Abstract
We carry out the canonical analysis of the $n$-dimensional Palatini action with or without a cosmological constant $(n\geq3)$ introducing neither second-class constraints nor resorting to any gauge fixing. This is accomplished by providing an expression for the spatial components of the connection that allows us to isolate the nondynamical variables present among them, which can later be eliminated from the action by using their own equation of motion. As a result, we obtain the description of the phase space of general relativity in terms of manifestly $SO(n-1,1)$ [or $SO(n)$] covariant variables subject to first-class constraints only, with no second-class constraints arising during the process. Afterwards, we perform, at the covariant level, a canonical transformation to a set of variables in terms of which the above constraints take a simpler form. Finally, we impose the time gauge and make contact with the $SO(n-1)$ ADM formalism.
Highlights
The canonical analysis of general relativity has a very long history starting with attempts by Dirac himself
It turns out that in the resulting Hamiltonian form of the action both N and Na play the role of Lagrange multipliers imposing the scalar and diffeomorphism constraints, respectively, whereas qab and its canonically conjugate momentum pab—an object related to the extrinsic curvature—constitute the canonical variables that label the points of the phase space
In this paper we extend the theoretical approach of Ref. [12] to higher dimensions and perform from scratch the canonical analysis of the n-dimensional Palatini action with a cosmological constant
Summary
The canonical analysis of general relativity has a very long history starting with attempts by Dirac himself (see for instance Refs. [1,2]). [10]—where the second-class constraints emerging from the canonical analysis of the Holst action [11] are explicitly solved in a manifestly SOð3; 1Þ [or SOð4Þ] covariant fashion—because that technique is generic and is not restricted to 4-dimensional spacetimes. [12] to higher dimensions and perform from scratch the canonical analysis of the n-dimensional Palatini action with a cosmological constant In this framework, the original frame variables eμI are parametrized in terms of the momentum variables, the lapse function, and the shift vector, whereas the original connection variables ωμIJ are expressed in terms of the configuration variables, some auxiliary fields, and some Lagrange multipliers. In Appendix A we discuss in detail the 3-dimensional Palatini action (for which the auxiliary fields are absent from the very beginning), and in Appendix B we depict an alternative approach for the 4-dimensional case
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