are used, which allow the behaviour of the solutions to be observed at the event horizon and at future null infinity as well. For the initial data, pure multipole configurations that have compact support and are either stationary or non-stationary are taken. It is found that with such initial data the solutions of the Fackerell-Ipser equation converge at late times either to a known static solution (up to a constant factor) or to zero. As the limit is approached, the solutions exhibit a quasinormal ringdown and finally a power-law decay. The exponents characterizing the power-law decay of the spherical harmonic components of the field variable are extracted from the numerical data for various values of the parameters of the initial data, and based on the results a proposal for a Price's law relevant to the Fackerell-Ipser equation is made. Certain conserved energy and angular momentum currents are used to verify the numerical implementation of the underlying mathematical model. In the construction of these currents a discrete symmetry of the Fackerell-Ipser equation, which is the product of an equatorial reflection and a complex conjugation, is also taken into account.
A bstract
We consider four-dimensional Euclidean gravity in a finite cavity. Dirichlet conditions do not yield a well-posed elliptic system, and Anderson has suggested boundary conditions that do. Here we point out that there exists a one-parameter family of boundary conditions, parameterized by a constant p , where a suitably Weyl rescaled boundary metric is fixed, and all give a well-posed elliptic system. Anderson and Dirichlet boundary conditions can be seen as the limits p → 0 and ∞ of these. Focussing on static Euclidean solutions, we derive a thermodynamic first law. Restricting to a spherical spatial boundary, the infillings are flat space or the Schwarzschild solution, and have similar thermodynamics to the Dirichlet case. We consider smooth Euclidean fluctuations about the flat space saddle; for p > 1/6 the spectrum of the Lichnerowicz operator is stable — its eigenvalues have positive real part. Thus we may regard large p as a regularization of the ill-posed Dirichlet boundary conditions. However for p < 1/6 there are unstable modes, even in the spherically symmetric and static sector. We then turn to Lorentzian signature. For p < 1/6 we may understand this spherical Euclidean instability as being paired with a Lorentzian instability associated with the dynamics of the boundary itself. However, a mystery emerges when we consider perturbations that break spherical symmetry. Here we find a plethora of dynamically unstable modes even for p > 1/6, contrasting starkly with the Euclidean stability we found. Thus we seemingly obtain a system with stable thermodynamics, but unstable dynamics, calling into question the standard assumption of smoothness that we have implemented when discussing the Euclidean theory.
We investigate the interior of a dynamical black hole as described by the Einstein–Maxwell-charged-Klein–Gordon system of equations with a cosmological constant, under spherical symmetry. In particular, we consider a characteristic initial value problem where, on the outgoing initial hypersurface, interpreted as the event horizon H + of a dynamical black hole, we prescribe: (a) initial data asymptotically approaching a fixed sub-extremal Reissner–Nordström–de Sitter solution and (b) an exponential Price law upper bound for the charged scalar field. After showing local well-posedness for the corresponding first-order system of partial differential equations, we establish the existence of a Cauchy horizon C H + for the evolved spacetime, extending the bootstrap methods used in the case Λ = 0 by Van de Moortel (Commun Math Phys 360:103–168, 2018. https://doi.org/10.1007/s00220-017-3079-3 ). In this context, we show the existence of C 0 spacetime extensions beyond C H + . Moreover, if the scalar field decays at a sufficiently fast rate along H + , we show that the renormalized Hawking mass remains bounded for a large set of initial data. With respect to the analogous model concerning an uncharged and massless scalar field, we are able to extend the known range of parameters for which mass inflation is prevented, up to the optimal threshold suggested by the linear analyses by Costa–Franzen (Ann Henri Poincaré 18:3371–3398, 2017. https://doi.org/10.1007/s00023-017-0592-z ) and Hintz–Vasy (J Math Phys 58(8):081509, 2017. https://doi.org/10.1063/1.4996575 ). In this no-mass-inflation scenario, which includes near-extremal solutions, we further prove that the spacetime can be extended across the Cauchy horizon with continuous metric, Christoffel symbols in L loc 2 and scalar field in H loc 1 . By generalizing the work by Costa–Girão–Natário–Silva (Commun Math Phys 361:289–341, 2018. https://doi.org/10.1007/s00220-018-3122-z ) to the case of a charged and massive scalar field, our results reveal a potential failure of the Christodoulou–Chruściel version of the strong cosmic censorship under spherical symmetry.
We review recent mathematical results concerning the high-frequency solutions to the Einstein vacuum equations and the limits of these solutions. In particular, we focus on two conjectures of Burnett, which attempt to give an exact characterization of high-frequency limits of vacuum spacetimes as solutions to the Einstein–massless Vlasov system. Some open problems and future directions are discussed.
This is the second paper in a series of papers addressing the characteristic gluing problem for the Einstein vacuum equations. We solve the codimension-10 characteristic gluing problem for characteristic data which are close to the Minkowski data. We derive an infinite-dimensional space of gauge-dependent charges and a 10-dimensional space of gauge-invariant charges that are conserved by the linearized null constraint equations and act as obstructions to the gluing problem. The gauge-dependent charges can be matched by applying angular and transversal gauge transformations of the characteristic data. By making use of a special hierarchy of radial weights of the null constraint equations, we construct the null lapse function and the conformal geometry of the characteristic hypersurface, and we show that the aforementioned charges are in fact the only obstructions to the gluing problem. Modulo the gauge-invariant charges, the resulting solution of the null constraint equations is for any specified integer in the tangential directions and in the transversal directions to the characteristic hypersurface. We also show that higher-order (in all directions) gluing is possible along bifurcated characteristic hypersurfaces (modulo the gauge-invariant charges).
We study the linear stability problem to gravitational and electromagnetic perturbations of the extremal , | Q | = M , Reissner–Nordström spacetime, as a solution to the Einstein–Maxwell equations. Our work uses and extends the framework [28, 32] of Giorgi, and contrary to the subextremal case we prove that instability results hold for a set of gauge invariant quantities along the event horizon H + . In particular, for associated quantities shown to satisfy generalized Regge–Wheeler equations we prove decay, non-decay, and polynomial blow-up estimates asymptotically along H + , the exact behavior depending on the number of translation invariant derivatives that we take. As a consequence, we show that for generic initial data, solutions to the generalized Teukolsky system of positive and negative spin satisfy both stability and instability results. It is worth mentioning that the negative spin solutions are significantly more unstable, with the extreme curvature component α ̲ not decaying asymptotically along the event horizon H + , a result previously unknown in the literature.
We consider solutions to the massless Vlasov equation on the domain of outer communications of the Schwarschild black hole. By adapting the -weighted energy method of Dafermos and Rodnianski, used extensively in order to study wave equations, we prove superpolynomial decay for a non-degenerate energy flux of the Vlasov field f through a well-chosen foliation. An essential step of this methodology consists in proving a non-degenerate integrated local energy decay. For this, we take in particular advantage of the redshift effect near the event horizon. The trapping at the photon sphere requires, however, to lose an of integrability in the velocity variable. Pointwise decay estimates on the velocity average of f are then obtained by functional inequalities, adapted to the study of Vlasov fields, which allow us to deal with the lack of a conservation law for the radial derivative.
Stability of Schwarzschild Family of Solutions in General Relativity
MSc Thesis defense presentation
by Abolfazl Chaman Motlagh
Supervisor: Dr Mohammad Safdari
In this paper, we initiate the study of characteristic event horizon gluing in vacuum. More precisely, we prove that Minkowski space can be glued along a null hypersurface to any round symmetry sphere in a Schwarzschild black hole spacetime as a solution of the Einstein vacuum equations. The method of proof is fundamentally nonperturbative and is closely related to our previous work in spherical symmetry [KU22] and Christodoulou's short pulse method [Chr09]. We also make essential use of the perturbative characteristic gluing results of Aretakis-Czimek-Rodnianski [ACR21a; CR22]. As an immediate corollary of our methods, we obtain characteristic gluing of Minkowski space to the event horizon of very slowly rotating Kerr with prescribed mass M and specific angular momentum a. Using our characteristic gluing results, we construct examples of vacuum gravitational collapse to very slowly rotating Kerr black holes in finite advanced time with prescribed M and . Our construction also yields the first example of a spacelike singularity arising from one-ended, asymptotically flat gravitational collapse in vacuum.
In this work, we compute the precise late-time asymptotics for the scalar field in the interior of a non-static subextreme Kerr black hole, based on recent progress on deriving its precise asymptotics in the Kerr exterior region. This provides a new proof of the generic H loc 1 H^1_{\text {loc}} -inextendibility of the Kerr Cauchy horizon against scalar perturbations that was first shown by Luk–Sbierski [J. Funct. Anal. 271 (2016), pp. 1948–1995]. The analogous results in Reissner–Nordström spacetimes are also discussed.
We study the massive scalar field equation on a stationary and spherically symmetric black hole g (including in particular the Schwarzschild and Reissner--Nordstr\"om black holes in the full sub-extremal range) for solutions projected on a fixed spherical harmonic. Our problem involves the scattering of an attractive long-range potential (Coulomb-like) and thus cannot be treated perturbatively. We prove precise (point-wise) asymptotic tails of the form , where f(t) is an explicit oscillating profile. Our asymptotics appear to be the first rigorous decay result for a massive scalar field on a black hole. Establishing these asymptotics is also an important step in retrieving the assumptions used in work of the third author regarding the interior of dynamical black holes and Strong Cosmic Censorship.
We give a short proof of the existence of a small piece of null infinity for (3+1)-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the standard wave coordinate gauge in which all non-physical metric degrees of freedom have strong decay at null infinity. Using a formulation of the gauge-fixed Einstein vacuum equations which implements constraint damping, we establish this strong decay regardless of the validity of the constraint equations. On a technical level, we use notions from geometric singular analysis to give a streamlined proof of semiglobal existence for the relevant quasilinear hyperbolic equation.
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