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 Asymptotically Flat Boundary Conditions for the U(1)3 Model for Euclidean Quantum Gravity

Universe ◽  
2021 ◽  
Vol 7 (3) ◽  
pp. 68
Author(s):  
Sepideh Bakhoda ◽  
Hossein Shojaie ◽  
Thomas Thiemann
Keyword(s):  
General Relativity ◽  
Gauge Theory ◽  
Quantum Gravity ◽  
Boundary Conditions ◽  
Crucial Role ◽  
Test Laboratory ◽  
Asymptotically Flat ◽  
Generally Covariant ◽  
Asymptotic Symmetries

A generally covariant U(1)3 gauge theory describing the GN→0 limit of Euclidean general relativity is an interesting test laboratory for general relativity, specially because the algebra of the Hamiltonian and diffeomorphism constraints of this limit is isomorphic to the algebra of the corresponding constraints in general relativity. In the present work, we the study boundary conditions and asymptotic symmetries of the U(1)3 model and show that while asymptotic spacetime translations admit well-defined generators, boosts and rotations do not. Comparing with Euclidean general relativity, one finds that the non-Abelian part of the SU(2) Gauss constraint, which is absent in the U(1)3 model, plays a crucial role in obtaining boost and rotation generators.

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.

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.

scholarly journals A NOVEL SPIN-STATISTICS THEOREM IN (2 + 1)DCHERN–SIMONS GRAVITY

2001 ◽  
Vol 16 (20) ◽  
pp. 1335-1347 ◽  
Author(s):  
A. P. BALACHANDRAN ◽  
E. BATISTA ◽  
I. P. COSTA E SILVA ◽  
P. TEOTONIO-SOBRINHO
Keyword(s):  
General Relativity ◽  
Quantum Gravity ◽  
Canonical Formalism ◽  
Irreducible Representations ◽  
First Order ◽  
Algebraic Description ◽  
Spin And Statistics

It has been known for some time that topological geons in quantum gravity may lead to a complete violation of the canonical spin-statistics relation: There may be no connection between spin and statistics for a pair of geons. We present an algebraic description of quantum gravity in a (2 + 1) D manifold of the form Σ × ℝ, based on the first-order canonical formalism of general relativity. We identify a certain algebra describing the system, and obtain its irreducible representations. We then show that although the usual spin-statistics theorem is not valid, statistics is completely determined by spin for each of these irreducible representations, provided one of the labels of these representations, which we call flux, is superselected. We argue that this is indeed the case. Hence, a new spin-statistics theorem can be formulated.

scholarly journals Brans-Dicke theory in Bondi-Sachs form: Asymptotically flat solutions, asymptotic symmetries, and gravitational-wave memory effects

2021 ◽  
Vol 103 (10) ◽  
Author(s):  
Shammi Tahura ◽  
David A. Nichols ◽  
Alexander Saffer ◽  
Leo C. Stein ◽  
Kent Yagi
Keyword(s):  
Gravitational Wave ◽  
Memory Effects ◽  
Asymptotically Flat ◽  
Asymptotic Symmetries

scholarly journals Clock-dependent spacetime

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Sung-Sik Lee
Keyword(s):  
Hilbert Space ◽  
General Relativity ◽  
Quantum Gravity ◽  
Hilbert Spaces ◽  
Coordinate Systems ◽  
Physical Laws

Abstract Einstein’s theory of general relativity is based on the premise that the physical laws take the same form in all coordinate systems. However, it still presumes a preferred decomposition of the total kinematic Hilbert space into local kinematic Hilbert spaces. In this paper, we consider a theory of quantum gravity that does not come with a preferred partitioning of the kinematic Hilbert space. It is pointed out that, in such a theory, dimension, signature, topology and geometry of spacetime depend on how a collection of local clocks is chosen within the kinematic Hilbert space.

scholarly journals QUANTUM GRAVITY BY THE COMPLEX CANONICAL FORMULATION

1992 ◽  
Vol 01 (03n04) ◽  
pp. 439-523 ◽  
Author(s):  
HIDEO KODAMA
Keyword(s):  
General Relativity ◽  
Quantum Gravity ◽  
Conventional Treatment ◽  
Quantum States ◽  
Hamiltonian Constraint ◽  
Canonical Formulation ◽  
Recent Developments ◽  
Canonical Approach ◽  
New Formulation ◽  
Basic Features

The basic features of the complex canonical formulation of general relativity and the recent developments in the quantum gravity program based on it are reviewed. The exposition is intended to be complementary to the review articles already available and some original arguments are included. In particular the conventional treatment of the Hamiltonian constraint and quantum states in the canonical approach to quantum gravity is criticized and a new formulation is proposed.

scholarly journals Plebański–Demiański solution of general relativity and its expressions quadratic and cubic in curvature: Analogies to electromagnetism

2015 ◽  
Vol 24 (10) ◽  
pp. 1550079 ◽  
Author(s):  
Jens Boos
Keyword(s):  
General Relativity ◽  
Electromagnetic Field ◽  
General Type ◽  
Coordinate Systems ◽  
Irreducible Decomposition ◽  
Asymptotically Flat ◽  
Type D ◽  
Curvature Invariants ◽  
Free Parameters ◽  
First Time

Analogies between gravitation and electromagnetism have been known since the 1950s. Here, we examine a fairly general type D solution — the exact seven parameter solution of Plebański–Demiański (PD) — to demonstrate these analogies for a physically meaningful spacetime. The two quadratic curvature invariants B2 - E2 and E⋅B are evaluated analytically. In the asymptotically flat case, the leading terms of E and B can be interpreted as gravitoelectric mass and gravitoelectric current of the PD solution, respectively, if there are no gravitomagnetic monopoles present. Furthermore, the square of the Bel–Robinson tensor reads (B2 + E2)2 for the PD solution, reminiscent of the square of the energy density in electrodynamics. By analogy to the energy–momentum 3-form of the electromagnetic field, we provide an alternative way to derive the recently introduced Bel–Robinson 3-form, from which the Bel–Robinson tensor can be calculated. We also determine the Kummer tensor, a tensor cubic in curvature, for a general type D solution for the first time, and calculate the pieces of its irreducible decomposition. The calculations are carried out in two coordinate systems: In the original polynomial PD coordinates and in a modified Boyer–Lindquist-like version introduced by Griffiths and Podolský (GP) allowing for a more straightforward physical interpretation of the free parameters.

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