Hamiltonian Gotay-Nester-Hinds analysis of the parametrized unimodular extension of the Holst action

Abstract

We give a detailed account of the Hamiltonian GNH analysis of the parametrized unimodular extension of the Holst action. The purpose of the paper is to derive, through the clear geometric picture furnished by the GNH method, a simple Hamiltonian formulation for this model and explain why it is difficult to arrive at it in other approaches. We will also show how to take advantage of the field equations to anticipate the simple form of the constraints that we find in the paper.

Concise symplectic formulation for tetrad gravity

“…The Hamiltonian description of tetrad gravity discussed here can be obtained from the Holst action [9] by using the geometrically inspired GNH method [8,[10][11][12]. Instead of following this approach, which is interesting in itself and will be presented in an upcoming publication [13], we will justify the validity of our formulation by deriving the real Ashtekar formulation from it.…”

“…iv) By removing the Γ i term from (13), it is straightforward to get the SO(1, 3)-ADM formulation by using the canonical variables K i ∶= ω 0i and E i ∶= ijk e j ∧ e k .…”

On-shell equivalence of general relativity and Holst theories with nonmetricity, torsion, and boundaries

“…with U K arbitrary. Plugging this solution into E ðtÞ KL ¼ 0 of (5.3) removes the dependence in S and the system becomes the ones studied in [13,41], where we found the solution ωIJ ¼ ω…”

“…When matter fields are not present, the equations of motion derived from the Palatini-Cartan and Holst actions are completely equivalent. This implies that the presence of the dual term does not change the Palatini space of solutions in a significant way [4][5][6][7][8][9][10][11][12][13][14]. When gravity is coupled to bosonic matter fields, the coupling terms are independent of the connection and therefore the field equations remain unchanged (see, for instance, [15]).…”

Proof of the equivalence of the symplectic forms derived from the canonical and the covariant phase space formalisms

“…Some noteworthy advantages are: symmetries are explicit, the study of null infinity is easier, there are methods to compute conserved quantities, and higher-derivative theories are treated on equal footing as 1st-order ones [7][8][9]. All this has important consequences in effective and perturbation theory, both of which are relevant to the of study string theories, edge modes, corner and BMS algebras, or the analysis of consistent deformations [10][11][12][13][14][15][16][17][18][19]. However, one drawback is that there are no known canonical structures on the spaces involved.…”