Hamiltonian structure of three-dimensional gravity in Vielbein formalism

Abstract

Considering Chern-Simons like gravity theories in three dimensions as first order systems, we analyze the Hamiltonian structure of three theories Topological massive gravity, New massive gravity, and Zwei-Dreibein Gravity. We show that these systems demonstrate a new feature of the constrained systems in which a new kind of constraints emerge due to factorization of determinant of the matrix of Poisson brackets of constraints. We find the desired number of degrees of freedom as well as the generating functional of local Lorentz transformations and diffeomorphism through canonical structure of the system. We also compare the Hamiltonian structure of linearized version of the considered models with the original ones.

Exotic massive 3D gravity

“…Without the invertibility assumption for e, we could interpret the equations (4.10) as constraints on the Lagrange multipliers (e 0 , h 0 , f 0 ), in accordance with Dirac's prescription for construction of the Hamiltonian [32]. The above constraints are therefore not "secondary" in Dirac's sense and must be dealt with differently [16,33]. Here we follow the procedure of [16] in which these constraints are omitted from the "total Hamiltonian"; consistency then requires certain conditions on the Poisson bracket relations of the 'secondary' constraints, but we find that these are satisfied.…”

Hamiltonian structure of bi-gravity, problem of ghost and bifurcation

“…In fact, constraints such as Γ should be viewed as a new kind of constraints, which are different from primary constraints (which emerge due to definition of momenta) and secondary constraints (which emerge from the Poisson brackets of the constraints with the canonical Hamiltonian). This kind of constraints which we denote them as 'new kind' are also familiar to us in the canonical analysis of Chern-Simons like theories in 3 dimensions [25] 5 .…”

“…In other words, it is not allowed to use the original constraints S i . To see an interesting case of this point see [25]. It is well known, on the other hand, that [22] given the constraints ϕ a , one can redefine them as ϕ a = M ab ϕ b provided that M ab is nonsingular on the constraint surface.…”

“…Sometimes the physically acceptable branch of a theory may imply to consider subregions where a given matrix of coefficients is singular. To see an interesting case see Ref [23]…”