There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at null infinity of an asymptotically flat spacetime in the Bondi-Sachs coordinates. The quasi-local mass leads to a local description of the radiation that is purely gravitational at null infinity. In particular, the quasi-local mass is evaluated in terms of the news function of the Bondi-Sachs coordinates.
The constraints on the asymptotic behavior of the conformal factor and conformal extrinsic curvature imposed by the initial value equations of general relativity on constant mean extrinsic curvature (CMC) hypersurfaces are analyzed in detail. We derive explicit formulas for the Bondi-Sachs energy and momentum in terms of coefficients of asymptotic expansions on CMC hypersurfaces near future null infinity. Precise numerical results for the Bondi-Sachs energy, momentum, and angular momentum are used to interpret physically Bowen-York solutions of the initial value equations on conformally flat C…
In this paper, we show that the peeling property still holds for Bondi-Sachs metrics with nonzero cosmological constant under the boundary condition given by Sommerfeld's radiation condition together with three nontrivial $Λ$-independent functions $B$, $a$, $b$. This should indicate the new boundary condition is natural. Moreover, we construct some nonstationary vacuum Bondi-Sachs metrics without Bondi news, which Newmann-Penrose quantities fall faster than usual. This provides a new feature of gravitational waves for nonzero cosmological constant.
We study the Bondi-Sachs rockets with nonzero cosmological constant. We observe that the acceleration of the systems arises naturally in the asymptotic symmetries of (anti-) de Sitter spacetimes. Assuming the validity of the concepts of energy and mass previously introduced in asymptotically flat spacetimes, we find that the emission of pure radiation energy balances the loss of the Bondi mass in certain special families of the Bondi-Sachs rockets, so in these there is no gravitational radiation.
We calculate the Hilbert action for the Bondi-Sachs metrics. It yields the Einstein vacuum equations in a closed form. Following the Dirac approach to constrained systems we investigate the related Hamiltonian formulation.
How does one compute the Bondi mass on an arbitrary cut of null infinity $\scri$ when it is not presented in a Bondi system? What then is the correct definition of the mass aspect? How does one normalise an asymptotic translation computed on a cut which is not equipped with the unit-sphere metric? These are questions which need to be answered if one wants to calculate the Bondi-Sachs energy-momentum for a space-time which has been determined numerically. Under such conditions there is not much control over the presentation of $\scri$ so that most of the available formulations of the Bondi ener…
The Bondi-Sachs formalism of General Relativity is a metric-based treatment of the Einstein equations in which the coordinates are adapted to the null geodesics of the spacetime. It provided the first convincing evidence that gravitational radiation is a nonlinear effect of general relativity and that the emission of gravitational waves from an isolated system is accompanied by a mass loss from the system. The asymptotic behaviour of the Bondi-Sachs metric revealed the existence of the symmetry group at null infinity, the Bondi-Metzner-Sachs group, which turned out to be larger than the Poinca…
A metric representing the Kerr geometry has been obtained by Pretorius and Israel. We make a coordinate transformation on this metric, thereby bringing it into Bondi-Sachs form. In order to validate the metric, we evaluate it numerically on a regular grid of the new coordinates. The Ricci tensor is then computed, for different discretizations, and found to be convergent to zero. We also investigate the behaviour of the metric near the axis of symmetry and confirm regularity. Finally we investigate a Bondi-Sachs representation of the Kerr geometry reported by Fletcher and Lun; we confirm numeri…
We construct the gravitational energy-momentum of the Bondi-Sachs space-time, in the famework of the teleparallel equivalent of general relativity (TEGR). The Bondi-Sachs line element describes gravitational radiation in the asymptotic region of the space-time, and is determined by the mass aspect and by two functions, $c$ and $d$, that yield the news functions, which are interpreted as the radiating degrees of freedom of the gravitational field. The standard expression for the Bondi-Sachs energy-momentum is constructed in terms of the mass aspect only. The expression that we obtain in the con…
We present a numerical study of the evolution of a non-linearly disturbed black hole described by the Bondi--Sachs metric, for which the outgoing gravitational waves can readily be found using the news function. We compare the gravitational wave output obtained with the use of the news function in the Bondi--Sachs framework, with that obtained from the Weyl scalars, where the latter are evaluated in a quasi-Kinnersley tetrad. The latter method has the advantage of being applicable to any formulation of Einstein's equations---including the ADM formulation and its various descendants---in additi…
This work is a pedagogical review dedicated to a modern description of the Bondi-Metzner-Sachs group. The curved space-times that will be taken into account are the ones that suitably approach, at infinity, Minkowski space-time. In particular we will focus on asymptotically flat space-times. In this work the concept of asymptotic symmetry group of those space-times will be studied. In the first two sections we derive the asymptotic group following the classical approach which was basically developed by Bondi, van den Burg, Metzner and Sachs. This is essentially the group of transformations bet…
The constrained Hamiltonian analysis of geometric actions is worked out before applying the construction to the extended Bondi-Metzner-Sachs group in four dimensions. For any Hamiltonian associated with an extended BMS$_4$ generator, this action provides a field theory in two plus one spacetime dimensions whose Poisson bracket algebra of Noether charges realizes the extended BMS$_4$ Lie algebra. The Poisson structure of the model includes the classical version of the operator product expansions that have appeared in the context of celestial holography. Furthermore, the model reproduces the evo…
We linearize the Einstein equations when the metric is Bondi-Sachs, when the background is Schwarzschild or Minkowski, and when there is a matter source in the form of a thin shell whose density varies with time and angular position. By performing an eigenfunction decomposition, we reduce the problem to a system of linear ordinary differential equations which we are able to solve. The solutions are relevant to the characteristic formulation of numerical relativity: (a) as exact solutions against which computations of gravitational radiation can be compared; and (b) in formulating boundary cond…
The relation between gravitational memory effects and Bondi-Metzner-Sachs symmetry of the asymptotically flat spacetimes is studied in the scalar-tensor theory. For this purpose, the solutions to the equations of motion near the future null infinity are obtained in the generalized Bondi-Sachs coordinates with a suitable determinant condition. It turns out that the Bondi-Metzner-Sachs group is also a semi-direct product of an infinitesimal dimensional supertranslation group and the Lorentz group as in general relativity. There are also degenerate vacua in both the tensor and the scalar sectors …
The asymptotic symmetries in Brans-Dicke theory are analyzed using Penrose's conformal completion method, which is independent of the coordinate system used. These symmetries indeed include supertranslations and Lorentz transformations for an asymptotically flat spacetime. With the Wald-Zoupas formalism, "conserved charges" and fluxes of the Bondi-Metzner-Sachs algebra are computed. The scalar degree of freedom contributes only to the Lorentz boost charge, although it plays a role in various fluxes. The flux-balance laws are further used to constrain displacement memory, spin memory and center…
Dray and Streubel proposed a definition of angular momentum in general relativity based on `Bondi-Metzner-Sachs (BMS) charges'. I show here that the natural definition of center of mass in this program has an infinite-dimensional ambiguity. (This seems not to have been noticed before because previous work has tacitly carried over a special-relativistic assumption.) A related point is that the natural definition of spin in this context is translation-, but not supertranslation-, invariant.
A formula is given for the variation of the Hawking energy along any one-parameter foliation of compact spatial 2-surfaces. A surface for which one null expansion is positive and the other negative has a preferred orientation, with a spatial or null normal direction being called outgoing or ingoing as the area increases or decreases respectively. A natural way to propagate such a surface through a hypersurface is to choose the foliation such that the null expansions are constant over each surface. For such uniformly expanding foliations, the Hawking energy is non-decreasing in any outgoing dir…
The peeling properties of a lightlike signal propagating through a general Bondi-Sachs vacuum spacetime and leaving behind another Bondi-Sachs vacuum space-time are studied. We demonstrate that in general the peeling behavior is the conventional one which is associated with a radiating isolated system and that it becomes unconventional if the asymptotically flat space-times on either side of the history of the light-like signal tend to flatness at future null infinity faster than the general Bondi-Sachs space-time. This latter situation occurs if, for example, the space-times in question are s…
In the gravitational context, Liouville theory is the two-dimensional conformal field theory that controls the boundary dynamics of asymptotically AdS_3 spacetimes at the classical level. By taking a suitable limit of the coupling constants of the Hamiltonian formulation of Liouville, we construct and analyze a BMS_3 invariant two-dimensional field theory that is likely to control the boundary dynamics at null infinity of three dimensional asymptotically flat gravity.
The excitation of a black hole by infalling matter or radiation has been studied for a long time, mostly in linear perturbation theory. In this paper we study numerically the response of a Schwarzschild black hole to an incoming gravitational wave pulse. We present a numerically well-posed initial boundary value problem for the generalized conformal field equations in which a wave profile for the ingoing wave is specified at an outer time-like boundary which then hits an initially static and spherically symmetric black hole. The non-linear interaction of the black hole with the gravitational w…
Gravitational radiation plays an important role in astrophysics. Based on the fact that our universe is expanding, the gravitational radiation when a positive cosmological constant is presented has been studied along with two different ways recently, one is the Bondi-Sachs (BS) framework in which the result is shown by BS quantities in the asymptotic null structure, the other is the perturbation approach in which the result is presented by the quadrupoles of source. Therefore, it is worth to interpret the quantities in asymptotic null structure in terms of the information of the source. In thi…
In the Bondi-Sachs gauge, the Einstein equations with a cosmological constant coupled to a scalar field in spherical symmetry are cast into a first order strongly hyperbolic formulation in which the lapse and shift are the fundamental variables. For this system of equations, the lapse and shift are ingoing characteristic fields, and the scalar field has three modes: ingoing, outgoing and static, respectively. A constraint-preserving initial boundary value problem is constructed by using Bianchi identity. Using this scheme, we find that any small perturbation of the scalar field at the boundary…
It is shown that certain kinds of behavior, which hitherto were expected to be characteristic for classical gravity and quantum field theory in curved spacetime, as the infinite dimensional Bondi-Metzner-Sachs symmetry, holography on event horizons and an area proportionality of entropy, have in fact an unnoticed presence in Minkowski QFT. This casts new light on the fundamental question whether the volume propotionality of heat bath entropy and the (logarithmically corrected) dimensionless area law obeyed by localization-induced thermal behavior are different geometric parametrizations which …
We exhibit a Hamiltonian formulation, both for electromagnetism and gravitation, in which it is not required that the Bondi "news" vanish, but only that the incoming news be equal to the outgoing ones. This requirement is implemented by defining the fields on a two-sheeted hyperbolic surface, which we term "the hourglass". It is a spacelike deformation of the complete lightcone. On it one approaches asymptotically (null) past and future infinity while remaining at a fixed (hyperbolic) time, by going to large spatial distances on its two sheets. The Hamiltonian formulation and -- in particular …
In the Bondi-Sachs formulation of General Relativity space-time is foliated via a family of null cones. If these null cones are defined such that their vertices are traced by a regular world-line then the metric tensor has to obey regularity conditions at the vertices. We explore these regularity conditions when the world line is a time-like geodesic. In particular, we solve the Einstein equations for the Bondi-Sachs metric near the vertices for axially symmetric vacuum space- times. The metric is calculated up to third order corrections with respect to a flat metric along the time-like geodes…
We study asymptotically flat spacetimes in five spacetime dimensions by Hamiltonian methods, focusing on spatial infinity and keeping all asymptotically relevant nonlinearities in the transformation laws and in the charge-generators. Precise boundary conditions that lead to a consistent variational principle are given. We show that the algebra of asymptotic symmetries, which had not been uncovered before, is a nonlinear deformation of the semi-direct product of the Lorentz algebra by an abelian algebra involving four independent (and not just one) arbitrary functions of the angles on the $3$-s…
An idealized observer of an astronomical event is situated at future null infinity, where light rays emitted from the source approach. Mathematically, null infinity corresponds to the portion of the spacetime boundary defined by equivalence classes of null geodesics. But what can we observe at future null infinity? In the note, we start by reviewing the description of null infinity in terms of the Bondi-Sachs coordinate system. In particular, we reformulate and extend the invariance of Bondi mass and the mass loss formula in terms of a modified mass aspect 2-form. At the end, we discuss new re…
Super-BMS$_4$ algebras -- also called BMS$_4$ superalgebras -- are graded extensions of the BMS$_4$ algebra. They can be of two different types: they can contain either a finite number or an infinite number of fermionic generators. We show in this letter that, with suitable boundary conditions on the graviton and gravitino fields at spatial infinity, supergravity on asymptotically flat spaces possesses as superalgebra of asymptotic symmetries a (nonlinear) super-BMS$_4$ algebra containing an infinite number of fermionic generators, which we denote SBMS$_4$. These boundary conditions are not on…
In this work the asymptotic structure of space-time and the main properties of the Bondi-Metzner-Sachs (BMS) group, which is the asymptotic symmetry group of asymptotically flat space-times, are analysed. Every chapter, except the fourth, begins with a brief summary of the topics that will be dealt through it and an introduction to the main concepts. The work can be divided into three principal parts. The first part includes the first two chapters and is devoted to the development of the mathematical tools that will be used throughout all of the work. In particular we will introduce the notion…
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…
Gravitational-wave memory effects are identified by their distinctive effects on families of freely falling observers: after a burst of waves pass by their locations, memory effects can cause lasting relative displacements of the observers. These effects are closely related to the infrared properties of gravity and other massless field theories, including their asymptotic symmetries and conserved quantities. In this paper, we investigate the connection between memory effects, symmetries, and conserved quantities in Brans-Dicke theory. We compute the field equations in Bondi coordinates, and we…
The details are presented of a new evolution algorithm for the characteristic initial-boundary value problem based upon an affine parameter rather than the areal radial coordinate used in the Bondi-Sachs formulation. The advantages over the Bondi-Sachs version are discussed, with particular emphasis on the application to the characteristic extraction of the gravitational waveform from Cauchy simulations of general relativistic astrophysical systems.
Scalaron Quasinormal and Quasi-Bound Modes of Schwarzschild Black Holes in the Bondi-Sachs Formulation of Metric $f(R)$ Gravity
University of Cape Town（开普敦大学）；Botswana International University of Science and Technology（博茨瓦纳国际科学与技术大学）；University of South Africa（南非大学）；University of Oslo（奥斯陆大学）
The nature of superluminal photon propagation in the gravitational field describing radiation from a time-dependent, isolated source (the Bondi-Sachs metric) is considered in an effective theory which includes interactions which violate the strong equivalence principle. Such interactions are, for example, generated by vacuum polarisation in conventional QED in curved spacetime. The relation of the resulting light-cone modifications to the Peeling Theorem for the Bondi-Sachs spacetime is explained.
The Tolman-Bondi and Vaidya solutions are two solutions to Einstein equations which describe dust particles and null fluid, respectively. We show that it is possible to match the two solutions in one single spacetime, the Tolman-Bondi--Vaidya spacetime. The new spacetime is divided by a null surface with Tolman-Bondi dust on one side and Vaidya fluid on the other side. The differentiability of the spacetime is discussed. By constructing a specific solution, we show that the metric across the null surface can be at least $C^1$ and the stress-energy tensor is continuous.
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.
We use zero angular momentum null geodesics in the Kerr-de Sitter spacetime to transform the metric in a generalised Bondi coordinate system. We write the metric components explicitly. Next, we choose the radial coordinate to be the areal coordinate and write the asymptotic metric in the Bondi-Sachs gauge.
When a system emits gravitational radiation, the Bondi mass decreases. If the Bondi energy is Hamiltonian, it can thus only be a time dependent Hamiltonian. In this paper, we show that the Bondi energy can be understood as a time-dependent Hamiltonian on the covariant phase space. Our derivation starts from the Hamiltonian formulation in domains with boundaries that are null. We introduce the most general boundary conditions on a generic such null boundary, and compute quasi-local charges for boosts, energy and angular momentum. Initially, these domains are at finite distance, such that there …
We study the geometric properties of holomorphic distributions of totally null $m$-planes on a $(2m+ε)$-dimensional complex Riemannian manifold $(\mathcal{M}, \bm{g})$, where $ε\in {0,1}$ and $m \geq 2$. In particular, given such a distribution $\mathcal{N}$, say, we obtain algebraic conditions on the Weyl tensor and the Cotton-York tensor which guarrantee the integrability of $\mathcal{N}$, and in odd dimensions, of its orthogonal complement. These results generalise the Petrov classification of the (anti-)self-dual part of the complex Weyl tensor, and the complex Goldberg-Sachs theorem from …
We discuss the Hamiltonian dynamics of general relativity with real connection variables on a null foliation, and use the Newman-Penrose formalism to shed light on the geometric meaning of the various constraints. We identify the equivalent of Sachs' constraint-free initial data as projections of connection components related to null rotations, i.e. the translational part of the ISO(2) group stabilising the internal null direction soldered to the hypersurface. A pair of second-class constraints reduces these connection components to the shear of a null geodesic congruence, thus establishing eq…
We consider smooth null cones in a vacuum spacetime that extend to future null infinity. For such cones that are perturbations of shear-free outgoing null cones in Schwarzschild spacetimes, we prove bounds for the Bondi energy, momentum, and rate of energy loss. The bounds depend on the closeness between the given cone and a corresponding cone in a Schwarzschild spacetime, measured purely in terms of the differences between certain weighted $L^2$-norms of the space-time curvature on the cones, and of the geometries of the spheres from which they emanate. A key step in this paper is the constru…
We obtain an energy inequality on null surfaces $u=const$ in the Bondi-Sachs formalism. We show that for a sufficiently regular event horizon $H$ there is an affine radial coordinate which is constant on $H$. Then the energy inequality can be prolongated to the horizon giving an estimation which is closely related to the Penrose inequality. We test it for the Kerr solution written in the Fletcher-Lun coordinates.
Consider compact objects --such as neutron star or black hole binaries-- in \emph{full, non-linear} general relativity. In the case with zero cosmological constant $Λ$, the gravitational radiation emitted by such systems is described by the well established, 50+ year old framework due to Bondi, Sachs, Penrose and others. However, so far we do not have a satisfactory extension of this framework to include a \emph{positive} cosmological constant --or, more generally, the dark energy responsible for the accelerated expansion of the universe. In particular, we do not yet have an adequate gauge inv…
We give a brief review of the definition of the Wang-Yau quasilocal mass and discuss the evaluation of which on surfaces of unit size at null infinity of an axi-symmetric spacetime in Bondi-van der Burg-Metzner coordinates.
The spacetime of a boosted Bondi-Sachs rotating black hole is considered as a proper background to examine electromagnetic configurations connected to analytic solutions of Maxwell equations. In our analysis, we first use the Bondi-Sachs transformations in order to bring the boosted rotating black hole metric into the Kerr-Schild form, from which zero angular momentum observers (ZAMOs) are constructed via the ADM formalism. In Kerr-Schild coordinates we obtain the Killing fields as sources of Maxwell electrodynamics, and we fix a ZAMO in order to evaluate the components of the electric and mag…
We present the argument that the past limit of the Trautman-Bondi mass is the ADM mass under weak hypotheses on the decay of the metric towards spatial infinity, without any smallness conditions on the initial data, assuming well defined energy, momentum, center of mass and angular moment. Part of the proof consists of a careful inspection of the proof of stability of Minkowski space-time, which is sketched. This is complemented by an interior gluing result for asymptotically flat initial data with well defined Poincaré charges, which is proved in detail.
We calculate the limits of the quasi-local angular momentum and center-of-mass defined by Chen-Wang-Yau \cite{CWY} for a family of spacelike two-spheres approaching future null infinity in an asymptotically flat spacetime admitting a Bondi-Sachs expansion. Our result complements earlier work of Chen-Wang-Yau \cite{CWY2}, where the authors calculate the limits of the quasi-local energy and linear momentum at null infinity. Finiteness of the center-of-mass limit requires that the spacetime be in the so-called center-of-mass frame, a mild assumption on the mass aspect function amounting to vanish…
A recent refinement of Penrose's conformal framework for asymptotically flat space-times is summarized. The key idea concerns advanced and retarded conformal factors, which allow a rigid description of infinity as a locally metric light cone. In the new framework, the Bondi-Sachs energy-flux integrals of ingoing and outgoing gravitational radiation decay at spatial infinity such that the total radiated energy is finite, and the Bondi-Sachs energy-momentum has a unique limit at spatial infinity, coinciding with the uniquely rendered ADM energy-momentum.
