New variables for classical and quantum gravity in all dimensions: II. Lagrangian analysis

Abstract

We rederive the results of our companion paper, for matching spacetime and internal signature, by applying in detail the Dirac algorithm to the Palatini action. While the constraint set of the Palatini action contains second class constraints, by an appeal to the method of gauge unfixing, we map the second class system to an equivalent first class system which turns out to be identical to the first class constraint system obtained via the extension of the ADM phase space performed in our companion paper. Central to our analysis is again the appropriate treatment of the simplicity constraint. Remarkably, the simplicity constraint invariant extension of the Hamiltonian constraint, that is a necessary step in the gauge unfixing procedure, involves a correction term which is precisely the one found in the companion paper and which makes sure that the Hamiltonian constraint derived from the Palatini Lagrangian coincides with the ADM Hamiltonian constraint when Gauß and simplicity constraints are satisfied. We therefore have rederived our new connection formulation of General Relativity from an independent starting point, thus confirming the consistency of this framework.

Alternative self-dual gravity in eight dimensions

“…Thus assuming a non-vanishing cosmological constant one can focus in the gauge symmetry group SO(1, 3) in the case of (1 + 3)-dimensions and SO(1, 7) in the case of (1 + 7)-dimensions. These comments can be clarified further recalling that in (1 + 3)-dimensions the algebra so(1, 3) can be written as so(1, 3) = su(2) × su (2). So, the curvature R ab can be decomposed additively [2]:…”

“…These comments can be clarified further recalling that in (1 + 3)-dimensions the algebra so(1, 3) can be written as so(1, 3) = su(2) × su (2). So, the curvature R ab can be decomposed additively [2]:…”

“…Higher dimensional Ashtekar formalism has shown to be a promising proposal [1][2][3][4][5] (see also Ref. [6][7][8][9][10]).…”

Black hole entropy from loop quantum gravity in higher dimensions

“…≈ means equality on the constraint surface, defined by the SO(D+1) Gauß law and the simplicity constraint π a[IJ π b|KL] = 0. The simplicity constraint ensures that π aIJ ≈ 2/β n [I e J] b q ab √ q, that is, it is the product of a normal n I and a densitized D + 1-bein orthogonal to n I , see [11,12,18] for further details, e.g. a topological sector in 3 + 1 dimensions.…”

“…The classical part of this derivation was extended to higher dimensions in [9] and to LanczosLovelock gravity in [10], where the recently introduced connection variables for higher-dimensional general relativity [11][12][13][14] were employed. While the horizon degrees of freedom can be rewritten in a form similar to a higherdimensional Chern-Simons theory, it turns out to be more economical to use a canonically conjugate pair of normals n I ands I as horizon degrees of freedom [9].…”