scholarly journals Lie theory for asymptotic symmetries in general relativity: The BMS Group

2022 ◽  
Author(s):  
David Nicolas Prinz ◽  
Alexander Schmeding
Keyword(s):  
General Relativity ◽  
Lie Group ◽  
Group Structure ◽  
Symmetry Groups ◽  
Asymptotic Symmetry ◽  
Infinite Dimensional ◽  
Real Analytic ◽  
Definition Of ◽  
Future Work ◽  
Asymptotic Symmetries

Abstract 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.

Kleinian characterization of asymptotic symmetries of real general relativity

1993 ◽  
Vol 441 (1913) ◽  
pp. 465-471 ◽  
Keyword(s):  
General Relativity ◽  
Automorphism Group ◽  
Group Action ◽  
Radon Transform ◽  
Symmetry Groups ◽  
Asymptotic Symmetry ◽  
The Real ◽  
The Given ◽  
Asymptotic Symmetries

According to Klein’s Erlanger programme, one may (indirectly) specify a geometry by giving a group action. Conversely, given a group action, one may ask for the corresponding geometry. Recently, I showed that the real asymptotic symmetry groups of general relativity (in any signature) have natural ‘projective’ classical actions on suitable ‘Radon transform’ spaces of affine 3-planes in flat 4-space. In this paper, I give concrete models for these groups and actions. Also, for the ‘atomic’ cases, I give geometric structures for the spaces of affine 3-planes for which the given actions are the automorphism group.

Non-abelian cohomology of abelian Anosov actions

2000 ◽  
Vol 20 (1) ◽  
pp. 259-288 ◽  
Author(s):  
ANATOLE KATOK ◽  
VIOREL NIŢICĂ ◽  
ANDREI TÖRÖK
Keyword(s):  
Discrete Time ◽  
Lie Group ◽  
Global Information ◽  
Infinite Dimensional ◽  
Anosov Actions ◽  
Geometric Character ◽  
Higher Rank ◽  
Real Analytic ◽  
A New Technique ◽  
Discrete Time Case

We develop a new technique for calculating the first cohomology of certain classes of actions of higher-rank abelian groups (${\mathbb Z}^k$ and ${\mathbb R}^k$, $k\ge 2$) with values in a linear Lie group. In this paper we consider the discrete-time case. Our results apply to cocycles of different regularity, from Hölder to smooth and real-analytic. The main conclusion is that the corresponding cohomology trivializes, i.e. that any cocycle from a given class is cohomologous to a constant cocycle. The principal novel feature of our method is its geometric character; no global information about the action based on harmonic analysis is used. The method can be developed to apply to cocycles with values in certain infinite dimensional groups and to rigidity problems.

scholarly journals Analytic Jet Parametrization for CR Automorphisms of Some Essentially Finite CR Manifolds

2008 ◽  
Vol 189 ◽  
pp. 155-168
Author(s):  
Sung-Yeon Kim
Keyword(s):  
Uniform Convergence ◽  
Lie Group ◽  
Group Structure ◽  
Cr Manifolds ◽  
Real Analytic ◽  
The Stability

AbstractIn this paper we construct analytic jet parametrizations for the germs of real analytic CR automorphisms of some essentially finite CR manifolds on their finite jet at a point. As an application we show that the stability groups of such CR manifolds have Lie group structure under composition with the topology induced by uniform convergence on compacta.

Real and complex asymptotic symmetries in quantum gravity, irreducible representations, polygons, polyhedra, and the A, D, E series

1992 ◽  
Vol 338 (1650) ◽  
pp. 271-299 ◽  
Keyword(s):  
General Relativity ◽  
Quantum Gravity ◽  
Irreducible Representations ◽  
Symmetry Groups ◽  
Regular Polygons ◽  
Lorentzian Space ◽  
Asymptotically Flat ◽  
Irreducible Unitary Representations ◽  
The Common ◽  
Asymptotic Symmetries

The Bondi-Metzner-Sachs 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. However, in quantum gravity, complexified or euclidean versions of general relativity are frequently considered, and the question arises: Are there similar symmetry groups for these versions of the theory? In this paper it is shown that there are such analogues of B and a variety of further ones, either real in any signature, or complex. The relationships between these various groups are described. Irreducible unitary representations (IRS) of the complexification CB of B itself are analysed. It is proved that all induced IRS of CB arise from IRS of compact 'little groups’. It follows that some IRS of CB are controlled by the IRS of the ‘ A,D,E ' series of finite symmetry groups of regular polygons and polyhedra in ordinary euclidean 3-space. Possible applications to quantum gravity are indicated.

scholarly journals Asymptotic symmetries in Carrollian theories of gravity

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Alfredo Pérez
Keyword(s):  
General Relativity ◽  
Symmetry Algebra ◽  
Einstein Gravity ◽  
Point Of View ◽  
Asymptotic Symmetry ◽  
Gravitational Theories ◽  
Theories Of Gravity ◽  
Spatial Rotations ◽  
Lapse Function ◽  
Asymptotic Symmetries

Abstract Asymptotic symmetries in Carrollian gravitational theories in 3+1 space and time dimensions obtained from “magnetic” and “electric” ultrarelativistic contractions of General Relativity are analyzed. In both cases, parity conditions are needed to guarantee a finite symplectic term, in analogy with Einstein gravity. For the magnetic contraction, when Regge-Teitelboim parity conditions are imposed, the asymptotic symmetries are described by the Carroll group. With Henneaux-Troessaert parity conditions, the asymptotic symmetry algebra corresponds to a BMS-like extension of the Carroll algebra. For the electric contraction, because the lapse function does not appear in the boundary term needed to ensure a well-defined action principle, the asymptotic symmetry algebra is truncated, for Regge-Teitelboim parity conditions, to the semidirect sum of spatial rotations and spatial translations. Similarly, with Henneaux-Troessaert parity conditions, the asymptotic symmetries are given by the semidirect sum of spatial rotations and an infinite number of parity odd supertranslations. Thus, from the point of view of the asymptotic symmetries, the magnetic contraction can be seen as a smooth limit of General Relativity, in contrast to its electric counterpart.

scholarly journals Asymptotic symmetries and charges at null infinity: from low to high spins

2018 ◽  
Vol 191 ◽  
pp. 06011 ◽  
Author(s):  
Andrea Campoleoni ◽  
Dario Francia ◽  
Carlo Heissenberg
Keyword(s):  
Symmetry Group ◽  
A Priori ◽  
Asymptotic Symmetry ◽  
Null Infinity ◽  
Infinite Dimensional ◽  
S Matrix ◽  
Higher Spin ◽  
Matrix Invariant ◽  
Higher Spins ◽  
Asymptotic Symmetries

Weinberg’s celebrated factorisation theorem holds for soft quanta of arbitrary integer spin. The same result, for spin one and two, has been rederived assuming that the infinite-dimensional asymptotic symmetry group of Maxwell’s equations and of asymptotically flat spaces leave the S-matrix invariant. For higher spins, on the other hand, no such infinite-dimensional asymptotic symmetries were known and, correspondingly, no a priori derivation of Weinberg’s theorem could be conjectured. In this contribution we review the identification of higher-spin supertranslations and superrotations in D = 4 as well as their connection to Weinberg’s result. While the procedure we follow can be shown to be consistent in any D, no infinite-dimensional enhancement of the asymptotic symmetry group emerges from it in D > 4, thus leaving a number of questions unanswered.

scholarly journals The Lie group of vertical bisections of a regular Lie groupoid

2020 ◽  
Vol 32 (2) ◽  
pp. 479-489
Author(s):  
Alexander Schmeding
Keyword(s):  
Lie Groups ◽  
Gauge Group ◽  
Lie Group ◽  
Group Structure ◽  
Lie Groupoids ◽  
Gauge Groups ◽  
Infinite Dimensional ◽  
Lie Groupoid ◽  
Regularity Properties ◽  
Infinite Dimensional Lie Group

AbstractIn this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and discuss regularity properties (in the sense of Milnor) for these Lie groups. If the groupoid is locally trivial, i.e., a gauge groupoid, the vertical bisections coincide with the gauge group of the underlying bundle. Hence, the construction recovers the well-known Lie group structure of the gauge groups. To establish the Lie theoretic properties of the vertical bisections of a Lie groupoid over a non-compact base, we need to generalise the Lie theoretic treatment of Lie groups of bisections for Lie groupoids over non-compact bases.

The Centralizer

2018 ◽  
Author(s):  
Ercüment H. Ortaçgil
Keyword(s):  
Lie Group ◽  
Group Structure ◽  
Diffeomorphism Group ◽  
Group Diff ◽  
Local Solutions

The pseudogroup of local solutions in Chapter 3 defines another pseudogroup by taking its centralizer inside the diffeomorphism group Diff(M) of a manifold M. These two pseudogroups define a Lie group structure on M.

scholarly journals Asymptotic symmetries and celestial CFT

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Laura Donnay ◽  
Sabrina Pasterski ◽  
Andrea Puhm
Keyword(s):  
Principal Series ◽  
Finite Energy ◽  
Continuous Series ◽  
Asymptotic Symmetry ◽  
Null Infinity ◽  
Conformal Dimension ◽  
Contour Integrals ◽  
Celestial Sphere ◽  
Goldstone Modes ◽  
Asymptotic Symmetries

Abstract We provide a unified treatment of conformally soft Goldstone modes which arise when spin-one or spin-two conformal primary wavefunctions become pure gauge for certain integer values of the conformal dimension ∆. This effort lands us at the crossroads of two ongoing debates about what the appropriate conformal basis for celestial CFT is and what the asymptotic symmetry group of Einstein gravity at null infinity should be. Finite energy wavefunctions are captured by the principal continuous series ∆ ∈ 1 + iℝ and form a complete basis. We show that conformal primaries with analytically continued conformal dimension can be understood as certain contour integrals on the principal series. This clarifies how conformally soft Goldstone modes fit in but do not augment this basis. Conformally soft gravitons of dimension two and zero which are related by a shadow transform are shown to generate superrotations and non-meromorphic diffeomorphisms of the celestial sphere which we refer to as shadow superrotations. This dovetails the Virasoro and Diff(S2) asymptotic symmetry proposals and puts on equal footing the discussion of their associated soft charges, which correspond to the stress tensor and its shadow in the two-dimensional celestial CFT.

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