No file available
To read the file of this research,
you can request a copy directly from the authors.
A canonical foliation on null infinity in perturbations of Kerr
Preprints and early-stage research may not have been peer reviewed yet.
Abstract
Kerr stability for small angular momentum has been proved in the series of works by Klainerman-Szeftel, Giorgi-Klainerman-Szeftel and Shen. Some of the most basic conclusions of the result, concerning various physical quantities on the future null infinity are derived in the work of Klainerman-Szeftel. Further important conclusions were later derived in An-He-Shen and Chen-Klainerman. In this paper, based on the existence and uniqueness results for GCM spheres by Klainerman-Szeftel, we establish the existence of a canonical foliation on the future null infinity for which the null energy, linear momentum, center of mass and angular momentum are well defined and satisfy the expected physical laws of gravitational radiation. The rigid character of this foliation eliminates the usual ambiguities related to these quantities in the physics literature. We also show that under the initial assumption of Klainerman-Szeftel, the center of mass of the black hole has a large deformation (recoil) after the perturbation.
ResearchGate has not been able to resolve any citations for this publication.
In 2003, Klainerman and Nicolò [14] proved the stability of Minkowski in the case of the exterior of an outgoing null cone. Relying on the method used in [14], Caciotta and Nicolò [2] proved the stability of Kerr spacetime in external regions, i.e. outside an outgoing null cone far away from the Kerr event horizon. In this paper, we give a new proof of [2]. Compared to [2], we reduce the number of derivatives needed in the proof, simplify the treatment of the last slice, and provide a unified treatment of the decay of initial data which contains in particular the initial data considered by Klainerman and Szeftel in [20]. Also, concerning the treatment of curvature estimates, similar to [25], we replace the vectorfield method used in [2, 14] by rp–weighted estimates introduced by Dafermos and Rodnianski in [8].
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.
This is a follow-up of [5] on the general covariant modulated (GCM) procedure in perturbations of Kerr. In this paper, we construct GCM hypersurfaces, which play a central role in extending GCM admissible spacetimes in [7] where decay estimates are derived in the context of nonlinear stability of Kerr family for |a|≪m. As in [4], the central idea of the construction of GCM hypersurfaces is to concatenate a 1–parameter family of GCM spheres of [5] by solving an ODE system. The goal of this paper is to get rid of the symmetry restrictions in the GCM procedure introduced in [4] and thus remove an essential obstruction in extending the results to a full stability proof of the Kerr family.
This the first in a series of papers whose ultimate goal is to establish the full nonlinear stability of the Kerr family for |a|≪m. The paper builds on the strategy laid out in [6] in the context of the nonlinear stability of Schwarzschild for axially symmetric polarized perturbations. In fact the central idea of [6] was the introduction and construction of generally covariant modulated (GCM) spheres on which specific geometric quantities take Schwarzschildian values. This was made possible by taking into account the full general covariance of the Einstein vacuum equations. The goal of this, and its companion paper [7], is to get rid of the symmetry restriction in the construction of GCM spheres in [6] and thus remove an essential obstruction in extending the result to a full stability proof of the Kerr family.
This is a follow-up of our paper (Klainerman and Szeftel in Construction of GCM spheres in perturbations of Kerr, Accepted for publication in Annals of PDE) on the construction of general covariant modulated (GCM) spheres in perturbations of Kerr, which we expect to play a central role in establishing their nonlinear stability. We reformulate the main results of that paper using a canonical definition of ℓ=1 modes on a 2-sphere embedded in a 1+31+3 vacuum manifold. This is based on a new, effective, version of the classical uniformization theorem which allows us to define such modes and prove their stability for spheres with comparable metrics. The reformulation allows us to prove a second, intrinsic, existence theorem for GCM spheres, expressed purely in terms of geometric quantities defined on it. A natural definition of angular momentum for such GCM spheres is also introduced, which we expect to play a key role in determining the final angular momentum for general perturbations of Kerr.
The final black hole left behind after a binary black hole merger can attain a recoil velocity, or a "kick," reaching values up to 5000 km/s. This phenomenon has important implications for gravitational wave astronomy, black hole formation scenarios, testing general relativity, and galaxy evolution. We consider the gravitational wave signal from the binary black hole merger GW200129_065458 (henceforth referred to as GW200129), which has been shown to exhibit strong evidence of orbital precession. Using numerical relativity surrogate models, we constrain the kick velocity of GW200129 to v_{f}∼1542_{-1098}^{+747} km/s or v_{f}≳698 km/s (one-sided limit), at 90% credibility. This marks the first identification of a large kick velocity for an individual gravitational wave event. Given the kick velocity of GW200129, we estimate that there is a less than 0.48% (7.7%) probability that the remnant black hole after the merger would be retained by globular (nuclear star) clusters. Finally, we show that kick effects are not expected to cause biases in ringdown tests of general relativity for this event, although this may change in the future with improved detectors.
We study how conserved quantities such as angular momentum and center of mass evolve with respect to the retarded time at null infinity, which is described in terms of a Bondi–Sachs coordinate system. These evolution formulae complement the classical Bondi mass loss formula for gravitational radiation. They are further expressed in terms of the potentials of the shear and news tensors. The consequences that follow from these formulae are (1) Supertranslation invariance of the fluxes of the CWY conserved quantities. (2) A conservation law of angular momentum à la Christodoulou. (3) A duality paradigm for null infinity. In particular, the supertranslation invariance distinguishes the CWY angular momentum and center of mass from the classical definitions.
We study the nonlinear stability of the (3+1)-dimensional Minkowski spacetime as a solution of the Einstein vacuum equation. Similarly to our previous work on the stability of cosmological black holes, we construct the solution of the nonlinear initial value problem using an iteration scheme in which we solve a linearized equation globally at each step; we use a generalized harmonic gauge and implement constraint damping to fix the geometry of null infinity. The linear analysis is largely based on energy and vector field methods originating in work by Klainerman. 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. Upon compactifying to a manifold with corners, with boundary hypersurfaces corresponding to spacelike, null, and timelike infinity, we show, using the framework of Melrose’s b-analysis, that polyhomogeneous initial data produce a polyhomogeneous spacetime metric. Finally, 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.
For a spacelike 2-surface in spacetime, we propose a new definition of
quasi-local angular momentum and quasi-local center of mass, as an element in
the dual space of the Lie algebra of the Lorentz group. Together with previous
defined quasi-local energy-momentum, this completes the definition of conserved
quantities in general relativity at the quasi-local level. We justify this
definition by showing the consistency with the theory of special relativity and
expectations on an axially symmetric spacetime. The limits at spatial infinity
provide new definitions for total conserved quantities of an isolated system,
which do not depend on any asymptotically flat coordinate system or asymptotic
Killing field. The new proposal is free of ambiguities found in existing
definitions and presents the first definition that precisely describes the
dynamics of the Einstein equation.
A simple and purely algebraic construction of super-energy (s-e) tensors for arbitrary fields is presented in any dimensions. These tensors have good mathematical and physical properties, and they can be used in any theory having as basic arena an n-dimensional manifold with a metric of Lorentzian signature. In general, the completely timelike component of these s-e tensors has the mathematical features of an energy density: they are positive definite and satisfy the dominant property, which provides s-e estimates useful for global results and helpful in other matters, such as the causal propagation of the fields. Similarly, super-momentum vectors appear with the mathematical properties of s-e flux vectors.
The classical Bel and Bel–Robinson tensors for the gravitational fields are included in our
general definition. The energy–momentum and super-energy tensors of physical fields are also obtained, and the procedure will be illustrated by writing down these tensors explicitly for the cases of scalar, electromagnetic, and Proca fields. Moreover, higher order (super)k-energy tensors are defined and shown to be meaningful and in agreement for the different physical fields. In flat spacetimes, they provide infinitely many conserved quantities.
In non-flat spacetimes, the fundamental question of the interchange of s-e quantities between different fields is addressed, and answered affirmatively. Conserved s-e currents are found for any minimally coupled scalar field whenever there is a Killing vector. Furthermore, the exchange of gravitational and electromagnetic super-energy is also shown by studying the propagation of discontinuities. This seems to open the door for new types of conservation physical laws.
Following Bekenstein's (1973) work on recoiling black holes, the
gravitational wave linear momentum flux from a binary system of two
point masses in Keplerian orbit is calculated. The quasi-Newtonian
approach is adopted, and the resulting motion of the center of mass is
calculated. As such a system decays via gravitational wave energy
losses, the size of the orbit decreases until the components merge or
become tidally disrupted. Thereafter, the center of mass moves with the
linear momentum necessary to balance that carried off by the
gravitational waves. In the case of a binary black hole system, the
velocity of the center of mass could be of astrophysical significance,
although numerical studies would be necessary to check this claim.
We show that, under stronger asymptotic decay and regularity properties than those used in Christodoulou D and Klainerman S (1993 The Global Non Linear Stability of the Minkowski Space (Princeton Mathematical Series vol 41) (Princeton, NJ: Princeton University Press)) and Klainerman S and Nicolò F (2003 The Evolution Problem in General Relativity (Progress in Mathematical Physics vol 25) (Boston: Birkhauser)), asymptotically flat initial datasets lead to solutions of the Einstein vacuum equations which have strong peeling properties consistent with the predictions of the conformal compactification approach of Penrose. More precisely we provide a systematic picture of the relationship between various asymptotic properties of the initial datasets and the peeling properties of the corresponding solutions.
We prove, outside the influence region of a ball of radius R
0 centred in the origin of the initial data hypersurface, Σ0, the existence of global solutions near to Kerr spacetime, provided that the initial data are sufficiently near to those
of Kerr. This external region is the “far” part of the outer region of the perturbed Kerr spacetime. Moreover, if we assume
that the corrections to the Kerr metric decay sufficiently fast, o(r
−3), we prove that the various null components of the Riemann tensor decay in agreement with the “Peeling theorem”.
In general relativity, the notion of mass and other conserved quantities at spatial infinity can be defined in a natural way via the Hamiltonian framework: Each conserved quantity is associated with an asymptotic symmetry and the value of the conserved quantity is defined to be the value of the Hamiltonian which generates the canonical transformation on phase space corresponding to this symmetry. However, such an approach cannot be employed to define `conserved quantities' in a situation where symplectic current can be radiated away (such as occurs at null infinity in general relativity) because there does not, in general, exist a Hamiltonian which generates the given asymptotic symmetry. (This fact is closely related to the fact that the desired `conserved quantities' are not, in general, conserved!) In this paper we give a prescription for defining `conserved quantities' by proposing a modification of the equation that must be satisfied by a Hamiltonian. Our prescription is a very general one, and is applicable to a very general class of asymptotic conditions in arbitrary diffeomorphism covariant theories of gravity derivable from a Lagrangian, although we have not investigated existence and uniqueness issues in the most general contexts. In the case of general relativity with the standard asymptotic conditions at null infinity, our prescription agrees with the one proposed by Dray and Streubel from entirely different considerations. Comment: 39 pages, no figures
This is the main paper in a series of three in which we prove the nonlinear stability of the Schwarzschild spacetime under axially symmetric polarized perturbations, i.e. solutions of the Einstein vacuum equations for asymptotically flat 1+3 dimensional Lorentzian metrics which admit a hypersurface orthogonal spacelike Killing vectorfield with closed orbits. While building on the remarkable advances made in last 15 years on establishing quantitative linear stability, the paper introduces a series of new ideas among which we emphasize the \textit{general covariant modulation} (GCM) procedure which allows us to construct, dynamically, the center of mass frame of the final state. The mass of the final state itself is tracked using the well known Hawking mass relative to a well adapted foliation itself connected to the center of mass frame. Our work here is the first to prove the nonlinear stability of Schwarzschild in a restricted class of nontrivial perturbations, i.e. perturbations for which new ideas, such as our GCM procedure are needed. To a large extent, the restriction to this class of perturbations is only needed to ensure that the final state of evolution is another Schwarzschild space. We are thus confident that our procedure may apply in a more general setting.
Isolated objects in asymptotically flat spacetimes in general relativity are
characterized by their conserved charges associated with the
Bondi-Metzner-Sachs (BMS) group. These charges include total energy, linear
momentum, intrinsic angular momentum and center-of-mass location, and in
addition an infinite number of supermomentum charges associated with
supertranslations. Recently it has been suggested that the BMS symmetry algebra
should be enlarged to include an infinite number of additional symmetries known
as superrotations. We compute the corresponding "super center-of-mass'' charges
and show that they are finite and well defined. The supermomentum charges are
associated with ordinary gravitational wave memory, and the super
center-of-mass charges are associated with total (ordinary plus null)
gravitational wave memory, in the terminology of Bieri and Garfinkle. Some of
these charges can give rise to black hole hair, as described by Strominger and
Zhiboedov. We clarify how this hair evades the no hair theorems.
This paper is divided into four parts. In part A, some general considerations about gravitational radiation are followed by a treatment of the scalar wave equation in the manner later to be applied to Einstein's field equations. In part B, a co-ordinate system is specified which is suitable for investigation of outgoing gravitational waves from an isolated axi-symmetric reflexion-symmetric system. The metric is expanded in negative powers of a suitably defined radial co-ordinate r, and the vacuum field equations are investigated in detail. It is shown that the flow of information to infinity is controlled by a single function of two variables called the news function. Together with initial conditions specified on a light cone, this function fully defines the behaviour of the system. No constraints of any kind are encountered. In part C, the transformations leaving the metric in the chosen form are determined. An investigation of the corresponding transformations in Minkowski space suggests that no generality is lost by assuming that the transformations, like the metric, may be expanded in negative powers of r. In part D, the mass of the system is defined in a way which in static metrics agrees with the usual definition. The principal result of the paper is then deduced, namely, that the mass of a system is constant if and only if there is no news; if there is news, the mass decreases monotonically so long as it continues. The linear approximation is next discussed, chiefly for its heuristic value, and employed in the analysis of a receiver for gravitational waves. Sandwich waves are constructed, and certain non-radiative but non-static solutions are discussed. This part concludes with a tentative classification of time-dependent solutions of the types considered.
Techniques are studied for treating asymptotic problems in special and ; general relativity. Attention is devoted to: the geometrical definition of ; asymptotic flatness; covariant definitions of incoming and outgoing gravitational ; and other radiation fields; the detailed asymptotic behavior of the Riemann ; tensor and other fields; characteristics of the total energy-momentum; ; unification of finite and asymptotic versions of the characteristic initial value ; problem; and the geometrical derivation of an asymptotic symmetry group. ; (T.F.H.);
Although considerable progress has been made in generalizing the concept of angular momentum to general relativity, until now no satisfactory definition that allows for the exchange of angular momentum has been given. I here give the first such definition. It is a definition at null infinity, the place and time where gravity waves reach in the limit far from all masses. The definition applies to any isolated system of masses including those that change their angular momentum L by emitting gravity waves. L˙ is given solely in terms of parameters in principle measurable directly by Michelson interferometer gravitational wave detectors such as LIGO or LISA.
Gravitational fields containing bounded sources and gravitational radiation are examined by analyzing their properties at spatial infinity. A convenient way of splitting the metric tensor and the Einstein field equations, applicable in any space-time, is first introduced. Then suitable boundary conditions are set. The group of co-ordinate transformations that preserves the boundary conditions is analyzed. Different possible gravitational fields are characterized intrinsically by a combination of (i) characteristic initial data, and (ii) Dirichlet data at spatial infinity. To determine a particular solution one must specify four functions of three variables and three functions of two variables; these functions are not subject to constraints. A method for integrating the field equations is given; the asymptotic behaviour of the metric and Riemann tensors for large spatial distances is analyzed in detail; the dynamical variables of the radiation modes are exhibited; and a superposition principle for the radiation modes of the gravitational field is suggested. Among the results are: (i) the group of allowed co-ordinate transformations contains the inhomogeneous orthochronous Lorentz group as a subgroup; (ii) each of the five leading terms in an asymptotic expansion of the Riemann tensor has the algebraic structure previously predicted from analyzing the Petrov classification; (iii) gravitational waves appear to carry mass away from the interior; (iv) time-dependent periodic solutions of the field equations which obey the stated boundary conditions do not exist. It was found that the general fields studied in the present work are in many ways very similar to the axially symmetric fields recently studied by Bondi, van der Burg & Metzner.
The definition of angular momentum recently given by R. Penrose [Proc. R. Soc. Lond. Ser. A 381, No. 1780, 53–63 (1982)] is analysed on I. It is shown that this definition is essentially a supertranslation of previous definitions [for example, the one by Streubel in Gen. Relativity Gravitation 9, No. 6, 551–561 (1978)]. The origin dependence of Penrose’s angular momentum is shown to have the correct form. Based on Penrose’s expression, the authors then define a momentum for all elements of the BMS Lie algebra, which is the first such expression with the property that its flux vanishes identically in Minkowski space.
In this paper, we sketch the proof of the extension of the stability theorem of the Minkowski space in General Relativity done explicitly in previous work by the present author. We discuss solutions of the Einstein vacuum (EV) equations. We solve the Cauchy problem for more general, asymptotically flat initial data than in the pioneering work `The Global Nonlinear Stability of the Minkowski Space' of D. Christodoulou and S. Klainerman or than in any other work. Moreover, we describe precisely the asymptotic behaviour. Our relaxed assumptions on the initial data yield a spacetime curvature which is not bounded in . As a major result, we encounter in our work borderline cases, which we discuss in this paper as well. The fact that certain of our estimates are borderline in view of decay indicates that the conditions in our main theorem are sharp in so far as the assumptions on the decay at infinity on the initial data are concerned. Thus, the borderline cases are a consequence of our relaxed assumptions on the data. They are not present in the other works, as all of them place stronger assumptions on their data. We work with an invariant formulation of the EV equations. Our main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime. This result generalizes the work `The Global Nonlinear Stability of the Minkowski Space' of D. Christodoulou and S. Klainerman. Comment: 50 pages
It is shown that gravitational waves from astronomical sources have a nonlinear effect on laser interferometer detectors on earth, an effect which has hitherto been neglected, but which is of the same order of magnitude as the linear effects. The signature of the nonlinear effect is a permanent displacement of test masses after the passage of a wave train.
We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach shows that the Einstein-vacuum and the Einstein-scalar field equations with general asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time. The proof utilizes the classical harmonic (also known as de Donder or wave coordinate) gauge.
We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with the Schwarzschild solution in the neighborhood of space-like infinity. The result contradicts previous beliefs that wave coordinates are “unstable in the large” and provides an alternative approach to the stability problem originally solved ( for unrestricted data, in a different gauge and with a precise description of the asymptotic behavior at null infinity) by D. Christodoulou and S. Klainerman.
Using the wave coordinate gauge we recast the Einstein equations as a system of quasilinear wave equations and, in absence of the classical null condition, establish a small data global existence result. In our previous work we introduced the notion of a weak null condition and showed that the Einstein equations in harmonic coordinates satisfy this condition.The result of this paper relies on this observation and combines it with the vector field method based on the symmetries of the standard Minkowski space.
In a forthcoming paper we will address the question of stability of Minkowski space for the Einstein vacuum equations in wave coordinates for all “small” asymptotically flat data and the case of the Einstein equations coupled to a scalar field.
The inspiral and merger of binary black holes will likely involve black holes with both unequal masses and arbitrary spins. The gravitational radiation emitted by these binaries will carry angular as well as linear momentum. A net flux of emitted linear momentum implies that the black hole produced by the merger will experience a recoil or kick. Previous studies have focused on the recoil velocity from unequal mass, non-spinning binaries. We present results from simulations of equal mass but spinning black hole binaries and show how a significant gravitational recoil can also be obtained in these situations. We consider the case of black holes with opposite spins of magnitude a aligned/anti-aligned with the orbital angular momentum, with a the dimensionless spin parameters of the individual holes. For the initial setups under consideration, we find a recoil velocity of V = 475 \KMS a. Supermassive black hole mergers producing kicks of this magnitude could result in the ejection from the cores of dwarf galaxies of the final hole produced by the collision. Comment: 8 pages, 8 figures, replaced with version accepted for publication in ApJ
- X An
- T He
- D Shen
X. An, T. He and D. Shen, Angular momentum memory effect, arXiv:2403.11133.
- X Chen
- S Klainerman
X. Chen and S. Klainerman, Regularity of the Future Event Horizon in Perturbations of
Kerr, arXiv:2409.05700.
The Global Initial Value Problem in General Relativity
- D Christodoulou
D. Christodoulou, The Global Initial Value Problem in General Relativity. In V. G.
Gurzadyan, R. T. Jantzen, and R. Ruffini, editors, The Ninth Marcel Grossmann Meeting,
44-54, December 2002.
- M Dafermos
- G Holzegel
- I Rodnianski
- M Taylor
M. Dafermos, G. Holzegel, I. Rodnianski and M. Taylor, The non-linear stability of the
Schwarzschild family of black holes, arXiv:2104.08222.
- E Giorgi
- S Klainerman
- J Szeftel
E. Giorgi, S. Klainerman, and J. Szeftel, A general formalism for the stability of Kerr,
arXiv:2002.02740 (2020).
- E Giorgi
- S Klainerman
- J Szeftel
E. Giorgi, S. Klainerman and J. Szeftel, Wave equations estimates and the nonlinear stability of slowly rotating Kerr black holes, arXiv:2205.14808, To appear in Pure Appl. Math.
Q..
- O Graf
O. Graf, Global nonlinear stability of Minkowski space for spacelike-characteristic initial
data, arXiv:2010.12434, To appear in Mémoires de la SMF.
The Evolution Problem in General Relativity
- S Klainerman
- F Nicolò
S. Klainerman and F. Nicolò, The Evolution Problem in General Relativity. Progress in
Mathematical Physics, 25, Birkhauser, Boston, 2003.
- D Shen
D. Shen, Global stability of Minkowski spacetime with minimal decay, arXiv:2310.07483.
- D Shen
D. Shen, Exterior stability of Minkowski spacetime with borderline decay, arXiv:2405.00735.