Ashtekar’s variables reexamined

Ashtekar’s variables for gravity are investigated in terms of exterior forms. The origin of the new variables is traced back to their role in eliminating the potential piece of the Lagrangian L=T-V which results in constraints polynomial in the canonically conjugate fields. The second-order general relativity and the real as well as complex (self-dual) Palatini first-order formalisms are collated with respect to this property. This analysis keeps track of the Hamiltonian through all these versions which are linked by canonical transformations; it presents a simple proof of the positive mass theorem, examines the possibility of scalar constraints linear in the momenta, and corrects a tetrad approach given by Ashtekar and collaborators. Also a new concise formulation of Einstein’s gravity theory is proposed. To account for Cartan’s calculus, a short but self-contained guide from Lagrangian to Hamiltonian formalisms in terms of forms is included.