We present a characterization of the asymptotics of all asymptotically flat stationary vacuum solutions of Einstein's field equations. This characterization is given in terms of two sequences of symmetric trace free tensors (we call them the `null data'), which determine a formal expansion of the solution, and which are in a one to one correspondence to Hansen's multipoles. We obtain necessary and sufficient growth estimates on the null data to define an absolutely convergent series in a neighbourhood of spatial infinity. This provides a complete characterization of all asymptotically flat stationary vacuum solutions to the field equations.
We study formal expansions of asymptotically flat solutions to the static vacuum field equations which are determined by minimal sets of freely specifyable data referred to as `null data'. These are given by sequences of symmetric trace free tensors at space-like infinity of increasing order. They are 1:1 related to the sequences of Geroch multipoles. Necessary and sufficient growth estimates on the null data are obtained for the formal expansions to be absolutely convergent. This provides a complete characterization of all asymptotically flat solutions to the static vacuum field equations.
By using the Bondi-Sachs-van der Burg formalism we analyze the asymptotic properties at null infinity of axisymmetric electrovacuum spacetimes with a translational Killing vector and, in general, an infinite ``cosmic string'' (represented by a conical singularity) along the axis. Such spacetimes admit only a local null infinity. There is a non-vanishing news function due to the existence of the string even though there is no radiation. We prove that if null infinity has a smooth compact cross section and the spacetime is not flat in a neighbourhood of null infinity, then the translational Kill…
We discuss how asymptotic quantities, originally introduced on null infinity in terms of Bondi-type gauge conditions, can be calculated near space-like infinity to any desired precision.
It is shown that the precession of a gyroscope can be used to elucidate the nature of the smoothness of the null infinity of an asymptotically flat spacetime (describing an isolated body). A model for which the effects of precession in the non-smooth null infinity case are of order $r^{-2}\ln r$ is proposed. By contrast, in the smooth version the effects are of order $r^{-3}$. This difference should provide an effective criterion to decide on the nature of the smoothness of null infinity.
We describe our present understanding of the relations between the behaviour of asymptotically flat Cauchy data for Einstein's vacuum field equations near space-like infinity and the asymptotic behaviour of their evolution in time at null infinity.
This paper has been withdrawn by the author. It has come to my attention that most of the results reported here have already been obtained by Ludvigsen.
We show that the spin-2 equations on Minkowski space in the gauge of the `regular finite initial value problem at space-like infinity' imply estimates which, together with the transport equations on the cylinder at space-like infinity, allow us to obtain for a certain class of initial data information on the behaviour of the solution near space-like and null infinity of any desired precision.
We show that for general relativity in odd spacetime dimensions greater than 4, all components of the unphysical Weyl tensor for arbitrary smooth, compact spatial support perturbations of Minkowski spacetime fail to be smooth at null infinity at leading nonvanishing order. This implies that for nearly flat radiating spacetimes, the non-smoothness of the unphysical metric at null infinity manifests itself at the same order as it describes deviations from flatness of the physical metric. Therefore, in odd spacetime dimensions, it does not appear that conformal null infinity can be in any way use…
It is shown how the gauge of the ``regular finite initial value problem at spacelike infinity'' can be used to construct a certain type of estimates for the Maxwell field propagating on a Schwarzschild background. These estimates are constructed with the objective of obtaining information about the smoothness near spacelike and null infinity of a wide class of solutions to the Maxwell equations.
We prove an inequality relating the trace of the extrinsic curvature, the total angular momentum, the centre of mass, and the Trautman-Bondi mass for a class of gravitational initial data sets with constant mean curvature extending to null infinity. As an application we obtain non-existence results for the asymptotic Dirichlet problem for CMC hypersurfaces in stationary space-times.
The null infinity limit of the gravitational energy-momentum and energy flux determined by the covariant Hamiltonian quasi-local expressions is evaluated using the NP spin coefficients. The reference contribution is considered by three different embedding approaches. All of them give the expected Bondi energy and energy flux.
Motivated by our attempt to understand the question of angular momentum of a relativistic rotating source carried away by gravitational waves, in the asymptotic regime near future null infinity of the Kerr metric, a family of null hypersurfaces intersecting null infinity in shearfree (good) cuts are constructed by means of asymptotic expansion of the eikonal equation. The geometry of the null hypersurfaces as well asthe asymptotic structure of the Kerr metric near null infinity are studied. To the lowest order in angular momentum, the Bondi-Sachs form of the Kerr metric is worked out. The Newm…
We propose a method to determine the topological structure of an event horizon far in the future of a spacetime from the geometrical information of its future null infinity. In the present article, we mainly consider spacetimes with two black holes. Although, in most of cases, the black holes coalesce and their event horizon is topologically a single sphere far in the future, there are several possibilities that the black holes do not coalesce eternally and such exact solutions. In our formulation, the geometrical structure of future null infinity is related to the topological structure of the…
We describe the hyperboloidal compactification for Teukolsky equations in Kerr spacetime. We include null infinity on the numerical grid by attaching a hyperboloidal layer to a compact domain surrounding the rotating black hole and the orbit of an inspiralling point particle. This technique allows us to study, for the first time, gravitational waveforms from large- and extreme-mass-ratio inspirals in Kerr spacetime extracted at null infinity. Tests and comparisons of our results with previous calculations establish the accuracy and efficiency of the hyperboloidal layer method.
We re-express the Kerr metric in standard Bondi-Saches' coordinate near null infinity ${\cal I}^+$. Using the uniqueness result of characteristic initial value problem, we prove the Kerr metric is the only asymptotic flat, stationary, axial symmetric, Type-D solution of vacuum Einstein equation. The Taylor series of Kerr space-time is expressed in terms of B-S coordinates and the N-P constants have been calculated.
To answer a question of how can one observed angular momentum radiation near null infinity, one can first transform the dynamical twisting vacuum solution and make it satisfy Bondi coordinate conditions in the asymptotical region of the null infinity. We then obtain the Bondi-Sachs news function and also find the relationship of how does the angular momentum contribute to the news functions from the exact solution sense. By using the Komar's integral of angular momentum, the gravitational angular momentum flux of the dynamical twisting space-time can be obtained. All of our results can be comp…
Expansions of the gravitational field arising from the development of asymptotically Euclidean, time symmetric, conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6. To this ends a certain representation of spatial infinity as a cylinder is used. This set up is based on the properties of conformal geodesics. It is found that these expansions suggest that null infinity has to be non-smooth unless the Newman-Penrose constants of the spacetime, and some other higher order quantities of the spacetime vanish. As a consequence of these results …
The Conformal Einstein equations and the representation of spatial infinity as a cylinder introduced by Friedrich are used to analyse the behaviour of the gravitational field near null and spatial infinity for the development of data which are asymptotically Euclidean, conformally flat and time asymmetric. Our analysis allows for initial data whose second fundamental form is more general than the one given by the standard Bowen-York Ansatz. The Conformal Einstein equations imply upon evaluation on the cylinder at spatial infinity a hierarchy of transport equations which can be used to calculat…
A new approach to space-time asymptotics is presented, refining Penrose's idea of conformal transformations with infinity represented by the conformal boundary of space-time. Generalizing examples such as flat and Schwarzschild space-times, it is proposed that the Penrose conformal factor be a product of advanced and retarded conformal factors, which asymptotically relate physical and conformal null (light-like) coordinates and vanish at future and past null infinity respectively, with both vanishing at spatial infinity. A correspondingly refined definition of asymptotic flatness at both spati…
A study of the linearised gravitational field (spin 2 zero-rest-mass field) on a Minkowski background close to spatial infinity is done. To this purpose, a certain representation of spatial infinity in which it is depicted as a cylinder is used. A first analysis shows that the solutions generically develop a particular type of logarithmic divergence at the sets where spatial infinity touches null infinity. A regularity condition on the initial data can be deduced from the analysis of some transport equations on the cylinder at spatial infinity. It is given in terms of the linearised version of…
The energy-momentum and angular momentum contained in a spacelike two-surface of spherical topology are estimated by joining the two-surface to null infinity via an approximate no-incoming-radiation condition. The result is a set of gauge-invariant formulas for energy-momentum and angular momentum which should be applicable to much numerical work; it also gives estimates of the finite-size effects.
We investigated the relation between the behavior of gravitational wave at late time and the limit structure of future null infinity tangent which will determine the topology of the event horizon far in the future. In the present article, we mainly consider a spacetime with two black holes. Although in most of cases, the black holes coalesce and its event horizon is topologically a single sphere far in the future, there are several possibilities that the black holes never coalesce and such exact solutions as examples. In our formulation, the tangent vector of future null infinity is, under con…
The asymptotic behavior of Einstein-Rosen waves at null infinity in 4 dimensions is investigated in {\it all} directions by exploiting the relation between the 4-dimensional space-time and the 3-dimensional symmetry reduction thereof. Somewhat surprisingly, the behavior in a generic direction is {\it better} than that in directions orthogonal to the symmetry axis. The geometric origin of this difference can be understood most clearly from the 3-dimensional perspective.
We present two methods to include the asymptotic domain of a background spacetime in null directions for numerical solutions of evolution equations so that both the radiation extraction problem and the outer boundary problem are solved. The first method is based on the geometric conformal approach, the second is a coordinate based approach. We apply these methods to the case of a massless scalar wave equation on a Kerr spacetime. Our methods are designed to allow existing codes to reach the radiative zone by including future null infinity in the computational domain with minor modifications. W…
We discuss the existence of asymptotically Euclidean initial data sets to the vacuum Einstein field equations which would give rise (modulo an existence result for the evolution equations near spatial infinity) to developments with a past and a future null infinity of different smoothness. For simplicity, the analysis is restricted to the class of conformally flat, axially symmetric initial data sets. It is shown how the free parameters in the second fundamental form of the data can be used to satisfy certain obstructions to the smoothness of null infinity. The resulting initial data sets coul…
The behaviour of geometric quantities close to geometric pathologies of a spacetime is relevant to deduce the physical behaviour of the system. In this work, we compute the quasi-local mass quantities - the Hawking mass, the Brown-York mass and the Liu-Yau mass in the maximal extensions of the spherically symmetric solutions of the Einstein equations inside the black hole region, at the singularity, the event horizon, and the null infinity, in the limiting sense of a geometric flow.
Linear perturbations on Minkowski space are used to probe numerically the remote region of an asymptotically flat space-time close to spatial infinity. The study is undertaken within the framework of Friedrich's conformal field equations and the corresponding conformal representation of spatial infinity as a cylinder. The system under consideration is the (linear) zero-rest-mass equation for a spin-2 field. The spherical symmetry of the underlying background is used to decompose the field into separate non-interacting multipoles. It is demonstrated that it is possible to reach null-infinity fr…
The recently suggested quasi-local spin-angular momentum expressions, based on the Bramson superpotential and on the holomorphic or anti-holomorphic spinor fields, are calculated for large spheres near the future null infinity of asymptotically flat Einsten-Maxwell spacetimes. It is shown that although the expression based on the anti-holomorphic spinors is finite and unambiguously defined only in the center-of-mass frame (i.e. it diverges in general), the corresponding Pauli-Lubanski spin is always finite, free of ambiguities, and is built only from the gravitational data. Thus it defines a g…
We study gravitational perturbations of Schwarzschild spacetime by solving a hyperboloidal initial value problem for the Bardeen-Press equation. Compactification along hyperboloidal surfaces in a scri-fixing gauge allows us to have access to the gravitational waveform at null infinity in a general setup. We argue that this hyperboloidal approach leads to a more accurate and efficient calculation of the radiation signal than the common approach where a timelike outer boundary is introduced. The method can be generalized to study perturbations of Kerr spacetime using the Teukolsky equation.
This article uses the conformal Einstein equations and the conformal representation of spatial infinity introduced by Friedrich to analyse the behaviour of the gravitational field near null and spatial infinity for the development of initial data which are, in principle, non-conformally flat and time asymmetric. This article is the continuation of the investigation started in Class. Quantum Grav. 21 (2004) 5457-5492, where only conformally flat initial data sets were considered. For the purposes of this investigation, the conformal metric of the initial hypersurface is assumed to have a very p…
We investigate the asymptotic symmetries of Rindler space at null infinity and at the event horizon using both systematic and ad hoc methods. We find that the approaches that yield infinite-dimensional asymptotic symmetry algebras in the case of anti-de Sitter and flat spaces only give a finite-dimensional algebra for Rindler space at null infinity. We calculate the charges corresponding to these symmetries and confirm that they are finite, conserved, and integrable, and that the algebra of charges gives a representation of the asymptotic symmetry algebra. We also use relaxed boundary conditio…
Various works have suggested that the Bondi--Sachs--Penrose decay conditions on the gravitational field at null infinity are not generally representative of asymptotically flat space--times. We have made a detailed analysis of the constraint equations for ``asymptotically hyperboloidal'' initial data and find that log terms arise generically in asymptotic expansions. These terms are absent in the corresponding Bondi--Sachs--Penrose expansions, and can be related to explicit geometric quantities. We have nevertheless shown that there exists a large class of ``non--generic'' solutions of the con…
The ``standard'' expressions for total energy, linear momentum and also angular momentum of asymptotically flat Bondi metrics at null infinity are also obtained from differential conservation laws on asymptotically flat backgrounds, derived from a quadratic Lagrangian density by methods currently used in classical field theory. It is thus a matter of taste and commodity to use or not to use a reference spacetime in defining these globally conserved quantities. Backgrounds lead to Nœther conserved currents; the use of backgrounds is in line with classical views on conservation laws. Moreover, t…
After describing in short some problems and methods regarding the smoothness of null infinity for isolated systems, I present numerical calculations in which both spatial and null infinity can be studied. The reduced conformal field equations based on the conformal Gauss gauge allow us in spherical symmetry to calculate numerically the entire Schwarzschild-Kruskal spacetime in a smooth way including spacelike, null and timelike infinity and the domain close to the singularity.
In this work, we present a work in progress towards an efficient and economical computational module which interfaces between Cauchy and characteristic evolution codes. Our goal is to provide a standardized waveform extraction tool for the numerical relativity community which will allow CCE to be readily applied to a generic Cauchy code. The tool provides a means of unambiguous comparison between the waveforms generated by evolution codes based upon different formulations of the Einstein equations and different numerical approximation.
Certain aspects of the behaviour of the gravitational field near null and spatial infinity for the developments of asymptotically Euclidean, conformally flat initial data sets are analysed. Ideas and results from two different approaches are combined: on the one hand the null infinity formalism related to the asymptotic characteristic initial value problem and on the other the regular Cauchy initial value problem at spatial infinity which uses Friedrich's representation of spatial infinity as a cylinder. The decay of the Weyl tensor for the developments of the class of initial data under consi…
We consider AdS gravity duals to CFT on background spacetimes with a null infinity. Null infinity on the conformal boundary may extend to an extremal horizon in the bulk. For example it does so for Poincare-AdS, although does not for planar Schwarzschild-AdS. If null infinity does extend into an extremal horizon in the bulk, we show that the bulk near-horizon geometry is determined by the geometry of the boundary null infinity. Hence the `infra-red' geometry of the bulk is fixed by the large scale behaviour of the CFT spacetime. In addition the boundary stress tensor must have a particular dec…
In previous work, the numerical solution of the linearized gravitational field equations near space-like and null-infinity was discussed in the form of the spin-2 zero-rest-mass equation for the perturbations of the conformal Weyl curvature. The motivation was to study the behavior of the field and properties of the numerical evolution of the system near infinity using Friedrich's conformal representation of space-like infinity as a cylinder. It has been pointed out by H.O. Kreiss and others that the numerical evolution of a system using second order wave equations has several advantages compa…
Charged Scalar Field at Future Null Infinity via Nonlinear Hyperboloidal Evolution
CENTRA, Departamento de Física do Instituto Superior Técnico (IST), Universidade de Lisboa（里斯本大学高级理论研究中心物理系技术研究所）；Instituto de Telecomunicações (IT), Universidade de Aveiro（阿威罗大学电信研究所）；Departament de Física, Universitat de les Illes Balears（巴利阿里群岛大学物理系）
We define the angular momentum at null infinity in higher dimensions. The asymptotic symmetry at null infinity becomes the Poincare group in higher dimensions. This fact implies that the angular momentum can be defined without any ambiguities such as supertranslation in four dimensions. Indeed we can show that the angular momentum in our definition is transformed covariantly with respect to the Poincare group.
The dual-frame formalism leads to an approach to extend numerical relativity simulations in generalized harmonic gauge (GHG) all the way to null infinity. A major setback is that without care, even simple choices of initial data give rise to logarithmically divergent terms that would result in irregular variables and equations on the compactified domain, which would in turn prevent accurate numerical approximation. It has been shown, however, that a suitable choice of gauge and constraint addition can be used to prevent their appearance. Presently we give a first order symmetric hyperbolic red…
A representation of spatial infinity based in the properties of conformal geodesics is used to obtain asymptotic expansions of the gravitational field near the region where null infinity touches spatial infinity. These expansions show that generic time symmetric initial data with an analytic conformal metric at spatial infinity will give rise to developments with a certain type of logarithmic singularities at the points where null infinity and spatial infinity meet. These logarithmic singularities produce a non-smooth null infinity. The sources of the logarithmic singularities are traced back …
We discuss the asymptotic structure of null infinity in five dimensional space-time. Since it is known that the conformal infinity is not useful for odd higher dimensions, we shall employ the coordinate based method like the Bondi coordinate firstly introduced in four dimensions. Then we define the null infinity and identify the asymptotic symmetry. We also derive the Bondi mass expression and show its conservation law.
I describe the Schwarzian behavior of the gravitational field at null infinity under superrotations, to be understood as an \textit{unobservable} gauge artifact. To this end I review the induced geometry of null infinity and the construction of the gauge-invariant News tensor that characterizes gravitational radiation. I discuss potentially important implications for the celestial holography program, suggesting in particular that the uplift of the AdS$_3$/CFT$_2$ correspondence directly relates to the unobservable gauge sector.
We analyze the radiative aspects of nonsymmetric gravity theory to show that, in contrast to General Relativity, its nonstationary solutions cannot simultaneously exhibit acceptable asymptotic behavior at both future and past null infinity: good behavior at future null infinity is only possible through the use of advanced potentials with concomitant unphysical behavior at past null infinity.
We compute and analyze the gravitational waveform emitted to future null infinity by a system of two black holes in the large mass ratio limit. We consider the transition from the quasi-adiabatic inspiral to plunge, merger, and ringdown. The relative dynamics is driven by a leading order in the mass ratio, 5PN-resummed, effective-one-body (EOB), analytic radiation reaction. To compute the waveforms we solve the Regge-Wheeler-Zerilli equations in the time-domain on a spacelike foliation which coincides with the standard Schwarzschild foliation in the region including the motion of the small bla…
When Einstein's equations for an asymptotically flat, vacuum spacetime are reexpressed in terms of an appropriate conformal metric that is regular at (future) null infinity, they develop apparently singular terms in the associated conformal factor and thus appear to be ill-behaved at this (exterior) boundary. In this article however we show, through an enforcement of the Hamiltonian and momentum constraints to the needed order in a Taylor expansion, that these apparently singular terms are not only regular at the boundary but can in fact be explicitly evaluated there in terms of conformally re…
Matching conditions at null infinity in the presence of logarithms: the role of advanced and retarded radiation
Facultad de Ingeniería, Universidad San Sebastián（圣塞巴斯蒂安大学工程学院）；Université Libre de Bruxelles（布鲁塞尔自由大学）；International Solvay Institutes（国际索尔维研究所）；Collège de France, Université PSL（法兰西公学院，巴黎文理研究大学）；Centro de Estudios Científicos (CECs)（科学研究中心）；Erwin Schrödinger Int. Institute for Mathematics and Physics, University of Vienna（维也纳大学埃尔温·薛定谔数学与物理国际研究所）
We consider a characteristic problem of the vacuum Einstein equations with part of the initial data given on a future complete null cone with suitable decay, and show that the solution exists uniformly around the null cone for general such initial data. We can then define a segment of the future null infinity. The initial data are not required to be small and the decaying condition inherits from the works of \cite{Ch-K} and \cite{K-N}.
We provide a mapping between past null and future null infinity in three-dimensional flat space, using symmetry considerations. From this we derive a mapping between the corresponding asymptotic symmetry groups. By studying the metric at asymptotic regions, we find that the mapping is energy preserving and yields an infinite number of conservation laws.
In this paper, using the Bondi coordinates, we discuss the angular momentum at null infinity in five dimensions and address the Poincare covariance of the Bondi mass and angular momentum. We also show the angular momentum loss/gain law due to gravitational waves. In four dimensions, the angular momentum at null infinity has the supertranslational ambiguity and then it is known that we cannot construct well-defined angular momentum there. On the other hand, we would stress that we can define angular momentum at null infinity without any ambiguity in higher dimensions. This is because of the non…
