Hamiltonian analysis of the double null 2+2 decomposition of Ashtekar variables

d’Inverno, Ray; Lambert, Pierre-Paul; Vickers, James

doi:10.1088/0264-9381/23/11/005

Cited by 12 publications

(21 citation statements)

References 20 publications

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“…The fact that the isometry group on a null hypersurface is one dimension larger than for other foliations is of course a well-known property, that led Dirac himself to suggest the use of null foliations as preferred ones. In the context of first-order general relativity with complex self-dual variables, it has for instance been pointed out in [25,26].…”

Section: Jhep11(2017)205mentioning

confidence: 99%

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Sachs’ free data in real connection variables

Paoli

Speziale

2017

J. High Energ. Phys.

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Abstract: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 equivalence with the second-order formalism, which we show in details at the level of symplectic potentials. A special feature of the first-order formulation is that Sachs' propagating equations for the shear, away from the initial hypersurface, are turned into tertiary constraints; their role is to preserve the relation between connection and shear under retarded time evolution. The conversion of wave-like propagating equations into constraints is possible thanks to an algebraic Bianchi identity; the same one that allows one to describe the radiative data at future null infinity in terms of a shear of a (non-geodesic) asymptotic null vector field in the physical spacetime. Finally, we compute the modification to the spin coefficients and the null congruence in the presence of torsion.

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Section: Jhep11(2017)205mentioning

confidence: 99%

“…28 26 To make the argument precise, we should embed the dynamical components into a covariant connection whose non-dynamical parts are put to zero by linear combinations of constraints, see e.g. [61] for an analogue treatment in the space-like case.…”

Section: Foliationmentioning

confidence: 99%

Sachs’ free data in real connection variables

Paoli

Speziale

2017

J. High Energ. Phys.

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“…The key result that emerges from these studies is that in the double null description of metric dynamics the gravitational degrees of freedom have a simple description in terms of the conformal 2-structure (or equivalently the trace free shears in the two null directions) [10,11]. The application of a double null decomposition to connection dynamics is more recent [12][13][14] and builds very strongly on earlier work of Goldberg et al [15,16] who looked at a foliation by null hypersurfaces.…”

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confidence: 82%

Double null hamiltonian dynamics and the gravitational degrees of freedom

Vickers

2011

Gen Relativ Gravit

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In this paper we review the Hamiltonian description of General Relativity using a double null foliation. We start by looking at the 2+2 version of geometrodynamics and show the role of the conformal 2-structure of the 2-metric in encoding (through the shear) the 2 gravitational degrees of freedom. In the second part of the paper we consider instead a canonical analysis of a double null 2+2 Hamiltonian description of General Relativity in terms of self-dual 2-forms and the associated SO(3) connection variables. The algebra of first class constraints is obtained and forms a Lie algebra that consists of two constraints that generate diffeomorphisms in the two surface, a constraint that generates diffeomorphisms along the null generators and a constraint that generates self-dual spin and boost transformations.

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“…It contains only 3 independent components, which can be totally determined by the Einstein field equations in 3-dimensional spacetime. The most simple coframe fields contain only 3 independent variables which are equal to the number of the metric variables in (21), and can be chosen as…”

Section: Metric and Coframementioning

confidence: 99%

“…in Bondi-like coordinate, which is obviously a special form of the metric (21). The metric components are…”

Section: Btz Spacetimementioning

confidence: 99%

Hamiltonian Analysis of 3-Dimensional Connection Dynamics in Bondi-like Coordinates

Huang¹,

Kong²

2017

Commun. Theor. Phys.

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The Hamiltonian analysis for a 3-dimensional connection dynamics of so(1, 2), spanned by {L−+, L−2, L+2} instead of {L01, L02, L12}, is first conducted in a Bondi-like coordinate system. The symmetry of the system is clearly presented. A null coframe with 3 independent variables and 9 connection coefficients are treated as basic configuration variables. All constraints and their consistency conditions, the solutions of Lagrange multipliers as well as the equations of motion are presented. There is no physical degree of freedom in the system. The Bañados-Teitelboim-Zanelli (BTZ) spacetime is discussed as an example to check the analysis. Unlike the ADM formalism, where only non-degenerate geometries on slices are dealt with and the Ashtekar formalism, where non-degenerate geometries on slices are mainly concerned though the degenerate geometries may be studied as well, in the present formalism the geometries on the slices are always degenerate though the geometries for the spacetime are not degenerate.

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Hamiltonian analysis of the double null 2+2 decomposition of Ashtekar variables

Cited by 12 publications

References 20 publications

Sachs’ free data in real connection variables

Sachs’ free data in real connection variables

Double null hamiltonian dynamics and the gravitational degrees of freedom

Hamiltonian Analysis of 3-Dimensional Connection Dynamics in Bondi-like Coordinates

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