Bondi-Metzner-Sachs Group Research Articles

Homogeneous Projective Coordinates for the Bondi–Metzner–Sachs Group

This paper studies the Bondi–Metzner–Sachs group in homogeneous projective coordinates because it is then possible to write all transformations of such a group in a manifestly linear way. The 2-sphere metric, the Bondi–Metzner–Sachs metric, asymptotic Killing vectors, generators of supertranslations as well as boosts and rotations of Minkowski spacetime are all re-expressed in homogeneous projective coordinates. Lastly, the integral curves of vector fields which generate supertranslations are evaluated in detail. This work paves the way for more advanced applications of the geometry of asymptotically flat spacetime in projective coordinates by virtue of the tools provided from complex analysis in several variables and projective geometry.

The BMS group in D = 6 spacetime dimensions

The asymptotic structure of gravity in D = 6 spacetime dimensions is described at spatial infinity in the asymptotically flat context through Hamiltonian (ADM) methods. Special focus is given on the Bondi–Metzner–Sachs (BMS) supertranslation subgroup. It is known from previous studies that the BMS group contains more supertranslations as one goes from D = 4 to D = 5. Indeed, while the supertranslations are described by one single function of the angles in D = 4, four such functions are neeeded in D = 5. We consider the case D = 6 with the aim of determining whether the number of supertranslations keeps increasing with the dimension or remains equal to the number found in D = 5. We show that even though there is apparent room for more supertranslations, their number remains equal to the D = 5 value (four): the potentially new supertranslations turn out to define proper gauge transformations corresponding to a redundancy in the description of the system. Critical in the analysis are the boundary conditions chosen to yield a well-defined canonical formalism. Given the computational (but not conceptual) complexity as one increases the dimension, we explicitly discuss the linearized theory and argue that asymptotically, this analysis provides the correct picture. We conclude by considering higher spacetime dimensions where we indicate that the number of physically relevant supertranslations remains equal to four independently of the dimension ⩾5 . Contribution to Stanley Deser memorial volume ‘Gravity, Strings and Beyond’

Fully nonlinear transformations of the Weyl-Bondi-Metzner-Sachs asymptotic symmetry group

The asymptotic symmetry group of general relativity in asymptotically flat spacetimes can be extended from the Bondi-Metzner-Sachs (BMS) group to the generalized BMS (GMBS) group suggested by Campiglia and Laddha, which includes arbitrary diffeomorphisms of the celestial two-sphere. It can be further extended to the Weyl BMS (BMSW) group suggested by Freidel, Oliveri, Pranzetti and Speziale, which includes general conformal transformations. We compute the action of fully nonlinear BMSW transformations on the leading order Bondi-gauge metric functions: specifically, the induced metric, Bondi mass aspect, angular momentum aspect, and shear. These results generalize previous linearized results in the BMSW context by Freidel et al., and also nonlinear results in the BMS context by Chen, Wang, Wang and Yau. The transformation laws will be useful for exploring implications of the BMSW group.

Lie theory for asymptotic symmetries in general relativity: The NU group

We study the Newman–Unti (NU) group from the viewpoint of infinite-dimensional geometry. The NU group is a topological group in a natural coarse topology, but it does not become a manifold and hence a Lie group in this topology. To obtain a manifold structure we consider a finer Whitney-type topology. This turns the unit component of the NU group into an infinite-dimensional Lie group. We then study the Lie theoretic properties of this group. Surprisingly, the group operations of the full NU group become discontinuous, whence the NU group does not support a Lie group structure. The NU group contains the Bondi–Metzner–Sachs (BMS) group as a subgroup, whose Lie group structure was constructed in a previous article. It is well known that the NU Lie algebra splits into a direct sum of Lie ideals of the Lie algebras of the BMS group and conformal rescalings of scri. However, the lack of a Lie group structure on the NU group implies that the BMS group cannot be embedded as a Lie subgroup therein.

Lie theory for asymptotic symmetries in general relativity: The BMS group

We study the Lie group structure of asymptotic symmetry groups in general relativity from the viewpoint of infinite-dimensional geometry. To this end, we review the geometric definition of asymptotic simplicity and emptiness due to Penrose and the coordinate-wise definition of asymptotic flatness due to Bondi et al. Then we construct the Lie group structure of the Bondi–Metzner–Sachs (BMS) group and discuss its Lie theoretic properties. We find that the BMS group is regular in the sense of Milnor, but not real analytic. This motivates us to conjecture that it is not locally exponential. Finally, we verify the Trotter property as well as the commutator property. As an outlook, we comment on the situation of related asymptotic symmetry groups. In particular, the much more involved situation of the Newman–Unti group is highlighted, which will be studied in future work.

Definitions of angular momentum and super angular momentum in asymptotically flat spacetimes: Properties and applications to compact-binary mergers

The symmetries of asymptotically flat spacetimes in general relativity are given by the Bondi-Metzner-Sachs (BMS) group, though there are proposed generalizations of its symmetry algebra. Associated with each symmetry is a charge and a flux, and the values of these charges and their changes can characterize a spacetime. The charges of the BMS group are relativistic angular momentum and supermomentum (which includes four-momentum); the extensions of the BMS algebra also include generalizations of angular momentum called ``super angular momentum.'' Several different formalisms have been used to define angular momentum, and they produce nonequivalent expressions for the charge. It was shown recently that these definitions can be summarized in a two-parameter family of angular momenta, which we investigate in this paper. We find that requiring that the angular momentum vanishes in flat spacetime restricts the two parameters to be equal. If we do not require that the angular momentum agrees with a common Hamiltonian definition, then we are left with a one-parameter family of angular momenta that includes the definitions from the several different formalisms. We then also propose a similar two-parameter family of super angular momentum. We examine the effect of the free parameters on the values of the angular momentum and super angular momentum from nonprecessing binary-black-hole mergers. The definitions of angular momentum differ at a high post-Newtonian order for these systems, but only when the system is radiating gravitational waves (not before and after). The different super-angular-momentum definitions occur at lower orders, and there is a difference in the change of super angular momentum even after the gravitational waves pass, which arises because of the gravitational-wave memory effect. We estimate the size of these effects using numerical-relativity surrogate waveforms and find they are small but resolvable.

On the structure and applications of the Bondi–Metzner–Sachs group

On the structure and applications of the Bondi–Metzner–Sachs group

On the structure and applications of the Bondi–Metzner–Sachs group

This work is a pedagogical review dedicated to a modern description of the Bondi–Metzner–Sachs (BMS) group. Minkowski space-time has an interesting and useful group of isometries, but, for a generic space-time, the isometry group is simply the identity and hence provides no significant informations. Yet symmetry groups have important role to play in physics; in particular, the Poincaré group describing the isometries of Minkowski space-time plays a role in the standard definitions of energy-momentum and angular-momentum. For this reason alone it would seem to be important to look for a generalization of the concept of isometry group that can apply in a useful way to suitable curved space-times. 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 between coordinate systems of a certain type in asymptotically flat space-times. In the third section the conformal method and the notion of “asymptotic simplicity” are introduced, following mainly the works of Penrose. This section prepares us for another derivation of the BMS group which will involve the conformal structure, and is thus more geometrical and fundamental. In the subsequent sections we discuss the properties of the BMS group, e.g. its algebra and the possibility to obtain as its subgroup the Poincaré group, as we may expect. The paper ends with a review of the BMS invariance properties of classical gravitational scattering discovered by Strominger, that are finding application to black hole physics and quantum gravity in the literature.

On the representation theory of the Bondi–Metzner–Sachs group and its variants in three space–time dimensions

The original Bondi–Metzner–Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian radiating 4-dim space–times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973, with this motivation, McCarthy classified all relativistic B-invariant systems in terms of strongly continuous irreducible unitary representations (IRS) of B. Here we introduce the analogue B(2, 1) of the BMS group B in 3 space–time dimensions. B(2, 1) itself admits thirty-four analogues both real in all signatures and in complex space–times. In order to find the IRS of both B(2, 1) and its analogues, we need to extend Wigner–Mackey’s theory of induced representations. The necessary extension is described and is reduced to the solution of three problems. These problems are solved in the case where B(2, 1) and its analogues are equipped with the Hilbert topology. The extended theory is necessary in order to construct the IRS of both B and its analogues in any number d of space–time dimensions, d≥3, and also in order to construct the IRS of their supersymmetric counterparts. We use the extended theory to obtain the necessary data in order to construct the IRS of B(2, 1). The main results of the representation theory are as follows: The IRS are induced from “little groups” which are compact. The finite “little groups” are cyclic groups of even order. The inducing construction is exhaustive notwithstanding the fact that B(2, 1) is not locally compact in the employed Hilbert topology.

Representations of the Bondi—Metzner—Sachs group in three space—time dimensions

The original Bondi–Metzner–Sachs group B is the common asymptotic symmetry group of all asymptotically at Lorentzian 4–dim space–times. As such, B is the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973, with this motivation, P. J. McCarthy classified all relativistic B–invariant systems in terms of strongly continuous irreducible unitary representations (IRS) of B. Here, we construct the IRS of B(2, 1), the analogue of B, in 3 space–time dimensions. The IRS are induced from ‘little groups’ which are compact. The finite ‘little groups’ are cyclic groups of even order. The inducing construction is exhaustive notwithstanding the fact that B(2, 1) is not locally compact in the employed Hilbert topology.

First results on the representation theory of the Ultrahyperbolic BMS group UHB(2, 2)

The Bondi–Metzner–Sachs (BMS) group B is the common asymptotic group of all asymptotically flat (lorentzian) space–times, and is the best candidate for the universal symmetry group of General Relativity (G.R.). B admits generalizations to real space–times of any signature, to complex space–times, and supersymmetric generalizations for any space— time dimension. With this motivation McCarthy constructed the strongly continuous unitary irreducible representations (IRs) of B some time ago, and he identified B(2,2) as the generalization of B appropriate to the to the ultrahyperbolic signature (+,+,−,−) and asymptotic flatness in null directions. We continue this programme by introducing a new group UHB(2, 2) in the group theoretical study of ultrahyperbolic G.R. which happens to be a proper subgroup of B(2, 2). We report on the first general results on the representation theory of UHB(2, 2). In particular the main general results are that the all little groups of UHB(2, 2) are compact and that the Wigner–Mackeys inducing construction is exhaustive despite the fact that UHB(2, 2) is not locally compact in the employed Hilbert topology.

Conformal Carroll groups and BMS symmetry

The Bondi–Metzner–Sachs group is shown to be the conformal extension of Lévy-Leblond’s ‘Carroll’ group. Further extension to the Newman–Unti group is also discussed in the Carroll framework.

The Bondi–Sachs 4-momentum in non-axisymmetric Robinson–Trautman spacetimes

We derive the Bondi–Sachs 4-momentum conservation laws that regulate the emission of gravitational waves in non-axisymmetric Robinson–Trautman spacetimes. Although Robinson–Trautman spacetimes describe the exterior vacuum gravitational field of a bounded system radiating gravitational waves, the Robinson–Trautman metrics do not satisfy the appropriate Bondi–Sachs boundary conditions formulated for radiating systems. We therefore derive the transformations between Robinson–Trautman and Bondi–Sachs coordinates which allow us to obtain some of the basic Bondi–Sachs physical quantities, namely the news functions and the Bondi–Sachs energy–momentum fluxes of the gravitational waves. These quantities are fundamental to the description of the radiative transfer processes involved in the generation and emission of gravitational waves by the system. The Bondi–Sachs 4-momentum conservation laws are covariant under the transformations of the Bondi–Metzner–Sachs group. The section of this paper containing the derivation of the coordinate transformations extends the work of von der Gönna and Kramer done for the case of axisymmetric Robinson–Trautman spacetimes. We discuss some possible applications of the conservation laws in numerical simulations of black hole systems emitting gravitational waves, and also future research of Bondi–Sachs conservation laws for twisting Robinson–Trautman-type spacetimes.

Localization and the interface between quantum mechanics, quantum field theory and quantum gravity II: The search of the interface between QFT and QG

The main topics of this second part of a two-part essay are some consequences of the phenomenon of vacuum polarization as the most important physical manifestation of modular localization. Besides philosophically unexpected consequences, it has led to a new constructive “outside-inwards approach” in which the pointlike fields and the compactly localized operator algebras which they generate only appear from intersecting much simpler algebras localized in noncompact wedge regions whose generators have extremely mild almost free field behavior. Another consequence of vacuum polarization presented in this essay is the localization entropy near a causal horizon which follows a logarithmically modified area law in which a dimensionless area (the area divided by the square of dR where dR is the thickness of a light-sheet) appears. There are arguments that this logarithmically modified area law corresponds to the volume law of the standard heat bath thermal behavior. We also explain the symmetry enhancing effect of holographic projections onto the causal horizon of a region and show that the resulting infinite dimensional symmetry groups contain the Bondi–Metzner–Sachs group. This essay is the second part of a partitioned longer paper.

On extracting physical content from asymptotically flat spacetime metrics

A major issue in general relativity, from its earliest days to the present, is how to extract physical information from any solution or class of solutions to the Einstein equations. Though certain information can be obtained for arbitrary solutions, e.g., via geodesic deviation, in general, because of the coordinate freedom, it is often hard or impossible to do. Most of the time information is found from special conditions, e.g. degenerate principle null vectors, weak fields close to Minkowski space (using coordinates close to Minkowski coordinates), or from solutions that have symmetries or approximate symmetries. In the present work, we will be concerned with asymptotically flat spacetimes where the approximate symmetry is the Bondi–Metzner–Sachs group. For these spaces the Bondi 4-momentum vector and its evolution, found from the Weyl tensor at infinity, describes the total energy–momentum of the interior source and the energy–momentum radiated. By generalizing the structures (shear-free null geodesic congruences) associated with the algebraically special metrics to asymptotically shear-free null geodesic congruences, which are available in all asymptotically flat spacetimes, we give kinematic meaning to the Bondi 4-momentum. In other words, we describe the Bondi vector and its evolution in terms of a center of mass position vector, its velocity and a spin vector, all having clear geometric meaning. Among other items, from dynamic arguments, we define a unique (at our level of approximation) total angular momentum and extract its evolution equation in the form of a conservation law with an angular momentum flux.

Generalizations of the BMS group and results

The ordinary Bondi-Metzner-Sachs (BMS) group B is the common asymptotic symmetry group of all radiating, asymptotically flat, Lorentzian space-times. As such, B is the best candidate for the universal symmetry group of General Relativity. However, in studying quantum gravity, space-times with signatures other than the usual Lorentzian one, and complex space-times, are frequently considered. Generalisations of B appropriate to these other signatures have been defined earlier. In particular, the generalization B(2, 2) appropriate to the ultrahyperbolic signature (+, +, −, −) has been described in detail, and the study of its irreducible unitary representations (IRs) has been initiated. The infinite little groups of B(2, 2) have been given explicitly but its finite little groups have only been partially described. All the information needed in order to construct the finite little groups is given. Possible connections with gravitational instantons are being put forward.

Holography in asymptotically flat spacetimes and the BMS group

In a previous paper (Arcioni G and Dappiaggi C 2003 Preprint hep-th/0306142) we have started to explore the holographic principle in the case of asymptotically flat spacetimes and analysed, in particular, different aspects of the Bondi–Metzner–Sachs (BMS) group, namely the asymptotic symmetry group of any asymptotically flat spacetime. We continue this investigation in this paper. Having in mind an S-matrix approach with future and past null infinity playing the role of holographic screens on which the BMS group acts, we connect the IR sectors of the gravitational field with the representation theory of the BMS group. We analyse the (complicated) mapping between bulk and boundary symmetries pointing out differences with respect to the anti-de Sitter (AdS)/CFT set up. Finally, we construct a BMS phase space and a free Hamiltonian for fields transforming with respect to BMS representations. The last step is supposed to be an explorative investigation of the boundary data living on the degenerate null manifold at infinity.

BMS field theory and holography in asymptotically flat space-times

We explore the holographic principle in the context of asymptotically flat space-times by means of the asymptotic symmetry group of this class of space-times, the so called Bondi-Metzner-Sachs (BMS) group. In particular we construct a (free) field theory living at future (or past) null infinity invariant under the action of the BMS group. Eventually we analyse the quantum aspects of this theory and we explore how to relate the correlation functions in the boundary and in the bulk.

The BMS group and generalized gravitational instantons

The ordinary Bondi–Metzner–Sachs (BMS) group B is the best candidate for the fundamental symmetry group of General Relativity. It has been shown that B admits generalizations to real space–times of any signature, and also to complex space–times. It has been suggested that certain continuous unitary irreducible representations (IRs) of B and of its generalizations correspond to gravitational instantons. Here I make this correspondence more precise and I take this suggestion one step further by arguing that a subclass of IRs of B and of its generalizations correspond to generalized gravitational instantons. Some of these generalized gravitational instantons involve in their definition certain subgroups of the Cartesian product group Cn×Cm, where Cr is the cyclic group of order r. With this motivation, I give the subgroups of Cn×Cm explicitly.

Exploring the holographic principle in asymptotically flat spacetimes via the BMS group

We explore the holographic principle in the context of asymptotically flat spacetimes. In analogy with the AdS/CFT scenario we analyse the asympotically symmetry group of this class of spacetimes, the so-called Bondi–Metzner–Sachs (BMS) group. We apply the covariant entropy bound to relate bulk entropy to boundary symmetries and find a quite different picture with respect to the asymptotically AdS case. We then derive the covariant wave equations for fields carrying BMS representations to investigate the nature of the boundary degrees of freedom. We find some similarities with 't Hooft S-matrix proposal and suggest a possible mechanism to encode bulk data.