Abstract
We perform the canonical analysis of the Holst action for general relativity with a cosmological constant without introducing second-class constraints. Our approach consists in identifying the dynamical and nondynamical parts of the involved variables from the very outset. After integrating out the nondynamical variables associated with the connection, we obtain the description of phase space in terms of manifestly $SO(3,1)$ [or $SO(4)$, depending on the signature] covariant canonical variables and first-class constraints only. We impose the time gauge on them and show that the Ashtekar-Barbero formulation of general relativity emerges. Later, we discuss a family of canonical transformations that allows us to construct new $SO(3,1)$ [or $SO(4)$] covariant canonical variables for the phase space of the theory and compare them with the ones already reported in the literature, pointing out the presence of a set of canonical variables not considered before. Finally, we resort to the time gauge again and find that the theory, when written in terms of the new canonical variables, either collapses to the $SO(3)$ ADM formalism or to the Ashtekar-Barbero formalism with a rescaled Immirzi parameter.
Highlights
In the first-order formalism, general relativity is described by the Holst action [1], which is made of the Palatini action coupled to the Holst term via the Immirzi parameter [2]
Is that so? In this paper we show that it is possible to perform the canonical analysis of the Holst action without introducing neither second-class constraints nor gauge fixings spoiling Lorentz invariance
Diffeomorphism and scalar constraints, respectively. It is really remarkable how our approach simplifies the canonical analysis of general relativity, leading to a manifestly SOð3; 1Þ [or SOð4Þ] covariant parametrization of the phase space of the theory, and allowing us to keep track of the role played by each one of the original variables involved in the Holst action
Summary
In the first-order formalism, (real) general relativity is described by the Holst action [1], which is made of the Palatini action coupled to the Holst term via the Immirzi parameter [2]. The derivation of these variables makes use of the so-called time gauge, which breaks the Lorentz group SOð3; 1Þ down to the SOð3Þ subgroup This gauge fixing avoids the introduction of second-class constraints, simplifying the resulting canonical theory at the expense of local Lorentz invariance. In this paper we show that it is possible to perform the canonical analysis of the Holst action without introducing neither second-class constraints nor gauge fixings spoiling Lorentz invariance. This is accomplished by providing a parametrization of the spatial part of the connection that separates its dynamical components from its nondynamical ones. ΕIJKL ð2Þ and satisfies ðP−1ÞIJKLPKLMN 1⁄4 δI1⁄2MδJN. “∧” and “d” stand for the wedge product of differential forms and the exterior derivative, correspondingly
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