119248889
Analysis of the Hamiltonian formulations of linearized general relativity
The different forms of the Hamiltonian formulations of linearized General Relativity/spin-2 theories are discussed in order to show their similarities and differences. It is demonstrated that in the linear model, non-covariant modifications to the initial covariant Lagrangian (similar to those modifications used in full gravity) are in fact unnecessary. The Hamiltonians and the constraints are different in these two formulations but the structure of the constraint algebra and the gauge invariance derived from it are the same. It is shown that these equivalent Hamiltonian formulations are related to each other by a canonical transformation, which is explicitly given. The relevance of these results to the full theory of General Relativity is briefly discussed.
119087671
Comments on ``The Einstein-hilbert Lagrangian density in a 2-dimensional spacetime is an exact differential'' by R. da Rocha and W.A. Rodrigues, Jr.
We argue that the recent result of da Rocha and Rodrigues that in two dimensional spacetime the Lagrangian of tetrad gravity is an exact differential [1], despite the claim of the authors, neither proves the Jackiw conjecture [2], nor contradicts to the conclusion of [3]. This demonstrates that the tetrad formulation is different from the metric formulation of the Einstein-Hilbert action.
118488928
Remarks on the “Non-canonicity Puzzle”: Lagrangian Symmetries of the Einstein-Hilbert Action
Given the non-canonical relationship between variables used in the Hamiltonian formulations of the Einstein-Hilbert action (due to Pirani, Schild, Skinner (PSS) and Dirac) and the Arnowitt-Deser-Misner (ADM) action, and the consequent difference in the gauge transformations generated by the first-class constraints of these two formulations, the assumption that the Lagrangians from which they were derived are equivalent leads to an apparent contradiction that has been called “the non-canonicity puzzle”. In this work we shall investigate the group properties of two symmetries derived for the Einstein-Hilbert action: diffeomorphism, which follows from the PSS and Dirac formulations, and the one that arises from the ADM formulation. We demonstrate that unlike the diffeomorphism transformations, the ADM transformations (as well as others, which can be constructed for the Einstein-Hilbert Lagrangian using Noether’s identities) do not form a group. This makes diffeomorphism transformations unique (the term “canonical” symmetry might be suggested). If the two Lagrangians are to be called equivalent, canonical symmetry must be preserved. The interplay between general covariance and the canonicity of the variables used is discussed.
119113011
On canonical transformations between equivalent hamiltonian formulations of general relativity
Two Hamiltonian formulations of general relativity, due to Pirani, Schild and Skinner (Phys. Rev. 87, 452, 1952) and Dirac (Proc. Roy. Soc. A 246, 333, 1958), are considered. Both formulations, despite having different expressions for the constraints, allow one to derive four-dimensional diffeomorphism invariance. The relation between these two formulations at all stages of the Dirac approach to constrained Hamiltonian systems is analyzed. It is shown that the complete sets of their phase-space variables are related by a transformation which satisfies the ordinary condition of canonicity known for unconstrained Hamiltonians and, in addition, converts one total Hamiltonian into another, thus preserving form-invariance of generalized Hamiltonian equations for the constrained systems.
119254285
Translational invariance of the Einstein–Cartan action in any dimension
We demonstrate that from the first order formulation of the Einstein– Cartan action it is possible to derive the basic differential identity that leads to translational invariance of the action in the tangent space. The transformations of fields is written explicitly for both the first and second order formulations and the group properties of transformations are studied. This, combined with the preliminary results from the Hamiltonian formulation (Kiriushcheva and Kuzmin in arXiv:0907.1553 [gr-qc]), allows us to conclude that without any modification, the Einstein–Cartan action in any dimension higher than two possesses not only rotational invariance but also a form of translational invariance in the tangent space. We argue that not only a complete Hamiltonian analysis can unambiguously give an answer to the question of what a gauge symmetry is, but also the pure Lagrangian methods allow us to find the same gauge symmetry from the basic differential identities.
119222199
The Hamiltonian formulation of tetrad gravity: Three-dimensional case
The Hamiltonian formulation of tetrad gravity in any dimension higher than two, using its first-order form where tetrads and spin connections are treated as independent variables, is discussed, and the complete solution of the three-dimensional case is given. For the first time, applying the methods of constrained dynamics, the Hamiltonian and constraints are explicitly derived and the algebra of Poisson brackets among all constraints is calculated. The algebra of Poisson brackets, among first-class secondary constraints, locally coincides with Lie algebra of the ISO(2,1) Poincaré group. All the first-class constraints of this formulation, according to the Dirac conjecture and using the Castellani procedure, allow us to unambiguously derive the generator of gauge transformations and find the gauge transformations of the tetrads and spin connections which turn out to be the same as found by Witten without recourse to the Hamiltonian methods [Nucl. Phys. B 311, 46 (1988)]. The gauge symmetry of the tetrad gravity generated by the Lie algebra of constraints is compared with another invariance, diffeomorphism. Some conclusions about the Hamiltonian formulation in higher dimensions are briefly discussed; in particular, that diffeomorphism invariance is not derivable as a gauge symmetry from the Hamiltonian formulation of tetrad gravity in any dimension where tetrads and spin connections are used as independent variables.
100638900
On Separation of Gagolinium's Isotopes by Means of Isotopically Selective ICRH