Canonical analysis of unimodular Pleba\'nski gravity

TL;DR

This paper performs a canonical analysis of various unimodular gravity theories within the Plebański formalism, exploring their Hamiltonian structure, constraints, and issues related to reality conditions and Lorentzian solutions.

Contribution

It provides a detailed canonical formulation of unimodular gravity in the Plebański formalism, including fixed and dynamical cosmological constant scenarios.

Findings

Hamiltonian density includes a constant part in the fixed volume form case

Dynamical cosmological constant is constrained to be constant by field equations

Discussion of reality conditions and Lorentzian solution challenges

Abstract

We present the canonical analysis of different versions of unimodular gravity defined in the Pleba\'nski formalism, based on a (generally complex) SO(3) spin connection and set of (self-dual) two-forms. As in the metric formulation of unimodular gravity, one can study either a theory with fixed volume form or work in a parametrised formalism in which the cosmological constant becomes a dynamical field, constrained to be constant by the field equations. In the first case, the Hamiltonian density contains a part which is not constrained to vanish, but rather constrained to be constant, again as in the metric formulation. We also discuss reality conditions and challenges in extracting Lorentzian solutions.